text:
\section{Introduction}

\IEEEPARstart{C}{yber}-physical systems (CPSs) integrate, coordinate and monitor the operations of both a physical process and the cyber world~\cite{rajkumar2010cyber}. They
are decisive in supporting fundamental infrastructures and smart applications including automotive systems, transportation systems, smart grids, etc. However, although those CPSs were advanced at the time when being constructed, they can be either economical inefficient or energy inefficient from today's viewpoint.
For example, aging electricity distribution infrastructures are becoming less reliable and less efficient~\cite{kim2013overview}. Also, there are societal and industrial needs for better control of CPSs. For example, the increasing penetration of renewable energy poses threats to the reliability of power grids: it is essential to design better control to quickly attenuate large fluctuations caused by those energy sources. Furthermore, autonomous vehicles and mobile robots, being operated in an uncertain environment without complete information, require better control for more safety and reliability.

Usually, better control can be achieved from two perspectives. One way is rebuilding a new controller for the whole system. For example,~\cite{deveci2016performance} shows that by incorporating numerical modeling and simulation to design the controller for a photovoltaic system, system performance and robustness can be improved. The other way is to redesign the existing controller by adding extra dynamics while maintaining the control structure of the existing controller, i.e., if the original control is centralized/distributed, then the redesigned control is also centralized/distributed. This is referred to as control reconfiguration and can be simply realized, for example, by using additional information produced by newly added sensors, without affecting the structure of the controlled system. For instance,~\cite{zhang2014improving} proposes a modification approach based on a penalty method for improving the performance and robustness to delays of Internet congestion control protocols, and \cite{nevsic2005lyapunov} redesigns controllers by adding extra terms obtained based on continuous-time systems and Lyapunov functions to enhance stability and robustness. In general, there are tradeoffs between these two methods: rebuilding a new controller can achieve a better result by implementing the state-of-the-art sensing, communication and computing technologies but with the expense of complexity and high investment, while redesigning the existing controller can be easier and  more convenient though the effect may not be as good as rebuilding.

Instead of designing a new controller, this paper focuses on modifying the existing control from an optimization perspective to improve the performance of the whole system. Inspired by recent research on re-engineering typical CPSs~\cite{Low1999Optimal,li2015connecting,chen2016reverse,zhang2018distributed,zhou2020reverse}, this work extends to study control of general CPSs based on a reverse- and forward-engineering framework, which serves as a tool to bridge the gap between engineering CPS applications and existing theoretical results on optimization. As will be shown later, the proposed framework shows great potential to handle large-scale system cases.

The idea of redesign using a reverse- and forward-engineering framework for optimality has been introduced for over ten years~\cite{chiang2007layering}. From existing protocols designed based on engineering instincts, utility functions are implicitly determined and can be extracted via reverse-engineering. Other works consider various congestion control protocols as distributed algorithms for solving network utility maximization problems~\cite{kelly1998rate,Kunniyur2000End,Low1999Optimal}. Based on the insights obtained from reverse-engineering, forward-engineering systematically improves the protocols~\cite{chiang2007layering,Low1999Optimal}. Recently, inspired by this idea,~\cite{li2015connecting} connects automatic generation control (AGC) and economic dispatch by reverse-engineering AGC to improve power system economic efficiency, and ~\cite{zhang2015distributed,zhang2018distributed} develop a reverse- and forward-engineering framework to redesign control for improved efficiency, achieving optimal steady-state performance in network systems.

Nevertheless, little attention has been paid to improve system performance in terms of convergence rates/speeds and transient behaviors. This paper utilizes the reverse- and forward-engineering framework, together with several acceleration techniques, to systematically improve the performance of two classes of dynamical systems in discrete time. These systems include but are not limited to existing protocols and controlled systems, e.g., Internet congestion control, distributed proportional-integral (PI) control.

The main contribution of this paper is twofold.
\begin{itemize}
\item  A control reconfiguration approach is proposed to systematically improve the performance of two classes of general dynamical systems while maintaining the original control structure. This is realized by designing extra dynamics and adding it to the original control based on a reverse- and forward-engineering framework.

\item
Two standard acceleration methods are utilized in each class of dynamical systems for control reconfiguration and are theoretically proven to exhibit improved convergence rates.

\end{itemize}

The rest of this paper is organized as follows. Section~\ref{se:prelim} presents Class-$\mathcal{O}$\footnotemark[1] and Class-$\mathcal{S}$\footnotemark[1] as our target systems, together with two motivating examples. Section~\ref{se:redesign} presents the reconfiguration steps for systems in Class-$\mathcal{O}$ and Class-$\mathcal{S}$ and analyzes the convergence rates of the original and redesigned systems. Section~\ref{se:simu} provides simulation results of the motivating examples to illustrate the effectiveness of the reconfiguration framework. Section~\ref{se:conclusion} concludes the paper.
\footnotetext[1]{Notation $\mathcal{O}$ stands for \emph{optimization algorithms} in contrast to $\mathcal{S}$ for \emph{saddle-point algorithms}.}

\noindent\textbf{Notations:} Throughout this paper, we use upper case roman letters to denote matrices.
Let $\text{diag}\{\star\}$ denote the diagonal matrix with corresponding entries $\star$ on the main diagonal. Let $A \succeq 0$ ($A \succ 0$) denote that a square matrix is positive semi-definite (positive definite). For two symmetric matrices $A_1$ and $A_2$, notation $A_1 \preceq A_2$ implies that $A_2 - A_1$ is positive semi-definite. Let $\left\langle { \cdot {\rm{ }},{\rm{ }} \cdot } \right\rangle$ represent the Euclidean inner product, and let $\left\| {{\rm{ }} \cdot {\rm{ }}} \right\|$ denote the Euclidean norm for vectors and the spectrum norm for matrices. Denote by $\mathbb{Z}^+$ the set of positive integers, by $\mathbb{R}^n$ the $n$-dimensional Euclidean space and by $A \in \mathbb{R}^{m \times n}$ an $m \times n$ real matrix. Let $x^T$ and $\nabla f$ denote the transpose of $x$ and the gradient of $f$ (as a column vector).
Let ${\sigma _{\max }(B)}$/${\sigma _{\min }(B) }$ denote the maximum/minimum singular values of $B$. Let ${\lambda(A)}$ denote the eigenvalues of a square matrix $A$. Denote by $\bf{H}$$_f$ the Hessian matrix of $f$ with elements $\bf{H}$$_{i,j}=\frac{\partial^2 f}{\partial x_i \partial x_j}$.

\section{Preliminaries} \label{se:prelim}
In this section, we introduce two special classes of dynamical systems as our focus, i.e., Class-$\mathcal{O}$ and Class-$\mathcal{S}$, and present one typical example in each class as the motivating applications. In particular, we show what kinds of conditions are required to determine whether or not a system belongs to these classes (when reverse-engineering works)~\cite{zhang2018distributed}, mainly for the discrete-time linear case.

Reverse-engineering seeks a proper optimization problem inherently from given dynamical equations. Different from the previous optimization-to-algorithm framework, it generates a reverse flow, i.e., algorithm-to-optimization.

Consider a linear time-invariant (LTI) system
\begin{align}
x_{k+1}=Ax_k+Cw \label{eq:LTI-sys}
\end{align}where $x_k\in\mathbb{R}^{n}$ is the state vector, $A \in \mathbb{R}^{n \times n}$, $C \in \mathbb{R}^{n \times p}$ and $w\in\mathbb{R}^{p}$ is the exogenous input, e.g., disturbances. Note that $w$ can be constant or time-varying. In general, any given discrete-time LTI closed-loop system with either static feedback or dynamic feedback can be rearranged to fit~\eqref{eq:LTI-sys}.

\subsection{Target Systems: Class-$\mathcal{O}$}

\noindent\textbf{Class-$\mathcal{O}$}:
System~\eqref{eq:LTI-sys} belongs to Class-$\mathcal{O}$ if there exists a convex function $f(x): \mathbb{R}^{n}\rightarrow\mathbb{R}$ and a positive definite matrix $P \in \mathbb{R}^{n \times n}$, such that the set $\{\nabla f(x)=\textbf{0}\}$ is nonempty and system~\eqref{eq:LTI-sys} is a gradient descent algorithm to solve an unconstrained convex optimization problem $\min_{x} f(x)$, i.e., $x_{k+1} = x_k - P \nabla f(x) |_{x=x_k}$.

For linear systems in Class-$\mathcal{O}$, the associated $f$ must be a convex quadratic function, i.e., $f=\frac{1}{2}x^TQx+x^TR(Cw)+s(w)$ for some matrices $Q \succeq 0$ and $R$, and $s(w)$ stands for terms only related to  $w$. Note that $f$ is regarded as a function of the variable $x$ and $w$ is treated as a parameter. Therefore, system~\eqref{eq:LTI-sys} belongs to Class-$\mathcal{O}$ if and only if there exist $P \succ 0 $ and $Q \succeq 0$ such that $A=I_n-PQ$ and $R=-P^{-1}$ hold. This results in the following theorem~\cite{zhang2018distributed}.

\begin{theo}\label{theo:class-O}
Let $w$ be constant in~\eqref{eq:LTI-sys} and the set $\{(A-I_n)x+Cw=\mathbf{0}\}$ be nonempty. System~\eqref{eq:LTI-sys} belongs to Class-$\mathcal{O}$ if and only if system~\eqref{eq:LTI-sys} is marginally or asymptotically stable\footnote[2]{All the eigenvalues of $A$ have negative or zero real parts and all the Jordan blocks corresponding to eigenvalues with zero real parts are $1 \times 1$.}, all the eigenvalues of $I_n-A$ are real numbers and $I_n-A$ is diagonalizable.
\end{theo}

\subsection{Target Systems: Class-$\mathcal{S}$}

\noindent\textbf{Class-$\mathcal{S}$}: System~\eqref{eq:LTI-sys} belongs to Class-$\mathcal{S}$ if there exists a function $f(x^{(1)},x^{(2)}):\mathbb{R}^{n}\rightarrow \mathbb{R}$ and positive definite matrices $P_{x^{(1)}}$\footnotemark[3], $P_{x^{(2)}}$\footnotemark[3] such that ${\bf{H}}_{f(x^{(1)})}\footnotemark[3] = \mathbf{0}$, ${\bf{H}}_{f(x^{(2)})} \succeq 0$, the saddle-point set $\{\nabla f(x)=\mathbf{0}\}$ is nonempty, and~\eqref{eq:LTI-sys} is a primal-dual gradient algorithm to solve $\max_{x^{(1)}} \min_{x^{(2)}} f$, i.e., $x_{k+1}=x_k + \text{diag}\{P_{x^{(1)}}, -P_{x^{(2)}}\}\nabla f|_{x=x_k}$. \footnotetext[3]{Here $P_{x^{(1)}}$/$P_{x^{(2)}}$ is a positive definite matrix with dimension in accordance with $x^{(1)}$/ $x^{(2)}$; $f(x^{(1)})$ means $f$ is regarded as a function of the variable $x^{(1)}$ and $x^{(2)}$ is treated as a parameter.}
\vspace{6pt}

In the above definition, state $x$ is partitioned into $x^{(1)}$ and $x^{(2)}$, and $f$ is linear in $x^{(1)}$ (similar to that the Lagrangian function of a constrained optimization problem is linear in the dual variables). Accordingly, we rearrange system~\eqref{eq:LTI-sys} as
\begin{align}\label{equ:linearsystemsplit}
\underbrace {\left[ {\begin{array}{*{20}{c}}
{{x_{k+1}^{(1)}}}\\
{{x_{k+1}^{(2)}}}
\end{array}} \right]}_{x_{k+1}} = \underbrace {\left[ {\begin{array}{*{20}{c}}
{{A_{11}}}&{{A_{12}}}\\
{{A_{21}}}&{{A_{22}}}
\end{array}} \right]}_A \underbrace {\left[ {\begin{array}{*{20}{c}}
{x_k^{(1)}}\\
{x_k^{(2)}}
\end{array}} \right]}_{x_k}+Cw
\end{align}
where $x_k^{(1)}\in\mathbb{R}^{n_1}$, $x_k^{(2)}\in\mathbb{R}^{n_2}$ and $n_1+n_2=n$. For linear systems in Class-$\mathcal{S}$, the associated function $f$ must be a convex quadratic function, i.e.,
\begin{align}\label{equ:classSopt}
f=\frac{1}{2}
x^T\underbrace{\left[ {\begin{array}{*{20}{c}}
{{Q_{11}}}&{{Q_{12}}}\\
{Q_{12}^T}&{{Q_{22}}}
\end{array}} \right]}_Qx+x^TR(Cw)+s(w)
\end{align
 where $Q_{11}=\mathbf{0}\in\mathbb{R}^{n_1\times n_1}$, $Q_{22}\in\mathbb{R}^{n_2\times n_2}$ is symmetric and positive semi-definite ($f$ is  linear in $x^{(1)}$ and convex in $x^{(2)}$), and $Q_{12}\in\mathbb{R}^{n_1\times n_2}$. According to the definition of Class-$\mathcal{S}$, the following theorem is obtained.

