text:
\section{Introduction}
Chemical processes are traditionally designed for and operated at a certain steady-state operating condition, where the process economy is optimised. Nowadays, supply chains are increasingly dynamic and the process industry needs to shift from the traditional mass production to more agile, cost-effective and flexible process operations, in response to the fluctuations in market demand for products with different specifications, and the costs and supply of raw materials and energy.  As such, the control systems need to be able to drive the process safely and efficiently to any feasible time-varying operational targets, e.g., setpoints determined by the Real-time Optimisation (RTO) layer.

While most chemical processes are nonlinear, linear controllers have been commonly designed and implemented based on linearised models. This approach is defensible for regulatory control around a predetermined steady state. However, flexible process operation warrants nonlinear control as the target operating conditions can vary significantly to optimise economic costs (see, e.g., \cite{heidarinjad2021EMPC,kumar2002recycle}). Existing nonlinear control methods, e.g., Lyapunov-based approaches typically require redesigning the control Lyapunov function and controller when the target  equilibrium changes, not suitable for flexible process operation with time varying targets. To achieve time-varying setpoint tracking requires \emph{incremental stability} \cite{angeli2002lyapunov,santoso2012}.  This has motivated increased interest for alternative approaches based on the contraction theory framework \citep{WangBao17,mccloy2021differential, McCloyBao2021}. Introduced by \cite{lohmiller1998contraction}, contraction theory facilitates stability analysis and control of nonlinear systems with respect to arbitrary, time-varying (feasible) references without redesigning the control algorithm \citep{manchester2017control,lopez2019contraction}. Instead of using the state space process model alone, contraction theory also exploits the differential dynamics, a concept borrowed from fluid dynamics, to analyse the local stability of systems. Thus, one useful feature of contraction theory is that it can be used to analyse the incremental stability/contraction of nonlinear systems, and simultaneously synthesise a controller that ensures offset free tracking of feasible target trajectories using control contraction metrics (or CCMs, see, e.g., \cite{manchester2017control}). Furthermore, the dissipativity analysis  (\cite{willems2007dissipativity,ydstie2002passivity,garcia2016passivity}) was extended to differential system dynamics.  The resulting differential dissipativity \citep{forni2013diffdiss, WangBao17,mccloy2021differential} provides a framework to study the incremental input-output stability condition through consideration of differential storage functions and supply rates. Moreover, as virtually all modern process control systems are developed and implemented in a discrete time setting \citep{goodwin2001control}, there is a natural motivation for the development of tools for analysing, designing and implementing contraction-based control for discrete-time nonlinear systems. However, the current contraction-based (and more generally, differential system) analysis and control synthesis (e.g., \cite{manchester2014control,manchester2017control,manchester2018robust}), is limited to continuous-time control-affine nonlinear systems.

In the author's preliminary work \cite{wei2021control}, an SOS programming approach for the synthesis of contraction-based feedback control of discrete-time nonlinear systems was developed, utilising definitions for contracting systems in \cite{lohmiller1998contraction}. Leveraging the above results, we develop herein, the rigorous systematic analysis, control synthesis and implementation of contraction-based control through the use of SOS programming. Specifically, this article extends our previous work in \cite{wei2021control}, and provides a complete development and proofs for the exponential incremental stability (and convergence rates) of contracting discrete-time nonlinear processes, via discrete-time control contraction metrics (DCCMs). The above result has been further extended to  differential dissipativity conditions that ensure closed-loop differential $\mathcal{L}_2$ gain, which are useful in control design for disturbance attenuation. Tractable control synthesis approaches have been developed to achieve the desired contraction (for incremental stability) and dissipativity (for disturbance rejection) conditions.

The structure of this article is as follows. In Section~\ref{sec:DCCM diss}, the contraction and differential dissipativity analysis of control affine discrete-time nonlinear processes are presented, including the contraction and exponential differential dissipativity conditions. Section~\ref{sec:tractable} transforms the contraction and differential dissipativity conditions into computationally tractable synthesis problems. Based on these conditions,  Section~\ref{sec:synthesis} utilises the SOS programming to synthesise controllers, including details for controller implementation. Section~\ref{sec:example} illustrates the proposed contraction-based approach via a CSTR simulation study. Conclusions are drawn in Section~\ref{sec:conclusion}.