\begin{theo}\label{theo:class-S}
Let $w$ be constant in~\eqref{equ:linearsystemsplit} and the set $\{(A-I_n)x+Cw=\mathbf{0}\} $ be nonempty. System~\eqref{equ:linearsystemsplit} belongs to Class-$\mathcal{S}$ if and only if the following conditions are satisfied: (i)  system~\eqref{equ:linearsystemsplit} is marginally or asymptotically stable; (ii) the eigenvalues of $A_{11}-I_{n_1}$ and $A_{22}-I_{n_2}$ are non-positive real; (iii) $A_{11}-I_{n_1}$ and $A_{22}-I_{n_2}$ are diagonalizable with the diagonal canonical forms given by $A_{11}-I_{n_1}=J_1\Lambda_1J_1^{-1}, J_1\in\mathbb{R}^{n_1\times n_1}, A_{22}-I_{n_2}=J_2\Lambda_2J_2^{-1}, J_2\in\mathbb{R}^{n_2\times n_2}$, and there exist $V_1, V_2$ that
\begin{align}\label{equ:SDPfeasibility}
&(J_1^{-1})^TV_1J_1^{-1}A_{12}+A_{21}^T(J_2^{-1})^TV_2J_2^{-1}=\mathbf{0} \nonumber\\
&V_1\Lambda_1=\Lambda_1V_1,\text{ }V_2\Lambda_2=\Lambda_2V_2 \nonumber\\
&V_1\prec0,\text{ }V_2\prec0.
\end{align}
\end{theo}

Note that there could be multiple optimization problems corresponding to the same dynamical system~\eqref{eq:LTI-sys} in either Class-$\mathcal{O}$ or Class-$\mathcal{S}$. The derivation procedure of those problems can be found in~\cite{zhang2018distributed}.

\subsection{Motivating Applications} \label{se:application}
Many existing systems fall into the two classes. For example, consensus in multi-agent systems, primal Internet congestion control protocols and heating, ventilation and air-conditioning (HVAC) system control are in Class-$\mathcal{O}$ (see~\cite{myICCPS} for a detailed description) while primal-dual Internet congestion control protocols, distributed PI control for single integrators and frequency control in power systems are in Class-$\mathcal{S}$. Here we present one typical example for each class.

\subsubsection{Internet Congestion Control} \label{se:ICC}
Internet congestion control regulates the data transfer and efficient bandwidth sharing between sources and links.
Here we consider a standard primal congestion control algorithm~\cite{Srikant2004The}, which belongs to Class-$\mathcal{O}$:
\begin{subequations}\label{eq:ICC_O}
\begin{align}
&x_i(k+1)=x_i(k)+\varepsilon k_{x_i}(U_i^{\prime}(x_i(k))-q_i(k)) \\
&q_i(k)=\sum_{l=1}^M R_{li}p_l(k) \\
&p_l(k)=f_l(y_l(k)) \\
&y_l(k)=\sum_{i=1}^NR_{li}x_i(k)
\end{align}
\end{subequations}where $\varepsilon$ is the step size, $U_i(x_i)$ is the utility function of user $i$, which is assumed to be a continuously differentiable, monotonically increasing, strictly concave function of the transmission rate $x_i$. $R$ is a routing matrix describing the interconnection where $R_{li} = 1$ if user $i$ uses link $l$; otherwise $R_{li} = 0$.
Also, $k_{x_i}>0$ is the rate gain, $p_l>0$ is the link price, $q_i$ is the aggregate price along user $i$'s path, $y_l$ is the total flow through link $l$, $f_l(y_l)$ is a barrier/penalty function satisfying $f_l(y_l),f_l^\prime(y_l)>0$ (usually, $f_l(y_l)$ forces the satisfaction of the inequality constraint $y_l \le c_l$ where $c_l>0$ is the link capacity of link $l$), $N$ and $M$ are the numbers of sources and links.
The above dynamics can be rearranged in a vector form as
\begin{align}
x(k+1)=x(k) + \varepsilon \text{diag}\{k_{x_i}\} \left( U^{\prime} (x(k) ) - R^Tf(Rx(k)) \right)   \label{eq:network_matrix}
\end{align}
where $x(k), U^{\prime} (x(k) )\in \mathbb{R}^N, f(Rx(k))\in \mathbb{R}^M$ are the corresponding vector forms of $x_i(k)$, $U_i^{\prime} (x_i(k) )$, $f_l (\sum\limits_{i = 1}^N {R_{li} x_i(k) } )$.

Suppose $U_i(x_i)$ is quadratic and $f_l(y_l)$ is linear, i.e., $U(x)=-\frac{1}{2}Q \text{diag}\{x_i\}x+R_1x+s_1$ and $f(Rx)=R_2Rx+s_2$ where $Q,R_1,R_2$ are diagonal, positive definite constant matrices, and $s_1,s_2$ are constant vectors. Then~\eqref{eq:network_matrix} can be rearranged as
\begin{align}
 &x(k+1)=x(k) - \varepsilon \text{diag}\{k_{x_i}\}(Qx(k)+R^TR_2Rx(k))+\cdots \nonumber\\
 &=\big(I_n-\varepsilon \text{diag}\{k_{x_i}\}(Q+R^TR_2R)\big)x(k)+\cdots \label{eq:ICC_matrix}
\end{align}
where $\cdots$ denotes the remaining constant terms. Compared with~\eqref{eq:LTI-sys}, it is straightforward to notice that $I_n-A$ in~\eqref{eq:ICC_matrix} is $\varepsilon \text{diag}\{k_{x_i}\}(Q+R^TR_2R)$.
It satisfies the conditions listed in Theorem~\ref{theo:class-O} and thus, the above dynamics~\eqref{eq:ICC_O} or \eqref{eq:network_matrix} can be reverse-engineered as a gradient descent algorithm to solve
\begin{align} \label{eq:opt_net}
\min_{x_i\ge0}-\sum_{i=1}^NU_i(x_i)+\sum_{l=1}^M\int_0^{\sum_{i=1}^NR_{li}x_i}f_l(\beta)d\beta.
\end{align}
\indent In general, the reverse-engineering from~\eqref{eq:network_matrix} to the above optimization problem always works if $U_i(x_i)$ is concave.
It is of interest to redesign the primal congestion control algorithm~\eqref{eq:network_matrix} for a faster convergence speed to improve data transfer.

\subsubsection{Distributed PI Control for Single Integrator Dynamics} \label{se:PI}
In the following, we propose distributed PI control for single integrators as a motivating example that belongs to Class-$\mathcal{S}$. Consider $n$ agents with single integrator dynamics
\begin{align*}
 \dot y_i  &= d_i  + u_i
\end{align*}
where $d_i$ is a constant disturbance and $u_i$ is the control input given by
\begin{align} \label{eq:integrator_u}
u_i  =  - \sum\limits_{j \in {\rm \mathcal{N}}_i } {\left( {\rho_2(y_i  - y_j ) + \rho_1 \int\limits_0^t {\big( {y_i \left( \tau  \right) - y_j \left( \tau  \right)} \big)d\tau } } \right)} - \delta \left( {y_i  - y_i(0)} \right)
\end{align}
where $\rho_1, \rho_2, \delta$ are positive constant parameters, $y_i(0)$ is the initial condition and $\mathcal{N}_i$ is the set of neighbors of agent $i$. This controller drives agents to reach consensus under static disturbances and any initial condition according to Theorem 6 in~\cite{andreasson2014distributed}. Introduce the integral action $\dot z_i = y_i - y_n$ and rearrange the dynamics after discretizing in a vector form as
\begin{align*}
\underbrace {\left[ {\begin{array}{*{20}c}
   {z(k + 1)}  \\
   {y(k + 1)}  \\
\end{array}} \right]}_{x(k + 1)} =& \underbrace {\left[ {\begin{array}{*{20}c}
   {I_{n-1} } & {\varepsilon_2 D}  \\
   { - \varepsilon_1 \rho_1 \tilde{L}} & {I_n  - \varepsilon_1 \rho_2 L - \varepsilon_1 \delta I_n }  \\
\end{array}} \right]}_A\underbrace {\left[ {\begin{array}{*{20}c}
   {z(k)}  \\
   {y(k)}  \\
\end{array}} \right]}_{x(k)} \\  \nonumber
&+ \underbrace {\left[ {\begin{array}{*{20}c}
   \textbf{0}  \\
   {\varepsilon_1 \big(d+\delta y(0)\big)}  \\
\end{array}} \right]}_{Cw}
\end{align*}
where $z(k) \in \mathbb{R}^{n-1}$, $y(k), d \in \mathbb{R}^{n}$, $x(k) \in \mathbb{R}^{2n-1}$, $\varepsilon_1$ and $\varepsilon_2$ are the step sizes and $D=[I_{n-1} \quad -\textbf{1}]$ ($\textbf{1}$ is a column vector of ones). $L \in \mathbb{R}^{n \times n}$ is the Laplacian of the connected agent network and $\tilde{L} \in \mathbb{R}^{n \times (n-1)}$ is obtained after removing the $n$th column of $L$. Compared with~\eqref{equ:linearsystemsplit}, it is straightforward to notice that $A_{11}-I_{n_1}$ is $\textbf{0}$ and $A_{22}-I_{n_2}$ is $- \varepsilon_1 \rho_2 L - \varepsilon_1 \delta I_n$. They satisfy the conditions listed in Theorem~\ref{theo:class-S} and therefore, the above dynamics can be reverse-engineered as a primal-dual gradient algorithm to solve
\begin{align} \label{eq:PI_f}
\begin{array}{l}
\mathop {\max }\limits_{z \in \mathbb{R}^{n-1} } \mathop {\min }\limits_{y \in \mathbb{R}^n } f
 = \frac{\rho_2}{2}y^T Ly + \rho_1 z^T \tilde{L}^T y + \frac{\delta }{2}y^T y - y^T d - y^T\delta y(0).
 \end{array}
\end{align}
It is of interest to redesign the distributed PI controller~\eqref{eq:integrator_u} to improve system performance.

\subsection{Problem Setup}
Motivated by the above examples, the desiderata is: for dynamical systems belonging to Class-$\mathcal{O}$ and Class-$\mathcal{S}$, improve their performance (convergence rates/speeds and transient behaviors) through redesigning the existing protocols and controls while maintaining their original control structures, i.e., the amount and topology of input channels remain unchanged.

\section{Redesign Methodology} \label{se:redesign}
In this section, we propose the redesign methodology for linear dynamical systems in Class-$\mathcal{O}$ and Class-$\mathcal{S}$. In particular, we analyze the convergence rates of the original systems and the redesigned systems.
The corresponding explicit forms of the extra dynamics in the redesigned systems are derived. For convenience, we only consider discrete-time cases here, and continuous-time cases will be discussed in a future paper.

\begin{defi} \label{defi:f(x)}
\textbf{Function class $\mathscr{F}_L$}: $f \in \mathscr{F}_L$ means that the function $f$: $\mathbb{R}^{n}\rightarrow\mathbb{R}$ is convex and has $L$-Lipschitz continuous gradient ($L \geq 0$), i.e., $\forall x,y \in \mathbb{R}^n$, we have
\begin{align}
\left\| {\nabla f(x) - \nabla f(y)} \right\| \le L \left\| {x - y} \right\|. \label{eq:lipschitz}
\end{align}

\noindent\textbf{Function class $\mathscr{S}_{\mu,L}$}: $f \in \mathscr{S}_{\mu,L}$ means that $f$ is $\mu$-strongly convex ($\mu >0$) and has $L$-Lipschitz continuous gradient ($L \geq 0$), i.e., $\forall x,y \in \mathbb{R}^n$, we have \eqref{eq:lipschitz} and
\begin{align}
f(y) \ge f(x) + \nabla f(x)^T (y - x) + \frac{\mu }{2}\left\| {y - x} \right\|^2.
\end{align}
\end{defi}

\begin{rem}
If the function $f \in \mathscr{S}_{\mu,L}$, there exist constants $L$ and $\mu$ such that $\forall x,y \in \mathbb{R}^n$,
\begin{align*}
\mu \le\frac{\| \nabla f(x) - \nabla f(y)\|}{\|x-y\|} \le L
\end{align*}

Furthermore, if the function $f$ is twice-differentiable, then its second derivative satisfies
\begin{align*}
\mu I_n \preceq \nabla^2 f(x) \preceq LI_n.
\end{align*}
\indent Therefore, through the above inequalities, parameters $L$ and $\mu$ can be obtained.
\end{rem}

\subsection{Systems in Class-$\mathcal{O}$}
\subsubsection{Convergence Rate of the Original Systems}
Any system~\eqref{eq:LTI-sys} in Class-$\mathcal{O}$ can be reverse-engineered as a gradient descent (GD) algorithm to solve an unconstrained convex optimization problem, so the convergence rate of system~\eqref{eq:LTI-sys} in Class-$\mathcal{O}$ follows that of a GD algorithm. Suppose the reverse-engineered optimization problem is
\begin{align}
 \min_{x \in \mathbb{R}^n } f\left( x \right) \label{eq:uncons}
\end{align}
where $x \in \mathbb{R}^n$ is the decision vector and $f$ is a convex, scalar differentiable function of $x$. Assume that a global minimizer of problem~\eqref{eq:uncons} exists and is denoted by $x^*$.

The GD algorithm is the simplest algorithm to solve problem~\eqref{eq:uncons} in discrete-time~\cite{nesterov2018lectures}, as shown in Algorithm 1, where $k=0, 1, \dots, N$ is the time step. Theorems~\ref{theo:GD} and \ref{theo:GD_stro} summarize the convergence property of the GD algorithm/original system when the objective function $f\in \mathscr{F}_L$ or $f\in \mathscr{S}_{\mu,L}$.
\begin{algorithm}
\caption{Gradient descent algorithm}
\label{alg:GD}
\textbf{Setting:} Choose appropriate positive step size $\varepsilon$, and $x_0 \in \mathbb{R}^n$.
\begin{align} \label{eq:GD}
{x_{k + 1}} = {x_k}-\varepsilon\nabla f\left( {{x_k}} \right)
\end{align}
\end{algorithm}

\begin{theo}  \label{theo:GD}
For any system~\eqref{eq:LTI-sys} in Class-$\mathcal{O}$, if the objective function $f$ obtained via reverse-engineering belongs to $\mathscr{F}_L$ and $0<\varepsilon<2/L$, then the convergence rate of this system is given by (See Theorem 2.1.14 in \cite{nesterov2018lectures} for the proof)
\begin{align}
f\left( {{x_k}} \right) - {f(x^*)} \le \frac{{2{{\left\| {{x_0} - {x^*}} \right\|}^2}}}{{\varepsilon k}} = O\left( {\frac{1}{{\varepsilon k}}} \right), \quad \forall k \in \mathbb{Z}^+. \nonumber
\end{align}
\end{theo}

\begin{theo}  \label{theo:GD_stro}
For any system~\eqref{eq:LTI-sys} in Class-$\mathcal{O}$, if the objective function $f$ obtained via reverse-engineering belongs to $\mathscr{S}_{\mu,L}$ and $0<\varepsilon \le \frac{2}{{\mu  + L}}$, then the convergence rate of this system is given b
\begin{align*}
\left\| {x_{k}  -  x^* } \right\| \le  (1 - \mu \varepsilon )^k \left\| {x_0  - x^* } \right\|=O({e^{ - k \mu\varepsilon}}), \quad \forall k \in \mathbb{Z}^+.
\end{align*}
\end{theo}

\begin{proof}
Theorem 2.1.15 in~\cite{nesterov2018lectures} and Theorem 3.12 in~\cite{Bubeck2014convex} showed that $\left\| {x_{k + 1}  -  x^* } \right\|^2$$\le$ $\left(1 - \frac{{2\varepsilon \mu L}}{{L + \mu}}\right)$\\$\times\left\| {x_k  -  x^* } \right\|^2$$+ \varepsilon \left (\varepsilon  - \frac{2}{{L + \mu }}\right)\left\| {\nabla f(x_k )} \right\|^2$. But Theorem~\ref{theo:GD_stro} actually provides a more strict upper bound compared with Theorem 2.1.15 in~\cite{nesterov2018lectures}. The step size should be in the range of $0<\varepsilon \le \frac{2}{{\mu  + L}}$, so that the sequence $\|x_k - x^*\|_{k\ge 0}$ is decreasing. Applying inequality $\left\|\nabla f(y) - \nabla f(x) \right\| \ge \mu \left\|y-x \right \|$ and optimality condition $\nabla f( x^* ) = 0$, we have
\begin{align*}
{\left\| {{x_{k + 1}} - {{ x}^*}} \right\|^2} \le& \left( {1 - \frac{{2\varepsilon \mu L}}{{L + \mu}} + \varepsilon {\mu^2}\left( {\varepsilon  - \frac{2}{{L + \mu }}} \right)} \right){\left\| {{x_k} - {{ x}^*}} \right\|^2}
=\left (1- \mu \varepsilon \right)^2{\left\| {{x_k} - {{ x}^*}} \right\|^2}.
\end{align*}
The value of $\left(1 - \mu \varepsilon \right)$ is non-negative when $0<\varepsilon \le \frac{2}{{\mu  + L}}$ and $\mu \le L$. Removing squares on both sides of the inequality completes the proof. Note that the last equation in this theorem is obtained via the inequality $(1-t)\le e^{-t}, \forall t$.
\end{proof}

\begin{coro} \label{coro:GD}
For any system~\eqref{eq:LTI-sys} in Class-$\mathcal{O}$, if the objective function $f$ obtained via reverse-engineering belongs to $\mathscr{S}_{\mu,L}$ and the step size $\varepsilon=\frac{2}{{\mu  + L}}$, then this system is with the optimal convergence rate given by~\cite{nesterov2018lectures}
\begin{align}
\left\| {x_{k}  -  x^* } \right\| \le \left(\frac{{L  - \mu}}{{L  + \mu}}\right)^k \left\| {x_0  -  x^* } \right\| =  O({e^{ - k \frac{2\mu}{L+\mu}}}), \forall k \in \mathbb{Z}^+.   \nonumber
\end{align}
\end{coro}

\subsubsection{Redesign via a Heavy Ball Method}
Once reverse-engineering a given system~\eqref{eq:LTI-sys} in Class-$\mathcal{O}$ as a GD algorithm to solve~\eqref{eq:uncons}, it is natural to apply faster algorithms to solve this problem, which can result in a redesigned system formula with improved performance.

In Algorithm~\ref{alg:GD}, the next point only depends on the current point like in a Markov chain. The heavy ball (HB) method, introduced by Polyak~\cite{polyak1987introduction}, utilizes a momentum term $x_k-x_{k-1}$ to incorporate the effect of second-order change, often leading to smoother trajectories and a faster convergence rate~\cite{polyak1987introduction}. Algorithm~\ref{alg:HB} shows its iterations.

\begin{algorithm}
\caption{Heavy ball method}
\label{alg:HB}
\textbf{Setting:} Choose appropriate positive step size $\varepsilon$, coefficient $\beta$ and let $x_1 = x_0$ be an arbitrary initial condition.
\begin{align}
{x_{k + 1}} = {x_k} - \varepsilon \nabla f({x_k}) + \beta ({x_k} - {x_{k - 1}}) \label{eq:HB}
\end{align}
\end{algorithm}

For any system~\eqref{eq:LTI-sys} in Class-$\mathcal{O}$, to improve system performance, apply the HB method to solve problem~\eqref{eq:uncons} to obtain
\begin{align}\label{eq:HB_uk}
    {x_{k + 1}} = {x_k} - \varepsilon \nabla f({x_k}) + \underbrace {\beta ({x_k} - {x_{k - 1}})}_{\Delta {u_k}}
\end{align}
where $\Delta {u_k} = \beta ({x_k} - {x_{k - 1}})$ is the explicit form of extra dynamics. It is clear that~\eqref{eq:HB_uk} is the sum of the GD formula~\eqref{eq:GD} and extra dynamics $\Delta u_k$.
Therefore, the redesigned system via the HB method is
\begin{align}\label{eq:HB_re}
x_{k+1}=Ax_k+Cw+\Delta u_k
\end{align}
where $\Delta {u_k} = \beta ({x_k} - {x_{k - 1}})$. Theorem~\ref{theo:HB} summarizes the convergence property when applying the HB method to redesign.

\begin{theo} \label{theo:HB}
For any system~\eqref{eq:LTI-sys} in Class-$\mathcal{O}$, if the objective function $f$ obtained via reverse-engineering belongs to $\mathscr{S}_{\mu,L}$ and is twice-differentiable, the step size $\varepsilon=\frac{4}{{{{\left( {\sqrt L  + \sqrt \mu  } \right)}^2}}}$ and $\beta = \frac{\sqrt{L}-\sqrt{\mu}}{\sqrt{L}+\sqrt{\mu}}$, then the redesigned system via the HB method~(\ref{eq:HB_re}) is with the optimal convergence rate given by (See Section 3.2.1 in~\cite{polyak1987introduction} for the proof):
\begin{align}
\left\| {x_k  -  x^* } \right\| \le \left(\frac{{\sqrt L- \sqrt{\mu}}}{{\sqrt L + \sqrt{\mu}}}\right)^k \left\| {x_0 -  x^* } \right\|=O({e^{-k\frac{2\sqrt{\mu}}{\sqrt L + \sqrt{\mu}}}}), \quad \forall k \in \mathbb{Z}^+.  \nonumber
\end{align}
\end{theo}

According to Theorem~\ref{theo:HB}, when implementing the HB method, constant coefficients can be adopted for simplification, i.e., $\varepsilon=\frac{4}{{{{\left( {\sqrt L  + \sqrt \mu  } \right)}^2}}}$, $\beta = \frac{\sqrt{L}-\sqrt{\mu}}{\sqrt{L}+\sqrt{\mu}}$.