\textbf{Notation.} Denote by $f_k = f(x_k)$ for any function $f$, $\mathbb{Z}$ represents the set of all integers, $\mathbb{Z}^+$ represents set of positive integers, $\mathbb{R}$ represents set of real numbers. $\Sigma$ represents a polynomial sum of squares function that is always non-negative, e.g. $\Sigma(x_k,u_k)$ is a polynomial sum of squares function of $x_k$ and $u_k$.

\section{Discrete-time Differential Dynamics based Convergence Analysis}\label{sec:DCCM diss}
In this section, we provide a complete treatment for the analysis and control of discrete-time control affine nonlinear systems using contraction theory. By extending to the controlled system setting through the use of discrete-time control contraction metrics (DCCMs), closed-loop stability of a contracting system is shown. Leveraging the differential framework, dissipativity properties are also assessed and translated into controller synthesis conditions.

\subsection{Discrete-time Contraction Analysis}\label{sec:discrete-time contraction analysis}
\begin{figure*}[tb]
    \begin{center}
        \includegraphics[width=0.55\linewidth]{./pics/TIKZriemanniangeo.pdf}
        \caption{Illustration of the differential dynamics of a discrete-time system along the state manifold $\mathcal{X}$.}
        \label{fig:geodesic representation}
    \end{center}
\end{figure*}
In the following section, we introduce the necessary differential system framework and Riemannian tools underpinning contraction-based analysis and control approaches.

To develop control approaches which are capable of delivering time-varying setpoints or reference profiles, we require reference-independent stability conditions, i.e., incremental stability (e.g., \cite{angeli2002lyapunov}).
Contraction theory~\cite{lohmiller1998contraction,manchester2017control} uses the concept of displacement dynamics or \textit{differential dynamics} to assess incremental stability. Consider a discrete-time nonlinear system of the form,
\begin{equation}\label{equ:dynamical system}
    x_{k+1} = f(x_k),
\end{equation}
where $x_k \in \mathcal{X} \subseteq \mathbb{R}^n$ is state vector at time instant $k$, $\mathcal{X}$ represents "restricted" state space of $x_k$ for all $k$ and $f$ is a smooth differentiable function.

Define the state trajectories of~\eqref{equ:dynamical system} as $\hat{x}=(x_0, \cdots, x_k, \cdots)$ for $k=0,\cdots,\infty$. The state trajectories can be parameterized by $s$ and thus described as $\hat{x}(s)$ $=$ $(x_1(s), x_2(s),$ $\cdots,$ $x_k(s),$ $\cdots)$, where $0 \leq s \leq 1$. For example, consider the trajectory $\hat{x}$, associated with $s=0$, and $\hat{x}^*$, associated with $s=1$, as shown in Figure~\ref{fig:geodesic representation}.
Then, consider two neighboring discrete-time trajectories separated by an infinitesimal displacement $\delta_{x_k}$. Formally, $\delta_{x_k}:=\frac{\partial x_k}{\partial s}$ is a vector in the tangent space $T_x\mathcal{X}$ at $x_k$.
In this paper, the state manifold $\mathcal{X}$ and the tangent space $T_x\mathcal{X}$  can both be identified with $\mathbb{R}^n$.