\subsubsection{Redesign via an Accelerated Gradient Descent Method}
Similar to the HB method, an accelerated gradient descent (AGD) method also uses a momentum term. But it constructs an arbitrary auxiliary sequence $y_k$.
AGD achieves the optimal convergence rate $O(1/\varepsilon k^2)$ when objectives belong to function class $\mathscr{F}_L$. Algorithm~\ref{alg:AGD} shows its iterations.
\begin{algorithm}
\caption{Accelerated gradient descent method}\label{alg:AGD}
\textbf{Setting:} Choose appropriate positive step size $\varepsilon$, and let $x_1 = x_0$ be an arbitrary initial condition.
\begin{subequations} \label{eq:AGD}
\begin{align}
{x_{k + 1}} &= {y_k} - \varepsilon \nabla f\left( {{y_k}} \right) \label{AGD_1} \\
{y_k} &= {x_k} + \beta_k \left( {{x_k} - {x_{k - 1}}} \right) \label{AGD_2}
\end{align}
\end{subequations}
\end{algorithm}

For any system~\eqref{eq:LTI-sys} in Class-$\mathcal{O}$, apply the AGD method to solve problem~\eqref{eq:uncons} obtained via reverse-engineering to improve system performance. To show that the implementation is equivalent to introducing an extra part to the original GD dynamics,~\eqref{eq:AGD} is rearranged by interchanging the notations of $x$ and $y$ and combining the two equations:
\begin{align} \label{eq:AGD_combine}
    x_{k + 1}  = x_k  - \varepsilon \nabla f(x_k ) + \underbrace {\beta_k (y_{k + 1}  - y_k )}_{\Delta u_k }
\end{align}
where $\Delta u_k  = \beta_k (y_{k + 1}  - y_k ) = \beta_k \big[x_k  - \varepsilon \nabla f(x_k ) - x_{k - 1}  + \varepsilon \nabla f(x_{k - 1} )\big]$.
It is clear that~\eqref{eq:AGD_combine} is the sum of the gradient descent formula~\eqref{eq:GD} and extra dynamics $\Delta u_k$.
Therefore, the redesigned system via the AGD method is
\begin{align}\label{eq:AGD_re}
x_{k+1}=Ax_k+Cw+\Delta u_k
\end{align}
where $\Delta u_k  =  \beta_k \big(Ax_k+Cw - Ax_{k-1}-Cw^{\prime} \big)$ and $w^{\prime}$ is the value of $w$ at time step $k-1$.  Theorems~\ref{theo:AGD} and~\ref{theo:AGD_stro} summarize the convergence property when applying the AGD method to redesign.

\begin{theo}\label{theo:AGD}
For any system~\eqref{eq:LTI-sys} in Class-$\mathcal{O}$, if the objective function $f$ obtained via reverse-engineering belongs to $\mathscr{F}_L$, the step size $0<\varepsilon \le 1/L$ and $\beta_k = \frac{{k - 1}}{{k + 2}}$, then the convergence rate of the redesigned system via the AGD method~(\ref{eq:AGD_re}) is (See Theorem 2.2.2 in \cite{nesterov2018lectures} for the proof)
\begin{align}
f\left( x_k \right) - {f(x^*)} \le \frac{{8\left\| {x_0  - x^* } \right\|^2 }}{3{\varepsilon (k + 1)^2 }} = O\left(\frac{1}{\varepsilon k^2}\right), \quad \forall k \in \mathbb{Z}^+. \nonumber
\end{align}
\end{theo}

\begin{theo}\label{theo:AGD_stro}
For any system~\eqref{eq:LTI-sys} in Class-$\mathcal{O}$, if the objective function $f$ obtained via reverse-engineering belongs to $\mathscr{S}_{\mu,L}$, the step size $\varepsilon=\frac{1}{L}$ and $\beta_k = \frac{\sqrt{L}-\sqrt{\mu}}{\sqrt{L}+\sqrt{\mu}}$ (constant step scheme III in~\cite{nesterov2018lectures}), then the redesigned system via the AGD method~(\ref{eq:AGD_re}) is with the optimal convergence rate~\cite{Ghadimi2015Global}
\begin{align}
\left\| {x_k  -  x^* } \right\| \le \left(1 - \sqrt{\frac{\mu}{L}}\right)^k \left\| {x_0 -  x^* } \right\| =O({e^{-k\frac{\sqrt{\mu}}{\sqrt L}}}), \quad \forall k \in \mathbb{Z}^+.\nonumber
\end{align}
\end{theo}

According to Theorem~\ref{theo:AGD_stro}, when implementing the AGD method, constant coefficients can be adopted for simplification, i.e., $\varepsilon=\frac{1}{L}$, $\beta_k = \frac{\sqrt{L}-\sqrt{\mu}}{\sqrt{L}+\sqrt{\mu}}$ in Algorithm~\ref{alg:AGD} to achieve the optimal convergence rate.

Note that although both Theorem~\ref{theo:HB} and Theorem~\ref{theo:AGD_stro} require $f \in \mathscr{S}_{\mu,L}$ and the constant coefficient $\beta/\beta_k$ is related to $\mu$, our HB/AGD-based redesign is not restricted to strongly convex functions (obtained via reverse-engineering) only. It is still effective for $f \in \mathscr{F}_{L}$, but the convergence rates are unclear when coefficients are constants~\cite{Ghadimi2015Global}. In this case, we can adjust $\beta/\beta_k$ manually. Note that increasing the value of $\beta/\beta_k$ enlarges the influence of the momentum term $x_k-x_{k-1}$ in~\eqref{eq:HB_uk} and $y_{k+1}-y_{k}$ in~\eqref{eq:AGD_combine}.

TABLE~\ref{table:conv} compares the convergence rates of the original and redesigned systems under HB and AGD methods.  For any system~\eqref{eq:LTI-sys} in Class-$\mathcal{O}$,
When the objective function $f$ obtained via reverse-engineering is in $\mathscr{F}_L$, the redesigned system via the AGD method has a guaranteed better convergence rate than the original system. When $f \in \mathscr{S}_{\mu,L}$, the optimal convergence rate of the redesigned system via the HB method is always better than that of the original system and the redesigned system via the AGD method.
\footnotetext[4]{For objectives in this class, we compare the optimal convergence rate for convenience.}

\begin{table}[hb]
\begin{center}
\caption{Convergence rate comparison.}\label{table:conv}
\begin{tabular}{c|c|c|c}

\hline \hline
$f$ Classes   & Original & Apply HB & Apply AGD  \\
\hline
$f \in \mathscr{F}_L$  & $O\left( {\frac{1}{{\varepsilon k}}} \right)$ & $\diagup$ & $O\left(\frac{1}{{\varepsilon k^2 }}\right)$   \\ \hline
$f \in \mathscr{S}_{\mu,L}$\footnotemark[4] & $O({e^{ - k \frac{2\mu}{L+\mu}}})$ & $O({e^{-k\frac{2\sqrt{\mu}}{\sqrt L + \sqrt{\mu}}}})$ &  $O({e^{-k\frac{\sqrt{\mu}}{\sqrt L}}})$ \\ \hline\hline
\end{tabular}
\end{center}
\end{table}

\subsection{Systems in Class-$\mathcal{S}$}
\subsubsection{Convergence Rate of the Original Systems}
Any system~\eqref{eq:LTI-sys} in Class-$\mathcal{S}$ can be reverse-engineered as a primal-dual gradient (PDG) algorithm to solve a convex-concave saddle-point problem, so the convergence rate of system~\eqref{eq:LTI-sys} or \eqref{equ:linearsystemsplit} follows that of a PDG algorithm. Specifically, we focus on $A_{11}=I_{n_1}$ in~\eqref{equ:linearsystemsplit} so that the optimization problem obtained via reverse-engineering is
\begin{align}
    \mathop {\max }\limits_{\lambda  \in \mathbb{R}^m}  \mathop {\min }\limits_{x \in \mathbb{R}^n} \mathcal{L}(x,\lambda ) = f(x) + {\lambda ^T}(Bx - b) \label{eq:dual}
\end{align}
where $x \in \mathbb{R}^n$, $\lambda \in \mathbb{R}^m$ is the Lagrangian multiplier vector (dual variable vector), $B \in \mathbb{R}^{m \times n}$ and $b \in  \mathbb{R}^{m}$. Here we have rearranged~(\ref{equ:classSopt}) as above and replaced $x^{(1)},x^{(2)}, n_1,n_2$ with $\lambda, x,m,n$. Notations $f,n$ are abused in contrast to $f,n$ in~(\ref{equ:classSopt}).

\begin{ass} \label{ass:rank_B}
Matrix B is full row rank.
\end{ass}

The corresponding primal problem of~\eqref{eq:dual} is an equality constrained convex optimization problem given by
\begin{align}
   \min_{x \in \mathbb{R}^n } f\left(x \right) \quad
 \text{s.t.} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} Bx = b. \label{eq:primal}
\end{align}
A PDG algorithm is the simplest algorithm to solve problem~\eqref{eq:dual}, as shown in Algorithm~\ref{alg:PDG}.
\begin{algorithm}
\caption{First-order primal-dual gradient method}
\label{alg:PDG}
\textbf{Setting:} Choose appropriate positive step sizes $\varepsilon_1$, $\varepsilon_2$, and let $x_0$ and $\lambda_0$ be arbitrary initial conditions.
\begin{subequations} \label{eq:PDG}
\begin{align}
{x_{k + 1}} &= {x_k} - {\varepsilon _1}(\nabla f(x_k) + {B^T}{\lambda _k}) \label{alg:PDG_1}\\
{\lambda _{k + 1}} &= {\lambda _k} + {\varepsilon _2}(B{x_k} - b) \label{alg:PDG_2}
\end{align}
\end{subequations}
\end{algorithm}

Assume that a saddle point of $\mathcal{L}$ exists and is denoted by $(x^*, \lambda^*)$, which satisfies the optimality conditions:
\begin{subequations} \label{eq:L_opt}
\begin{align}
\nabla_x {\mathcal{L}} &= \nabla f({x^*}) + {B^T}{\lambda ^*} = 0 \label{eq:optimality_x}\\
\nabla_{\lambda} {\mathcal{L}} &= Bx^*-b = 0.
\end{align}
\end{subequations}

Furthermore, assume the objective function obtained via reverse-engineering $f \in \mathscr{S}_{\mu,L}$. Since the primal problem is strongly convex and the constraint is affine, strong duality holds. Therefore, $x^*$ is unique and is an optimal solution of the primal problem~\eqref{eq:primal}. When Assumption~\ref{ass:rank_B} holds, $\lambda^*$ is unique.

Introduce the conjugate function of $f(x)$: $f^{\star} (y) = \mathop {\sup }\limits_{x \in \mathbb{R}^n} \left\{ {\left\langle {x,y} \right\rangle  - f(x)} \right\}$ for all $y \in \mathbb{R}^n$, and~\eqref{eq:dual} can be further expressed as
$\mathop {\min }\limits_{\lambda \in \mathbb{R}^m}  {\rm{ }}{f^{\star}}( - {B^T}\lambda ) + {\lambda ^T}b$. Theorem~\ref{theo:PDG_conv} summarizes the convergence rate of the PDG algorithm/original system.

 \begin{theo} \label{theo:PDG_conv}
For any system~\eqref{eq:LTI-sys} in Class-$\mathcal{S}$, if the objective function $f$ in~(\ref{eq:dual}) obtained via reverse-engineering belongs to $\mathscr{S}_{\mu,L}$, Assumption~\ref{ass:rank_B} holds and step sizes $\varepsilon _1 , \varepsilon _2$ satisfy $0 < \varepsilon _1  \le \frac{2}{{L + \mu }}$, $0 < \varepsilon _2  \le \frac{2}{{\sigma _{\min }^2(B) /L + \sigma _{\max }^2(B) /\mu }}$ and $c< 1$, then this system converges to the unique saddle point $(x^*,\lambda^*)$ exponentially. Let $a_k=\left\| {x_k  - \nabla f^{\star} ( - B^T \lambda _k )} \right\|$, $b_k = \left\| {\lambda _k  - \lambda ^* } \right\|$ and define a potential function $V_k=\gamma a_k+b_k$, then for some constants $c, \gamma$ that depend on $\varepsilon_1, \varepsilon_2$, we have
\begin{align} \label{conv:PDG}
V_{k + 1}  \le c V_k, \quad  \forall k \in \mathbb{Z}^+.
\end{align}
where $\gamma>0$ and $c=\max \{ c_1 ,c_2 \}$ with $c_1  =  1 - \mu \varepsilon _1   + \frac{{\varepsilon _2 \sigma _{\max }^2(B) }}{\mu } + \frac{{\varepsilon _2 \sigma _{\max }(B) }}{\gamma }$ and $c_2  = 1 - \frac{{\varepsilon _2 \sigma _{\min }^2(B) }}{L} + \frac{{\varepsilon _2 \gamma \sigma _{\max }^3(B) }}{{\mu ^2 }}$.
\end{theo}

\begin{proof}
See the Appendix.
\end{proof}

Note that the potential function $V_k$ decreases at a geometric rate and the error of $\left\| {{x_k} - {x^*}} \right\|$ and $\left\| {{\lambda _k} - {\lambda ^*}} \right\|$ are bounded by $V_k$: $\left\| {{\lambda _k} - {\lambda ^*}} \right\| \le {V_k}$ and $\left\| {{x_k} - {x^*}} \right\| \le \left\| {{x_k} - \nabla {f^{\star}}( - {B^T}{\lambda _k})} \right\| + \left\| {\nabla {f^{\star}}( - {B^T}{\lambda _k}) - {x^*}} \right\|
 \le {a_k} + \frac{{{\sigma _{\max }(B)}}}{\mu }{b_k}
 \le \text{max}\left\{ {\frac{1}{\gamma },\frac{{{\sigma _{\max }(B)}}}{\mu }} \right\}{V_k}$. Therefore, as $V_k$ approaches zero, $(x_k,\lambda_k)$ approaches the saddle point $(x^*,\lambda^*)$.

\begin{coro} \label{coro:opt_stepsize}
When the step sizes $\varepsilon_1 = \frac{2}{L+\mu}$ and $\varepsilon_2 = \left( \frac{\sigma_{\max}^2(B)}{\mu} + \frac{\sigma_{\max}(B)}{\gamma}+\frac{\sigma_{\min}^2(B)}{L}  -\frac{\gamma \sigma_{\max}^3(B)}{\mu^2} \right)^{-1} \frac{2\mu}{L+\mu}$, the optimal convergence rate is attained for the original system.
\end{coro}

\begin{proof}
Since the geometric factor is determined by $c$, the convergence rate is optimal when $c$ is minimized. For a specific problem, parameters including $\mu, L, \gamma, \sigma_{\max}(B),$ $\sigma_{\min}(B)$ are fixed and we are able to adjust step sizes only. It is straightforward to notice that $c_1  =  1 - \mu \varepsilon _1   + \frac{{\varepsilon _2 \sigma _{\max }^2(B) }}{\mu } + \frac{{\varepsilon _2 \sigma _{\max }(B) }}{\gamma }$ is monotonically decreasing in $\varepsilon_1$ and increasing in $\varepsilon_2$ while $c_2  = 1 - \frac{{\varepsilon _2 \sigma _{\min }^2(B) }}{L} + \frac{{\varepsilon _2 \gamma \sigma _{\max }^3(B) }}{{\mu ^2 }}$ is monotonically decreasing in $\varepsilon_2$ since $\gamma < \frac{\mu^2 \sigma_{\min}^2(B)}{L \sigma_{\max}^3(B)}$ (due to $c<1$). Thus, $c$ is minimized when $\varepsilon _1$ takes its upper limit $\frac{2}{{L + \mu }}$ and $c_1 = c_2$, from which we obtain the value of $\varepsilon_2$ as given in this corollary.
\end{proof}

\begin{coro} \label{coro:opt_rate}
Suppose $\gamma {\rm{ = }}\frac{{\mu ^{\rm{2}} \sigma _{\min }^2 (B)}}{{2L\sigma _{\max }^3 (B)}} $ and step sizes are chosen as in Corollary~\ref{coro:opt_stepsize}, then we have $c\le 1 - \frac{1}{{\kappa ^3 (4\tau ^2 + 2\tau + 1 )}} $, where $\kappa = \frac{L}{\mu} $ is the condition number and $\tau = \frac{\sigma_{\max}^2(B)}{\sigma_{\min}^2(B)}$.
\end{coro}

\begin{proof}
First we show $\varepsilon _2$ satisfies the bound in Theorem~\ref{theo:PDG_conv}.
\begin{align*}
\varepsilon _2  &= 4\mu ^3 L\sigma _{\min }^2 (B)(\mu  + L)^{ - 1} \big(4L^2 \sigma _{\max }^4 (B) + \mu ^2 \sigma _{\min }^4 (B) + 2L\mu \sigma _{\min }^2 (B)\sigma _{\max }^2 (B)\big)^{ - 1}  \\
  &\le \frac{{4\mu ^3 L}}{{2\mu \big((4L^2  + 2L\mu )\sigma _{\max }^2 (B) + \mu ^2 \sigma _{\min }^2 (B)\big)}} \\
  &\le \frac{2}{{\frac{{6\sigma _{\max }^2 (B)}}{\mu } + \frac{{\sigma _{\min }^2 (B)}}{L}}} \\
  &\le \frac{2}{{\frac{{\sigma _{\max }^2 (B)}}{\mu } + \frac{{\sigma _{\min }^2 (B)}}{L}}}.
\end{align*}
Then we show the upper bound of $c$. According to Corollary~\ref{coro:opt_stepsize}, when $\gamma {\rm{ = }}\frac{{\mu ^{\rm{2}} \sigma _{\min }^2 (B)}}{{2L\sigma _{\max }^3 (B)}}$, we have $c_1  = c_2  = 1 - \frac{{\varepsilon _2 \sigma _{\min }^2 (B)}}{{2L}}$. Therefore,
\begin{align*}
c &=1 - 2\mu ^3 \sigma _{\min }^4 (B) (\mu  + L)^{ - 1} \big(4L^2 \sigma _{\max }^4 (B) + \mu ^2 \sigma _{\min }^4 (B) + 2L\mu \sigma _{\min }^2 (B)\sigma _{\max }^2 (B)\big)^{ - 1} \\
  &\le 1 - \frac{{\mu ^3 }}{{L(4L^2 \tau ^2  + \mu ^2  + 2L\mu \tau )}} \\
  &\le 1 - \frac{{\mu ^3 }}{{L^3 (4\tau ^2 + 2\tau + 1 )}} \\
  &\le 1 - \frac{1}{{\kappa ^3 (4\tau ^2 + 2\tau + 1  )}}.
\end{align*}
\end{proof}

Corollary~\ref{coro:opt_rate} shows that the optimal convergence rate of the original system is related to the condition number $\kappa$ and $\tau$. A large $\kappa$ implies a slow convergence rate since the iterations oscillate back and forth\cite{ortega2000iterative}.
Thus, we utilize augmented Lagrangian (AL) and hat-x methods to change the convergence rate by changing the value of $\kappa$ of the objective. Applying those methods to solve problem~\eqref{eq:dual} obtained via reverse-engineering results in a redesigned system formula with improved performance.

\subsubsection{Redesign via an Augmented Lagrangian Method}
Adding the square of the equality constraints as penalty terms can change the condition number of the original problem while the optimal solution stays unchanged. This method is known as the AL method.
The corresponding AL function for problem~\eqref{eq:primal} is
\begin{align}\label{eq:AL}
\mathcal{L}_a  = f(x) + \lambda ^T (Bx - b) + \frac{\alpha }{2}\left\| {Bx - b} \right\|^2
\end{align}
where $\alpha >0$ is a scalar. Equation~\eqref{eq:AL} is the Lagrangian for
\begin{align}\label{eq:AL_problem}
    \mathop {\min }\limits_{x \in \mathbb{R}^n} f(x) + \frac{\alpha }{2}\left\| {Bx - b} \right\|^2 \quad
 \text{s.t.} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} Bx = b
\end{align}
which has the same minimum and optimal solution as the original problem~\eqref{eq:primal}~\cite{bertsekas1997nonlinear}.

For any system~\eqref{eq:LTI-sys} in Class-$\mathcal{S}$, applying the AL method to solve the problem obtained via reverse-engineering leads to a redesigned system with improved performance:
\begin{align*}
x_{k + 1}  = x_k  - \varepsilon _1 \left( {\nabla f(x_k ) + B^T \lambda _k } \right)\underbrace { - \varepsilon _1 \alpha B^T (Bx_k  - b)}_{\Delta u_k }
\end{align*}
where $\Delta u_k={ - \varepsilon _1 \alpha B^T (Bx_k  - b)}$. Note that the extra part is added to primal variables only.

Let $g(x)=f(x) + \frac{\alpha }{2}\left\| {Bx - b} \right\|^2$ and we compare the condition numbers of $f$ and $g$.
The Hessian matrix of $g(x)$ is $\bf{H}$$_g ={\bf{H}}_f + \alpha B^TB$ and $\lambda_{\min}({\bf{H}}_g)I_n \preceq {\bf{H}}_g \preceq \lambda_{\max}({\bf{H}}_g)I_n$. The condition number of $g(x)$, denoted by $\kappa_g$, is $\frac{\lambda_{\max}({\bf{H}}_g)}{\lambda_{\min}({\bf{H}}_g)}$. Denote by $\kappa_0$ the condition number of $f$ and $\kappa_0=L/\mu$. For a specific problem, we are able to obtain the numerical form of matrix ${\bf{H}}_g$. In this case, we can calculate $\kappa_g$ and $\kappa_0$ precisely. If $\kappa_g < \kappa_0$, applying the AL method is able to improve the convergence rate; otherwise, $\alpha$ should be set to zero to avoid the influence of penalty terms. Note that matrix $B$ has a significant influence on the effectiveness of this method. The matrix product $B^TB$ will not change the topological structure of the original system as the redesigned system can be rearranged as $x_{k + 1}  = x_k  - \varepsilon _1  \nabla f(x_k ) -\varepsilon _1B^T \big(\lambda _k  +\alpha (Bx_k  - b)\big)$ where $Bx_k  - b$ can be obtained from the original update of $\lambda$.

Another benefit of applying the AL method is the convexification when the objective $f$ obtained via reverse-engineering is not strongly convex in $\mathbb{R}^n$.

\begin{theo}~\label{pro:convexification}
Assume that ${\bf{H}}_f$ is positive definite on the nullspace of $B^TB$, i.e., $y^T {\bf{H}}_f y >0$ for all $y \neq 0$ with $y^TB^TBy=0$. Then there exists a scalar $\bar{\alpha}$ such that for $\alpha>\bar{\alpha}$, we have (See Theorem 4.2 in~\cite{anstreicher2000note} for the proof)
\begin{align*}
{\bf{H}}_f + \alpha B^TB \succ 0.

\end{align*}
\end{theo}

\subsubsection{Redesign via a Hat-x method}
Another way is to introduce a free variable $\hat x \in \mathbb{R}^n$ to prevent a dramatic change of $x$. This leads to
\begin{align} \label{eq:hat}
 \mathop {\min }\limits_{x, \hat{x} \in \mathbb{R}^n} f(x) + \frac{\alpha }{2}\left\| {x - \hat x} \right\|^2 \quad
 \text{s.t.} \quad Bx = b
\end{align}
where $\alpha$ is a scalar. This problem is equivalent to
\begin{align*}
  \mathop {\min }\limits_{z \in \mathbb{R}^{2n}} h(z) \qquad  \text{s.t.} \quad\bar Bz = b
\end{align*}
where $z = \left[ {\begin{array}{*{20}c}
   x  \\
   {\hat x}  \\
\end{array}} \right]$, $h(z) = f(x) + \frac{\alpha }{2}z^T \left[ {\begin{array}{*{20}c}
   I_n & { - I_n}  \\
   { - I_n} & I_n \\
\end{array}} \right]z$, $\bar B = \left[ {\begin{array}{*{20}c}
   B & \mathbf{0}  \\
\end{array}} \right] \in \mathbb{R}^{m \times 2n} $. Suppose Assumption 1 holds, then $\bar B$ is also full row rank and the maximal and minimal singular values of $\bar B$ are the same as those of $B$. The Lagrangian is
\begin{align*
\mathcal{L}_h  = f(x) + \lambda ^T (Bx - b) + \frac{\alpha }{2}\left\| {x - \hat x} \right\|^2.
\end{align*}
Applying an optimality condition, we have
\begin{subequations} \label{eq:L_h_opt}
\begin{align}
\nabla_x {\mathcal{L}_h} &= \nabla f({x^*}) + {B^T}{\lambda ^*} + \alpha (x^*- \hat x^* )= 0 \label{eq:optimality_hatx}\\
\nabla_{\hat x} {\mathcal{L}_h} &= \alpha (\hat x^* - x^*) = 0 \label{eq:optimality_hatx_2}\\
\nabla_{\lambda} {\mathcal{L}_h} &= Bx^*-b = 0.
\end{align}
\end{subequations}
Then $\hat x^* = x^*$.
Therefore, $x^*,\lambda^*$ in the optimal solution $(x^*,\hat x^*,\lambda^*)$ of~\eqref{eq:hat} are the same as that of~\eqref{eq:primal}.