The differential dynamics (or displacement dynamics) for system (\ref{equ:dynamical system}) are  defined as
\begin{equation}\label{equ:differential dynamical system}
    \delta_{x_{k+1}}= A_k\delta_{x_k},
\end{equation}where $A_k : = \frac{\partial f(x_k)}{\partial x_k}$ represents the Jacobian matrix of function $f$.
In the context of Riemannian geometry \cite{do1992riemannian}, a generalized infinitesimal displacement, $\delta_{z_k}$,  can be defined using a coordinate transformation described by a mapping from $\delta_{x_k}$ to $\delta_{z_k}$, using a square state-dependent matrix function $\Theta$, i.e.,
\begin{equation}\label{equ:coordinate transformation}
    \delta_{z_k} = \Theta(x_k)\delta_{x_k}.
\end{equation}
If we consider the evolution of the infinitesimal squared distance for system (\ref{equ:differential dynamical system}) at time step $k$ as $\delta_{x_k}^\top\delta_{x_k}$; then, naturally, the generalized squared distance for (\ref{equ:differential dynamical system}) can be defined using (\ref{equ:coordinate transformation}) as
\begin{equation}\label{equ:generalised distance}
\begin{aligned}
    V(x_k,\delta_{x_k}) &= \delta_{z_k}^\top\delta_{z_k} = \delta_{x_k}^\top\Theta(x_k)^\top\Theta(x_k)\delta_{x_k} \\ &=: \delta_{x_k}^\top M(x_k)\delta_{x_k},
\end{aligned}
\end{equation}
where the \textit{metric} $M(x_k) :=\Theta(x_k)^\top\Theta(x_k)$ is a symmetric positive-definite matrix function and is uniformly bounded, i.e.,
\begin{equation}\label{inequ:metric bound}
    \alpha_1 I \leq M(x_k) \leq \alpha_2 I \ \ \forall x_k \in \mathcal{X},
\end{equation}
for some positive constants $\alpha_2 \geq \alpha_1 > 0$.
We then have the following definition for a contracting system:

\begin{definition}[\cite{lohmiller1998contraction}]
\label{def:contraction}
    System~\eqref{equ:dynamical system} is contracting, with respect to a uniformly bounded, positive definite metric, denoted by $M_k = M(x_k)$ and $M_{k+1} = M(x_{k+1})$, provided, $\forall x \in \mathcal{X}$ and $\forall \delta_x \in T_x\mathcal{X}$,
    \begin{equation}\label{inequ:contraction condition}
        \begin{aligned}
            \delta^\top_{x_{k}} (A_k^\top M_{k+1} A_k - M_{k})\delta_{x_{k}}&\leq -\beta \delta_{x_k}^\top M_{k}\delta_{x_k} < 0,
        \end{aligned}
    \end{equation}
     for some constant $0 < \beta \leq 1$, where~\eqref{inequ:contraction condition} describes the region of contraction. In addition, the function $V_k$ in \eqref{equ:generalised distance} can be understood as a discrete-time Lyapunov function for the differential dynamics~\eqref{equ:differential dynamical system}, i.e.,
    \begin{equation}
    \label{eqe:Vdotcondition}
    V_{k+1} - V_k \leq - \beta V_k < 0, \quad \forall x \in \mathcal{X}, \forall \delta_x \in T_x\mathcal{X}.
    \end{equation}
\end{definition}
The contraction region defined in~\eqref{eqe:Vdotcondition} provides a generalized convergence condition for the discrete-time nonlinear system~\eqref{equ:dynamical system}, with respect to the metric $M_k$. Then, following some additional definitions from Riemannian geometry, incremental stability can be established.

For a smooth curve function, $c(s): [0,1] \rightarrow \mathcal{X}$ (see, Figure~\ref{fig:geodesic representation}), connecting any pair of points, $x,x^* \in \mathcal{X}$ (such that $c(0) = x$ and $c(1) = x^*$), we define the Riemannian distance, $d(x,x^*)$, and energy, $E(x,x^*)$, as (see, e.g., \cite{do1992riemannian})
\begin{equation}\label{equ:Riemannian distance and energy}
    \begin{aligned}
    d(x,x^*) = d(c) :=\int_0^1 \sqrt{\delta^\top_{c(s)}M(c(s))\delta_{c(s)}}ds,\\
    E(x,x^*) = E(c) :=\int_0^1\delta^\top_{c(s)}M(c(s))\delta_{c(s)}ds,
    \end{aligned}
\end{equation}
where $\delta_{c(s)} := \frac{\partial c(s)}{\partial s}$.
The shortest path in Riemannian space, or \textit{geodesic}, between $x$ and $x^*$ is defined as
\begin{equation}\label{equ:geodesic}
    \gamma(s) :=\argmin_{c(s)} {d(x,x^*)}.
\end{equation}

\subsection{Discrete-time Control Contraction Metrics (DCCMs)}\label{sec:DCCM intro}
In this section, we complete an explicit proof for exponential incremental convergence analysis of contracting discrete-time nonlinear systems, and a controller structure for discrete-time contraction is developed (as an extension to existing \sloppy continuous-time results). We consider a discrete-time nonlinear system in control affine form as
\begin{equation}\label{equ:control affine}
    x_{k+1} = f(x_k) + g(x_k)u_k,
\end{equation}
where $x_k \in \mathcal{X} \subseteq \mathbb{R}^n$ and $u_k \in \mathcal{U} \subseteq \mathbb{R}^m$ are the state and control vectors, respectively, $\mathcal{X}$ and $\mathcal{U}$ represent the restricted state and control spaces, and, $f$ and $g$ are smooth functions.