For any system~\eqref{eq:LTI-sys} in Class-$\mathcal{S}$, apply the hat-x method to solve the optimization problem obtained via reverse-engineering to improve system performance. The iterations of the primal variables are
\begin{subequations}
\begin{align*}
x_{k + 1}  &= x_k  - \varepsilon _1 \left( {\nabla f(x_k ) + A^T \lambda _k } \right)\underbrace { - \varepsilon _1 \alpha (x_k  - \hat x_k )}_{\Delta u_k } \\
 \hat x_{k + 1}  &= \hat x_k  + \varepsilon _1 \alpha (x_k  - \hat x_k )
\end{align*}
\end{subequations}
where $\Delta u_k={ - \varepsilon _1 \alpha (x_k  - \hat x_k )}$. Note that the extra part is added to the primal variables only.

Denote by $\kappa_h$ and ${\bf{H}}_h$ the condition number and the Hessian matrix of $h(z)$. Then
$\kappa_h = \frac{\lambda_{\max}({\bf{H}}_h)}{\lambda_{\min}({\bf{H}}_h)}$. Next, we compare $\kappa_0$ and $\kappa_h$. To do this, we first obtain the range of $\kappa_h$, and then compare the lower and upper bounds of $\kappa_h$ with $\kappa_0$.

\begin{lem} \label{lem:preceq}
For symmetric matrices $A, B \in \mathbb{R}^{n \times n}$, if $A \preceq B$, then $\lambda_{\max}(A) \le \lambda_{\max}(B)$ and $\lambda_{\min}(A) \le \lambda_{\min}(B)$ hold.
\end{lem}

\begin{proof}
For any $x \in \mathbb{R}^{n}$, we have $x^TAx \le x^TBx$. Assume $x^*=\mathop {\text{arg max}}\limits_{\|x\|=1} x^TAx$. Then $\lambda_{\max}(A)={x^*}^TAx^* \le {x^*}^TBx^* \le \mathop {\max}\limits_{\|x\|=1} x^TBx=\lambda_{\max}(B)$. On the other hand, assume $\bar x=\mathop {\text{arg min}}\limits_{\|x\|=1} x^TBx$. Then $\lambda_{\min}(B)= \bar x^TB \bar x \ge {\bar x}^TA\bar x \ge \mathop {\min}\limits_{\|x\|=1}  x^TAx = \lambda_{\min}(A)$.
\end{proof}

\begin{theo}
Assume the objective obtained via reverse-engineering in~\eqref{eq:primal} $f \in \mathscr{S}_{\mu,L}$, then $\underline{\kappa_h} \le  \kappa_h  \le \overline{\kappa_h}$, where $\underline{\kappa_h} = \frac{2\alpha+\mu+\sqrt{\mu^2+4\alpha^2}}{2\alpha+L-\sqrt{L^2+4\alpha^2}}$ and $\overline{\kappa_h}  = \frac{2\alpha+L+\sqrt{L^2+4\alpha^2}}{2\alpha+\mu-\sqrt{\mu^2+4\alpha^2}}$.
\end{theo}

\begin{proof}
Since $f \in \mathscr{S}_{\mu,L}$, we have $\mu I_n \preceq \nabla^2 f(x) \preceq LI_n$ and the Hessian of $h(z)$ is
$H = \left[ {\begin{array}{*{20}c}
   \nabla^2 f(x)+\alpha I_n & -\alpha I_n  \\
   -\alpha I_n & \alpha I_n  \\
\end{array}} \right]
$. Then $\underline{H} \preceq  H \preceq \overline{H}$, where $\underline{H}=\left[ {\begin{array}{*{20}c}
   (\mu+\alpha) I_n & -\alpha I_n  \\
   -\alpha I_n & \alpha I_n  \\
\end{array}} \right]$ and $\overline{H}=\left[ {\begin{array}{*{20}c}
   (L+\alpha) I_n & -\alpha I_n  \\
   -\alpha I_n & \alpha I_n  \\
\end{array}} \right]$. Their eigenvalues are $\lambda (\underline{H})=\alpha+\frac{\mu}{2} \pm \frac{1}{2}\sqrt{\mu^2+4\alpha^2}$ and $\lambda(\overline{H})=\alpha+\frac{L}{2} \pm \frac{1}{2}\sqrt{L^2+4\alpha^2}$. By applying Lemma~\ref{lem:preceq}, we have $\lambda_{\min} (\underline{H}) \le \lambda_{\min}(H) \le \lambda_{\min} (\overline{H})$ and $\lambda_{\max} (\underline{H}) \le \lambda_{\max}(H) \le \lambda_{\max} (\overline{H})$. According to the definition of $\kappa_h$, we obtain the range of $\kappa_h$.
\end{proof}

Now we can compare the range of $\kappa_h$ with $\kappa_0$. Let $M=\underline{\kappa_h}-\kappa_0$, $w=\frac{\mu}{L}$ and $v=\frac{\alpha}{L}$ with $0<w \le 1$ and $v>0$, then $M =\frac{2v+w+\sqrt{w^2+4v^2}}{2v+1-\sqrt{1+4v^2}}-\frac{1}{w}.$
When $w$ tends to $0$, $M<0$; otherwise, $M>0$. On the other hand, $\overline{\kappa_h}-\kappa_0=\frac{1}{4\alpha \mu}\big[(2\alpha+\sqrt{L^2+4\alpha^2})(2\alpha+\mu+\sqrt{\mu^2+4\alpha^2})+L(\mu+\sqrt{\mu^2+4\alpha^2}-2\alpha)\big]$.
Since $\mu+\sqrt{\mu^2+4\alpha^2}>2\alpha$, $\overline{\kappa_h} > \kappa_0$. To conclude, $\kappa_h$ can be smaller than $\kappa_0$, depending on specific problems. Moreover, the hat-x method increases the dimension of the original system and works like a low-pass filter. This method could slow down system dynamics and smooth the trajectories, as demonstrated in Section~\ref{se:app_PI}.
\subsection{Reconfiguration Steps}

To summarize, the redesign approach is consist of the following three steps:

\begin{enumerate}
    \item \textbf{Reverse-engineering:} For a given system~\eqref{eq:LTI-sys}, apply Theorem~\ref{theo:class-O} or~\ref{theo:class-S} to reverse-engineer it as a GD algorithm~\eqref{eq:GD} or a PDG algorithm~\eqref{eq:PDG} for solving an unconstrained convex optimization problem~\eqref{eq:uncons} or a saddle-point problem~\eqref{eq:dual}, i.e., system dynamics~\eqref{eq:LTI-sys} can be rewritten as the form~\eqref{eq:GD} or~\eqref{eq:PDG}.

    \item \textbf{Acceleration:} Apply an HB/AGD method or an AL/hat-x method to solve the optimization problem obtained in Step 1), which results in a redesigned system

    \item \textbf{Implementation:} Rearrange the redesigned system and compare it with the original system~\eqref{eq:GD}/\eqref{eq:PDG} or~\eqref{eq:LTI-sys} to obtain the implementation of the extra dynamics.
\end{enumerate}

This approach can be implemented by any dynamical system belonging to Class-$\mathcal{O}$ or Class-$\mathcal{S}$ to systematically improve its performance. With the extra dynamics added to the original controllers, the convergence rates/speeds and transient behaviors will be improved while the original control structures remain. Also, this approach is able to handle network systems as shown in Section~\ref{se:simu}, whose scale can be large.

\section{Examples Revisited}\label{se:simu}

\subsection{Internet Congestion Control}
Following the redesign procedure, applying Algorithm~\ref{alg:HB} and Algorithm~\ref{alg:AGD} to solve problem~\eqref{eq:opt_net}, we obtain the same form of redesigned dynamics but with different $\Delta u(k)$ given by
\begin{align*}
x(k+1)&=x(k) + \varepsilon \text{diag}\{k_{x_i}\} \left( U^{\prime} (x(k) ) - R^Tf(Rx(k)) \right) +\Delta u(k).
\end{align*}
For HB-based redesign, $\Delta u(k) = \beta \big ( x(k)-x(k-1)\big)$; while for AGD-based redesign, $\Delta u(k) = \beta_k \Big\{x (k) -x (k-1) + \varepsilon \text{diag}\{k_{x_i}\} \big[U ^\prime  (x (k)) - q(k)-U ^\prime (x (k-1)) + q(k-1) \big] \Big\}$, where $q$ is the corresponding vector form of $q_i$ and $q=R^Tf(Rx)$.

Consider a $2$-link network
as shown in Fig.~\ref{fig:case}. Choose utility functions as $U_i(x_i)=\log x_i$, $k_{x_i}=0.1$, $\beta =0.54$ and $\beta_k =0.6$. Choose the penalty function as in~\cite{kelly1998rate}: $f_l (y_l ) = \frac{{(y_l  - c_l  + \sigma)^ +  }}{{\sigma ^2 }}$, where $\sigma $ is a small positive number and $(y_l  - c_l  + \sigma )^ + =\max\{y_l  - c_l  + \sigma,0\}$. Initially, the capacities of links A and B are $2, 4$ respectively and they change to $3, 1$ after some time to simulate sudden change of the link capacity. As shown in Fig.~\ref{fig:ICC}, redesigned dynamics via HB and AGD methods perform better than the original dynamics.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=10cm]{case.pdf}
\caption{Left: A $2$-link network shared by $3$ users. Right: A regular network of $6$ agents.}
\label{fig:case}
\end{center}
\end{figure}

\begin{figure*}[!t]
\centering
\begin{subfigure}[!t]{0.7\textwidth}
\centering
\includegraphics[width=1.1\textwidth]{sim_icc_o.pdf}
\end{subfigure}
\begin{subfigure}[!t]{0.29\textwidth}
\centering
\includegraphics[width=1.1\textwidth]{error_icc.pdf}
\end{subfigure}
\caption{Internet congestion control. Left \& Middle: Simulation results (``OD'' stands for original dynamics; ``HB" and ``AGD" stand for redesigned dynamics via HB and AGD). Right: State error measured by $\left\| {{\rm{ }} x-x^* {\rm{ }}} \right\|_2$.}
\label{fig:ICC}
\end{figure*}

\begin{figure*}[!t]
\centering
\begin{subfigure}[!t]{0.7\textwidth}
\centering
\includegraphics[width=1.05\textwidth]{sim_pi.pdf}
\end{subfigure}
\begin{subfigure}[!t]{0.29\textwidth}
\centering
\includegraphics[width=1.05\textwidth]{error_pi.pdf}
\end{subfigure}
\caption{Distributed PI control. Left \& Middle: Simulation results (``OD'' stands for original dynamics; ``AL" and ``HAT" stand for redesigned dynamics via AL and hat-x). Right: State error measured by $\left\| {{\rm{ }} y-y^* {\rm{ }}} \right\|_2$.}
\label{fig:PI}
\end{figure*}

\subsection{Distributed PI Control for Single Integrator Dynamics}\label{se:app_PI}
Problem~\eqref{eq:PI_f} obtained via reverse-engineering is equal to
\begin{align*}
\mathop {\min }\limits_{y \in \mathbb{R}^n } &f= \frac{\rho_2}{2}y^T Ly + \frac{\delta }{2}y^T y - y^T d - \delta y^T y(0) \\
&\text{s.t.} \quad \rho_1 \tilde{L}^T y = \textbf{0}.
\end{align*}
Since ${\bf{H}}_f = \rho_2L+\delta I_n \succ 0$, $f \in \mathscr{S}_{L,u}$. Following the redesign procedure, applying augmented Lagrangian and hat-x method, we obtain the same form of redesigned dynamics but with different $\Delta u(k)$ given by
\begin{align*}
 y(k + 1) =& y(k) - \varepsilon _1 \big(\rho_2 Ly(k) + \delta y(k) - d - \delta y(0)  + \rho_1\tilde{L} z(k)\big)+ \Delta u(k) \\
 z(k+1) =& z(k)+ \varepsilon _{\rm{2}} D y(k).
\end{align*}
For the AL-based redesign, $\Delta u(k) =  - \varepsilon _1 \alpha \tilde{L} \tilde{L}^T y(k)$; while for the hat-x-based redesign, $\Delta u(k) =  - \varepsilon _1 \alpha \big(y(k) - \hat y(k)\big)$, where $\hat y(k + 1) = \hat y(k) + \varepsilon _1 \alpha (y(k) - \hat y(k) \big)$.

Consider a network of $6$ agents with the topology shown in Fig.~\ref{fig:case}. In addition, communication delay is considered in this network since this could happen in reality.
Let the constant disturbance $d=[0, 2, 0, 0, 0, 0]^T$, the initial condition $x(0)=[5,-6, 8, 2, -4, 0]^T$, the integral gain $\rho_1=10$, the static gain $\rho_2=0.5$, $\delta =1$. As shown in Fig.~\ref{fig:PI}, redesigned dynamics via the AL and hat-x methods perform better than the original dynamics, while the redesigned dynamics via hat-x is the most smooth.

\section{Conclusion and Future work}\label{se:conclusion}
This paper has proposed a control reconfiguration approach to improve the performance of two classes of dynamical systems. This approach is to firstly reverse-engineer a given dynamical system as a gradient descent or a primal-dual gradient algorithm to solve a convex optimization problem. Then, by utilizing several acceleration techniques, the extra control term is obtained and added to the original control structure. Under this retrofit procedure, system performance is improved while the control structure remains, as demonstrated by both theoretical results and practical applications.

In the future, we will investigate the implementation of the redesign in a continuous-time setting as well as the framework for nonlinear systems. Also, the case when ${\bf{H}}_{f(x^{(1)})} \preceq 0$ in the definition of Class-$\mathcal{S}$ will be studied. Last but not least, we will consider redesigning to improve other properties of systems, for example, robustness to delays.

redpajama_set_name: RedPajamaArXiv


text:
Q: Storage & Network device drivers source code for Nexus 6 and Samsung Galaxy S6 For university research purposes, I am searching for a specific part of Nexus 6, and Samsung Galaxy S6 source code. Particularly, I am interested in "network" & "storage" driver source codes. I have visited developer resources for both Samsung (which are said to be open-sourced), and Motorolla, but was unable to find the desired code.
I also have the pure Android source code on this website, but I guess the drivers for "network" & "storage" might be hardware-specific.
I am having a hard time locating "network" & "storage" drivers for the mentioned two devices. In case the code I am looking for is open-sourced, can anybody please let me know where they are located?

A: Have a look at the code for drivers(from kernel) of-

*

*Samsung Galaxy S6 - Network and Storage.

*Nexus 6 - Network and Storage

*Nexus 6P - Network and Storage
But, as FunkSoulBrother pointed out, each batch of manufacturing and every model variation(Europe, US specific carrier models, International model etc.,) will have different drivers. As chances are internals also change. So, this can be a starting point for your research for sure :) Happy coding!

A: Have you checked this site http://androidxref.com/.You can browse most of the android source code-Not very sure of specific drivers.

redpajama_set_name: RedPajamaStackExchange


text:
August 26 famous birthdays in India Indian celebrity stars
In this post -- Famous Birthdays August 26 Bollywood Actor Actress Cricket Politics, check who are sharing your birthday in India August 26's famous birthday personality in India like Bollywood actor, actress, cricketer, politician and all famous Indian celebrity.
August 26 famous birthdays in India
Neeru Bajwa
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Birth Date: 26-Aug-1980
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Maneka Gandhi
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Rubina Dilaik is an Indian actress. She played Radhika, the main character in the TV series Choti Bahu, and was playing the same role in Choti Bahu 2. Rubina was born in Shimla on August 26, 1987. Rubina is from Himachal Pradesh. She has studied in St. Bedes, Shimla. She had also participated and won two local beauty pageants. She has been Miss Shimla 2006 and also Miss North. She also participate...
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redpajama_set_name: RedPajamaCommonCrawl


text:
Our main goal is defined by the professionalisation and promotion of investigative journalism, almost non-existent in Romania.
LINX benefits from the support, expertise and reputation of CRJI, a founding member of the Global Investigative Journalism Network and collaborator of the International Consortium of Investigative Journalists. LINX has access to resources and databases to which regular media organizations do not have.
Russian agents
Who's behind the anti-Western speech in Romania and Bulgaria
Discurs anti-occidental
Bulgarian media in May: the EU's end is near
The Bulgarian president, Rumen Radev, feels good in Putin's presence. Source: Facebook public page
In view of the upcoming Brexit and of the concern in many EU countries, in Bulgarian media are often noticed comments, analyzes and news regarding the disintegration of the alliance.
Тhe political scientist Ivan Krastev, a contributing opinion writer for the International New York Times, explains the changes and tendencies in the EU and predicts future problems for the Union. In an interview for Deutsche Welle, Krastev says that the end of the current EU is near. Krastev also shares his opinion in an analysis for the French newspaper "Le Monde". He says that Europe is afraid of the future and believes that the possibility of disintegration of the EU is real.
Of the same opinion as Krastev is Gergana Passy, former Minister of European Affairs in the government of Sergey Stanishev (he's now the leader of the Party of European Socialists, back then he held the prime minister position with the support of the Bulgarian Socialists Party). "Europe today is very different from before. Now it consists of people and countries which don't listen to one another. We saw a mini-version of this lack of understanding in Brexit", explains Passy.
About this bad feeling concerning the EU future that people tend to have is also speaking Vesela Cherneva, director of the Bulgarian Office of the European Council for Foreign Policy. She analyzes a study conducted in 14 EU countries, study which examines the way people in Europe felt about the European Parliament elections. "A lot of the people have come to their senses, and the Bulgarians are no exception", explains Cherneva in an interview for the Bulgarian National Radio.
The president of the European Commission, Jean-Claude Juncker, also refers to the problems that EU the has to deal with at the moment. "We don't love each other and we've lost our collective libido", says Juncker in an interview for the German Haldelsblatt, quoted by Politico and Guardian and also published in the Bulgarian media dnevnik.bg. Juncker also says that it gets more and more difficult for the EU leaders to convince the people that populism is no solution.
Russia, my love
All media in Bulgaria were extremely focused on the European Parliament elections held on the 26th of May.
Many newspapers, online portals etc published comments, columns and interviews given by politicians and various experts who do not support many of the the EU policies and therefor do not hesitate to express their opinion on that.
For Volen Siderov, the leader of the nationalist party "ATAKA" – as he tried to explain during a meeting (that took place in the Bulgarian Parliament) of the Bulgaria-Russia Parliamentary Friendship Group – some of the issues that prevent the EU and Bulgaria to get along are in regard to the sanctions against Russia: "Bulgaria needs to be bolder in its decisions and to clarify, in relation with Brussels, that our country is losing economically because sanctions were imposed to Russia."
The leader of the Party of European Socialists and a mеmber of the European Parliament, Sergey Stanishev criticized the European People's Party on the issue of "a huge misuse of the EU funds" and stated that this is scandalous.
Former vice prime minister of the current government of Bulgaria, Valeri Simeonov, who is also the leader of the nationalist political party "National Front for the Salvation of Bulgaria" and a candidate (in May) for a seat in the European Parliament, is quoted in an interview for NOVA that the EU policy simply isn't good for Bulgaria, due to the fact that the EU regulations are, in fact, just a set-back for the country who is under constant outside pressure.
The Bulgarian president, Rumen Radev, who is supported by the Bulgarian Socialists Party, is also usually expressing his opinion against the Bulgarian government and respectively is criticizing the EU policy. In a meeting with the Germasn Minister of the Foreign Affairs, Heiko Maas, Radev stated that the EU needs more effective and working mechanisms to address the current challenges.
Stefan Tafrov, candidate for a seat in the European Parliament (supported by the "Democratic Bulgaria" party), also criticizes the EU, in an interview for NOVA. He believes that the EU is in crisis and the nationalist parties that become more and more popular are a consequence of Russia's role in the EU policies.
The US – China trade war
One of the most commonly found topics in Bulgarian media in May is the US – China trade war. Many of the Bulgarian media outlets publish news and analyses every day on the subject.
24 chasa.bg has republished an analysis drafted by Virginie Montet, initially published by France Presse, according to which the price for the sanctions imposed by the US to China will eventually be paid by the American people.
Christine Lagarde, Chairman of the International Monetary Fund, has also shared her concerns about the US – China commercial issue: Lagarde warns that, should the Washington – Beijing trade tensions continue, the world economy is threatened.
The media outlet investor.bg has translated an article drawn up by an expert, Dan Kopf (initially published by the economic edition of "Quartz"), who argues that the US are going to be in trouble precisely because of the sanctions imposed to China. Kopf pointed out that the big problems is in connection with US imported goods, meaning that American companies, as importers, are to pay the tariffs, not the Chinese ones.
In May, the authors of this study examined 20 Bulgarian media outlets, the most relevant in terms of readership.
Most of the articles were provided by bnr.bg and 24chasa.bg (20 articles for each).
Dnevnik.bg is ranked on the second place with 18 articles.
On the third place is fakti.bg with 14 articles.
93 of the collected articles could be classified as "neutral". 47 of those are news, the other 46 – analyses.
28 articles are labeled "biased" (17 of them are news, 11 are analyses).
The leader of the biased articles in May is fakti.bg (7 news and 7 analyses, both anti-EU and anti-US articles included). Right under fakti.bg comes nova.bg (6 news and 7 analyses, all of them anti-US articles).
Biased articles were also published by bnr.bg, dnevnik.bg, clubz.bg and baricada.org.
* The authors of this analysis/study divided the media articles into two categories (news and analysis), each classified as biased or neutral. A geographical criterion was also used to link the media articles to the EU and the US. According to this study, a piece of news is nothing more but a short article covering a fact or a statement whereas an "analysis" may be a column, an investigative piece or any other type of article that is based on several facts and statements that are premises for the conclusion that the author of the article wishes to make public. The neutral characteristic is attributed to those news and analyses that use actual quotes (and not made up or out of context ones), rely on fact checking and logical syllogisms, provide side relevant data (context) in order for the public to get the bigger picture. News and analyses are labelled as biased when, on the contrary, the journalist's work is not compliant with all of the above: quotes are partial and/or manipulated/manipulative, there's no vetting process, neutral context is not added (and when there's some context provided, facts are selected to match the conclusion – which, in this sort of cases, is pre-set – and not the other way round).
Based on this report:
Discursul anti-Occident – interviu cu jurnalistul Cătălin PRISACARIU
Антиевропейски настроения в медиите през май, свързани с евроизборите и търговската война САЩ-Китай
Dilema veche la Timpul prezent – Discursul anti-Occident
Discursul anti-Occident: Aspiratoarele zburătoare americane și spionii ruși
Discursul anti-Occident: Președintele Bulgariei se simte captiv în UE
EU's freedom of goods policy opened door to Paris terror attacks
SwissLeaks. Wafer and Swiss accounts-flavored love
Academia SRI blochează de 5 luni un verdict în cazul doctoratului suspect de plagiat al premierului
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Heavy metal in Moldova: Dangerous traffic
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The Nowness collection is chic and modern. Whether you have a business meeting or going out to dance in the city, this model makes every outfit complete. This collection has a selection of different colours of mesh straps.


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The Government's Horizon Scanning Group is gearing up to create a third review of technologies with the potential to increase UK productivity and increase public solutions. The focus of the day was to work on our thinking and conceptual abilities as possible future researchers in education and (emerging) technologies in order to equip us to frame, revise, conceptualise and restate the problem statement that you intend to investigation. We have to use computing technologies to not only disseminate understanding to but also to expand our reach to non-conventional learners.
As we delve additional into issues relevant to educational investigation with a focus on emerging technologies as issue, I hope it will grow to be clearer which investigation design will prove to be the greatest fit" for the outcomes I have in thoughts – the answer(s)" to my query … or clues and hints which will point me into a distinct path. There had been other imprints from other sessions that I will blog about in due time but for now I want to share my take on the queries we had been tasked with, as students in the Post Graduate course on Educational Technologies. The main implication of usability and accessibility of technologies is innovation.
Computing technologies provide students radically new ways to seek information outside the classroom of the standard setting. Buzzetto-A lot more (2015) also discovered that gender had no effect on the perceived value of YouTube in the teaching and studying process. Cloud computing is normally defined as a kind of computing that relies on sharing computing resources rather than getting neighborhood servers or individual devices to manage applications (Beal, 2015).
Cloud computing permits a number of learners to access remotely a server to save, update, and retrieve their information with out acquiring licenses for diverse applications (Beal, 2015). In this info age, computing technologies have transformed the way we teach and understand. The major relationship between neighborhood constructing and successful instruction is the creation of a finding out atmosphere. Knight (2015) claimed to have selected FLEXspace since of its flexibility and innovation.
FLEXspace is an application that aids users to personalize their bookings, workspace and room coordination capabilities along with calendaring, catering and IT requests management amongst other folks (Optimus BT, 2015). As an open and not for profit platform run by volunteers, she claims that FLEXspace might be accessed by community colleges to view a number of visual examples of redesigned studying spaces in use at other face-to-face institutions, rather than visiting a number of institutions".