We define a triplet solution sequence $(x_k^*,u_k^*,x_{k+1}^*)$ for system \eqref{equ:control affine} and consider the discrete-time control input, $u_k = u_k(x_k,x^*_k,u^*_k)$. The differential dynamics for system (\ref{equ:control affine}) satisfy \begin{equation}\label{equ:differential control affine}
    \delta_{x_{k+1}} = A(x_k)\delta_{x_k} + B(x_k)\delta_{u_k},
\end{equation} where, denoting $h(x_k,u_k) = f(x_k) + g(x_k)u_k$, the matrices $A:=\frac{\partial h(x_k,u_k)}{\partial x_k}$ and $B:=\frac{\partial h(x_k,u_k)}{\partial u_k}$ are the Jacobian matrices of functions $f$ and $g$ in \eqref{equ:control affine} respectively.   Consider a differential state-feedback control law \begin{equation}\label{equ:differential feedback}
    \delta_{u_k} = K(x_k) \delta_{x_k},
\end{equation} which leads to the following state-feedback law (by integrating \eqref{equ:differential feedback} over the geodesic $\gamma(x,x^*)$ \eqref{equ:geodesic}):
\begin{equation}\label{equ:control integral}
    u_k = u^*_k + \int_0^1K(\gamma(s))\frac{\partial \gamma(s)}{\partial s}\,ds.
\end{equation}
Note that this particular formulation is reference-independent, since the target trajectory variations do not require structural redesign of the feedback controller and is naturally befitting to the flexible manufacturing paradigm. Moreover, the discrete-time control input, $u_k$~\eqref{equ:control integral}, is a function with arguments ($x_k$, $x^*_k$, $u^*_k$) and hence the control action is computed from  the current state and target trajectory.
\begin{theorem}
\label{theorem:incremental stability}
Consider a discrete-time nonlinear system~\eqref{equ:control affine} with differential dynamics \eqref{equ:differential control affine} and a differential state-feedback controller \eqref{equ:differential feedback} (implemented as a state-feedback controller \eqref{equ:control integral}). The above closed-loop system is contracting, with respect to a uniformly bounded, positive definite DCCM, $M(x_k)$, if
\begin{equation}\label{inequ: control affine contraction condition}
    (A_k+B_kK_k)^\top M_{k+1}(A_k+B_kK_k) - (1-\beta)M_{k} < 0,
\end{equation}
for some constant $0 < \beta \leq 1$.

Furthermore, for any feasible reference trajectory  $(\hat{x}^*, \hat{u}^*)$ (where $\hat{x}^*=(x^*_0, \cdots, x^*_k, \cdots)$ and $\hat{u}^*=(u^*_0, \cdots, u^*_k, \cdots)$) and an initial value  $x_0$ in the contraction region given by Definition  \ref{def:contraction}, the above closed-loop system is