Hoffman Park has 15 camping spots with electric hook-up and 5 tent sites. Camping is open from approximately April 15 through October 15 each year. (Exact dates vary based on weather - please call for updates.) Bathrooms, shower, water, and dump station are available. Electric site fee is $20 per night. Tent site fee (non-electric) is $10 per night.
Maximum stay is 2 weeks in electric sites and 5 days for tents.
Campsites are on a first come, first served basis. We do not accept reservations.
Campers are required to prepay into the yellow mailbox as sites are checked daily for valid occupancy.
View the Hoffman Park Camping Map for more information.


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Nick Cave and The Bad Sees - The Road To God Knows Where / Live At The Paradiso (DVD)
Renaissance man Nick Cave is an actor, novelist, screenplay writer and poet but he's best known of course as frontman of The Bad Seeds. The undisputed King Of The Goths released a double album Abatoir Blues / The Lyre Of Orphans and a DVD of one of their concerts from that tour would have been ideal, but that's yet to materialize. Until that is inevitably released, to keep fans happy a double disc DVD is on its way.
Disc One The Road To god Knows Where is a documentary filmed in 1989 on the bands American / Canadian tour. The band are captured in glorious black and white, sometimes grainy with footage of the guys backstage, clips of live performances, unexpected joviality on the tour bus, the banality of soundchecks and photoshoots in seedy looking dressing rooms. The music is great throughout although the sound quality is a little patchy. Director Schweppel nails the tedium and boredom of touring with the distance of a casual observer, especially the scene which shows Nick lovingly reading out a fan letter. Extras include The Song which is much better quality showing a youthful, energetic skinny Cave rehearsing. A narration option would have benefited this short, but it's fascinating to see this volatile character at work in his natural environment. Finishing off the first of the two discs is city of refuge which over the song "You Better Run" shows a montage of the many wondrous sights experienced through the windows of a tour bus.
Disc Two is a more straightforward, but dazzlingly exciting document of a concert filmed at The Paradiso in 192. Filmed in colour with plenty of close ups, this dispenses with unnecessary flashy visuals and just displays Nick Cave And The Bad Seeds in exactly the way you'd view it as a fan on the front row.
"Jack The Ripper" visually matches the punky ferociousness of the song with occasional shaky camera work. This is an exercise in controlled anger with harmonies from The Bad Seeds and growling, angry lead vocal from Cave. "The Weeping Song" is a melodic piano heavy song sung in part by Bad Seeds guitarist Mick Harvey. A more reflective bad sees, restrained and poetic, sensitive and poetic with great slide guitar. "New Morning" sees Nick sit down by the piano and croon like a Goth Sinatra with gospel undertones and a gentle country feel.
This double DVD package is essential for all Nick Cave completists and is an interesting insight into a man who has never compromised his artistic vision
Nicholas Paul Godkin
Click here to leave your Nick Cave comments on the Message Board


Why a Stormwater Management Roof?
Water storage in a standard green roof build-up cannot be increased arbitrarily because more water could result in changes to the vegetation used.
However, this is possible with a Stormwater Management Roof which offers this option in addition to all the other features of a green roof build-up.
Up to approx. 80 l/m² of precipitation can be retained with a Stormwater Management Roof and then released into the sewerage system over a pre-defined period (between 24 hours and a few days). The space required for the increased precipitation is created by a grid spacer situated beneath the actual green roof build-up.


Kevin Pho, MD is a practicing board-certified internal medicine physician and a health care social media leader since 2004. He is co-author of the book, "Establishing, Managing, and Protecting Your Online Reputation: A Social Media Guide for Physicians and Medical Practices." Dr. Pho will take you through his decade-long social media journey, and show how social media can make a profound difference in 3 important ways. First, how social media can facilitate the connection between health professionals and patients. Second, how social media can proactively define and protect an online reputation. And finally, how social media can share the stories of those who intersect with our health care system, but are rarely heard from. Not only will you see how social media made a difference in Dr. Pho's life, you will also be inspired to use social media and make your own difference in health care.


We all know there are two sides to every story under the sun and that includes the controversial issue of owning and caring for exotic animals. Some creatures that fall within this category might be more suitable and less dangerous than others, but for the most part, exotic animals do not make good pets, and lions, tigers, wolves, primates, bears and reptiles do not thrive outside their natural habitat.
Why don't exotic animals make good pets?
The average person cannot provide the specal care, housing and maintenance that exotic animals need in order to flourish as living things. Often they suffer at the hands of private individuals due to malnutrtion and inadequate housing, and they can be very dangerous both to the safety and health of those in their proximity. These animals are wild and meant to be free; cages bruise their souls and crush their spirits.
In the forefront of those pushing to end the private ownership of wild animals is a non-profit organization called Born Free USA. According to My Health News Daily, this advocacy organization has documented 1,500 attacks, including 75 human deaths caused by exotic animals since 1990. There have also been more than a few harmful if not fatal confrontations, including an incident in Texas where a four-year-old boy was hospitalized after being mauled by a pet mountain lion.
Society at large has much to fear from exotic animals that are not properly cared for. When these animals are isolated from others of their kind and forced into small enclosures unable to roam freely, over time they can and often do detonate with all the deadly force of a hand grenade. Many exotic animals are carriers of diseases, such as Herpes B, Monkey Pox and Salmonellosis, all of which can be transmitted to humans.
The problem with keeping large cats as pets is that they are completely unpredictable. They can be friendly and loving one minute, and in an instant ferocious and deadly. While lions, tigers, leopards and cougars are adorable when they are kittens, their potential to kill or maim people and other animals as they grow cannot be ignored.
According to veterinarian, Terry Mattive of T & D Mountain Range Menagerie, a sanctuary for unwanted, abused and exploited jungle cats in Penn Creek, Pennsylvania: "People buy these large cats when they are kittens and dont' have the foresight to see in four years that kitten is going to weigh 500 pounds, and instead of two bottles it is going to need 30 to 50 pounds of meat a day. They try to make a pet out of something that will never be a pet."
The US Endangered Species Act of 1973 prohibits the possession of pure wolves. For the most part, despite the image above, pure wolves are unpredictable and can become aggressive towards small animals and children. Their predatory behavior is genetically coded and although it can sometimes be suppressed, it can never be eliminated by socialization or training. Wolves are pack animals and although they are often shy with humans other than their owners, even when they seem friendly they have been known to attack people as they would other wolves in certain situations.
Monkeys are the most common primates owned by private individuals. In one instance, Judi Zazula, an occupational therapist and rehabilitation engineer, founded an organization back in 1979 called Helping Hadns-Monkey Helpers. The organization trains monkeys to help paralized or otherwise disabled individuals to "eat, scratch an itch, reposition a hand or a foot after a muscle spasm, assist with use of a telephone and computer or handle a DVD." The training takes between 18 and 24 months and costs $35,000 per monkey. Since its inception, the organization has trained and successfully placed more than 100 monkeys.
Monkeys can be very affectionate pets but can become extremely agressive upon reaching two years of age. Males particularly have been known to establish dominance, but females bite to defend themselves as well. Monkey bites can cause severe infections that can penetrate the bone and result in permanent deformity. Yet, to date, there is very little research that has been done on the subject.
According to the Center For Disease Control (CDC), many owners of pet macaques don't report attacks and bites for fear that the animal they love will be taken away and euthanized. Monkey bites are serious business such as ilustrated above.The woman pictured not only had the canine teeth removed on her pet capuchin, she had raised it from that time it was an infant abandoned by its mother. In her own words: "The nerves in my hand and wrist were so severely severed that i will likely never regain use despite all of the surgeries I have endured."
Harmless to monkeys but often fatal in humans is the Herpes B-Virus or Simian B. Between 80 to 90% of all macaque monkeys are infected with Herpes B, which generally occurs when the animal is ill, under stress or during breeding season. There has never been any kind of scientific study detailing the frequency of the Herpes B infection in humans, making it extremely difficult to calculate how mnay people have actually been affected. Those who live with or work with infected monkeys are at risk for contracting the virus.
About 3% of American households posses 7.3 million pet reptiles with snakes being the most common. According to the University of Florida, more than 7,000 venomous snakebites are reported annually in the United States, 15 of which result in death. (It is not known how many of these bites were made by snakes that were pets.) Most pet snakes lead very cramped lives in tanks so small they can't even stretch out. Pythons, boa constrictors, anaconda and other large snakes are capable of killing an adult human, and they should not be kept as pets. Corn snakes, like the one indicated below, on the other hand, make wonderful pets and are very popular.
It is estimated that 90% of all reptiles carry and shed salmonella in their feces. Iguanas, snakes, lizards and turtles are common carriers of the bacterium, which does not at all affect the animals's health. In humans, it can cause severe diarrhea, headache, malaise, nausea, fever, vomiting, abdominal cramps and even death in some instances. Individuals who eat after failing to wash their hands properly after handling a reptile or objects the reptile contaminated are subject to this malady.
Although the American Veterinary Medical Association, the United States Department of Agriculture (USDA) and the CDC have all expressed opposition to the possession of certain exotic animals, in some instances such adoptions do work, with best results occurring with wild birds such as parrots and macaws. Generally speaking, a genuine bond can occur when individuals possessing exotic animals do not attempt to alter their nature or their environment.
Animal Planet has featured several true life examples where wild animals have integrated positively with their human caretakers. Wilfd Thing, featured above, is a buffalo that roams free on aTexas farm with house privleges. He is destructive, but his owners accept his nature for what it is; that of a wild beast. In a similar story, a grizzly bear named Brutus served as best man at his caretaker's wedding.
Although there are laws governing the ownership and sale of exotic animals, regulations are not standardized: they vary from community to community. For this reason, no one can say how many exotic animals are owned privately in the United States. Estimates are staggering; 6,000 to 7,000 tigers alone.
There are three federal laws that pertain to the possession of exotic animals. They are: the Endangered Species Act, the Public Heath Service Act and the Lacey Act. Their limitations are that they only regulate the importation of exotic animals into the United States and not private possession. State laws have the auhtority to regulate ownership, although there are variations between them.
The private possession of large cats, wolves, bears, monkeys and dangerous reptiles is banned in thirteen states. They are: Vermont, Utah, Arkansas, New Hamshire, California, Colorado, Georgia, Maine, New Jersey, New Mexico, Tennessee, Wtyoming and Hawaii. Other states have partial bans and still others neither prohibit nor require a license or proof that an exotic animal was legally acquired. Local laws are stricter. It is the City or County Council that determines the threat to the health, safety and welfare of its residents, but some skirt the law by getting licensed as breeders, rezoning their property or moving elsewhere.
Few things are easier than buying an exotic animal over the more than 1,000 sites on the Internet. Of course, there is never any mention of laws regulating possession or the importance of understanding the needs of the animal they are selling. Unfortunately, the exotic pet trade is big bussiness and one of the largest sources of ill-gotten gains, behind only arms smuggling and drug trafficking. It is the animals, however, that pay the price with their suffering and maltreatment.
Don't buy an exotic animal for a pet. Wild animals need to thrive with members of their own species as they have complex behavioral, social, nutritional and psychological needs.They have lived for thousands of years without human interaction and they do not belong in your home with your children and other pets.
While 'fatal attraction" might best be remebered as the title of a suspense thriller from a few decades ago, it also is an apt disclaimer that applies to the perilous possession of exotic animals.