    exponentially incrementally  stable, i.e.,
    \begin{equation}
    \label{equ:exp_stability}
        |x_k-x^*_k| \leq  R e^{-\lambda k \Delta_t} |x_0 - x^*_0|,
\end{equation}
for some constants $R$ and $\lambda$, where $x_k$ is the $k$-th step of the closed-loop state.
\end{theorem}
\begin{proof}
    Firstly, substituting \eqref{equ:differential feedback} into \eqref{equ:differential control affine} yields, the closed-loop differential system
    \begin{equation}\label{eq:diffcl}
        \delta_{x_{k+1}} = \left(A(x_k) + B(x_k)K(x_k)\right)\delta_{x_k}.
    \end{equation}
    Substituting the closed-loop differential dynamics \eqref{eq:diffcl} and generalised squared distance \eqref{equ:generalised distance} (as a differential Lyapunov candidate) into the differential Lyapunov condition of \eqref{eqe:Vdotcondition} provides the contraction condition in \eqref{inequ: control affine contraction condition} $\forall x ,\delta_x \in \mathcal{X}$.
    Then, consider $(x_k,x_{k+1})$ and $(x^*_{k},x^*_{k+1})$ as two solution pairs of system~\eqref{equ:dynamical system}. The shortest path connecting each of these solutions at time step $k$, denoted by the geodesic $\gamma_k(s)$ in~\eqref{equ:geodesic}, is defined with $\gamma_k(0) = x_k$ and $\gamma_k(1) = x_k^*$, and $c_{k+1}(s)$ denotes a path (not necessarily a geodesic) at the next time step, connecting $c_{k+1}(0) = x_{k+1}$ and $c_{k+1}(1)=x_{k+1}^*$. Hence, from \eqref{equ:dynamical system}, we have $c_{k+1}(0) = x_{k+1} = f(x_k) =f(\gamma_k(0))$ and $c_{k+1}(1) = x_{k+1}^* = f(x_k^*) = f(\gamma_k(1))$ as shown in Figure~\ref{fig:geodesic representation},
    or, without loss of generality,
    \begin{equation}\label{equ:geodesic k to path k+1}
        \begin{aligned}
            c_{k+1}(s) = f(\gamma_k(s)).
        \end{aligned}
    \end{equation}
    From \eqref{equ:Riemannian distance and energy}, \eqref{equ:geodesic} and the Hopf-Rinow Theorem, (e.g., \cite{manchester2017control}), we have
    \begin{equation}\label{equ:Hopf-Rinow}
        E(\gamma) = d(\gamma)^2 \leq d(c)^2 \leq E(c),
    \end{equation}
    which, given \eqref{inequ:contraction condition} and \eqref{equ:geodesic k to path k+1}, yields decreasing Riemannian energy, i.e.,
    \begin{equation}{\label{equ:energy bound}}
        \begin{aligned}
        E(\gamma_{k+1})
        & \leq \int_0^1 {\frac{\partial c_{k+1}(s)}{\partial s}}^\top M(c_{k+1}(s))\frac{\partial c_{k+1}(s)}{\partial s} ds \\
        & \leq \int_0^1 (1-\beta) {\frac{\partial \gamma_k(s)}{\partial s}}^\top M(\gamma_k(s))\frac{\partial \gamma_k(s)}{\partial s} \\
        &= (1-\beta)E(\gamma_k).
        \end{aligned}
    \end{equation}
    Taking the square root of \eqref{equ:energy bound} gives
    \begin{equation}\label{inequ:distance bound}
        d(\gamma_{k+1}) \leq (1-\beta)^{\frac{1}{2}} d(\gamma_k).
    \end{equation}
    Since $M_k$ is by definition uniformly bounded (by two positive constants $\alpha_1$ and $\alpha_2$), then from \eqref{equ:Riemannian distance and energy} and \eqref{equ:geodesic}, we have
    \begin{equation}\label{inequ:a1 a2 bound}
        \sqrt{\alpha_1}|x(k)-x^*(k)| \leq d(\gamma(k)),\ \
        d(\gamma(0)) \leq \sqrt{\alpha_2}|x(0)-x^*(0)|.
    \end{equation}
    Combining \eqref{inequ:distance bound} and \eqref{inequ:a1 a2 bound} we have
    \begin{equation}
        |x_k-x^*_k| \leq (1-\beta)^{\frac{k}{2}} \sqrt{\frac{\alpha_2}{\alpha_1}} |x_0 - x^*_0|,
    \end{equation}
    which for $R = \sqrt{\frac{\alpha_2}{\alpha_1}} $ yields \begin{equation}
    \label{equ:asym_stability}
        |x_k-x^*_k| \leq (1-\beta)^{\frac{k}{2}} R|x_0 - x^*_0|.
    \end{equation}
    Since the contraction rate, $\beta$, is by definition bounded as $ 0 < \beta \leq 1$, let $e^{-\lambda k \Delta_t} = (1-\beta)^{\frac{k}{2}}$ for some positive constant $\lambda$. Furthermore, we can express \eqref{inequ:distance bound} with respect to a discrete-time interval, $\Delta_t$, i.e.,
    \begin{equation}
    \label{equ:asy2exp}
        d(\gamma_{k\Delta_t}) \leq (1-\beta)^{\frac{k}{2}} d(\gamma_0) = e^{-\lambda k \Delta_t} d(\gamma_0).
    \end{equation}
    Then, provided the $\lambda$ satisfying \eqref{equ:asy2exp}, e.g., for $\lambda =-\ln(1-\beta)/(2\Delta_t)$, we have the result in \eqref{equ:exp_stability} by substituting \eqref{equ:asy2exp} into \eqref{inequ:a1 a2 bound}.

Now, we consider the controlled system. The contraction condition in \eqref{inequ: control affine contraction condition} is obtained by substituting the corresponding differential dynamic equation \eqref{equ:differential control affine} and differential feedback control law \eqref{equ:differential feedback} into the autonomous contraction condition \eqref{inequ:contraction condition}. The control law in \eqref{equ:control integral} can be obtained by integrating the differential controller (\ref{equ:differential feedback}) along the geodesic, with respect to metric, $M_k$, in \eqref{equ:generalised distance} and the stability result is straightforward from the above proof by replacing the contraction condition \eqref{inequ:contraction condition} by \eqref{inequ: control affine contraction condition}.
\end{proof}

To summarize, a suitably designed contraction-based controller ensures that the length of the minimum path (i.e., geodesic) between any two trajectories (e.g., the plant state, $x$, and desired state, $x^*$, trajectories), with respect to the metric $M$, shrinks with time, i.e., provided that the contraction condition \eqref{inequ: control affine contraction condition} holds for the discrete-time nonlinear system \eqref{equ:control affine},
we can employ a stabilizing feedback controller \eqref{equ:control integral} to ensure convergence to feasible operating targets.

\subsection{Exponential Differential Dissipativity}
In the following, we extend differential dissipativity results \cite{WangBao17} to the discrete-time setting and use this property to determine bounded disturbance characteristics of the corresponding nonlinear system. Consider the discrete-time nonlinear system with disturbance, $\nu$, in the form of
\begin{equation}\label{eq:sys_d}
x_{k+1} = f(x_k)x_k + g_u(x_k)u_k + g_{\nu}(x_k)\nu_k,
\end{equation}
with corresponding differential dynamics described by \begin{equation}\label{eq:sys_d diff}
    \delta_{x_{k+1}} = A\delta_{x_k} +  B_{u,k}\delta_{u_k} +  B_{\nu,k}\delta_{\nu_k},
\end{equation}
where $A = \frac{\partial h(x_k,u_k,\nu_k)}{\partial x_k}$, $B_u = \frac{\partial h(x_k,u_k,\nu_k)}{\partial u_k}$ and $B_\nu = \frac{\partial h(x_k,u_k,\nu_k)}{\partial \nu_k}$ and $h(x_k,u_k,\nu_k) = f(x_k)x_k + g_u(x_k)u_k + g_{\nu}(x_k)\nu_k$.
\begin{definition}\label{def:diff diss}
System \eqref{eq:sys_d}--\eqref{eq:sys_d diff} is exponentially differentially dissipative with respect to the differential supply rate $\sigma(\delta_x,\delta_d)$, if there exists a differential storage function $V(x,\delta_x)$ such that
\begin{equation}\label{eq:diff_diss}
    V_{k+1} - (1 - \beta) V_k\leq \sigma_k.
\end{equation}
Moreover, the system is called exponentially differentially \sloppy (Q,S,R)-dissipative if \eqref{eq:diff_diss} is satisfied with respect to the partitioned supply rate
\begin{equation}\label{eq:qsr def}
    \sigma =
\begin{bmatrix}
\delta_x \\ \delta_\nu
\end{bmatrix}^\top
\begin{bmatrix} Q & S\\S^\top & R \end{bmatrix}
\begin{bmatrix}
\delta_x \\ \delta_\nu
\end{bmatrix}.
\end{equation}
\end{definition}

One particular application for the differential dissipativity framework lies in systematically determining physical properties of the closed-loop system. For example, satisfying \eqref{eq:diff_diss} with $Q=0$, $S=0.5I$, $R=0$, indicates the system is differentially passive. Furthermore, differential dissipativity with respect to a supply rate $Q<0$, $S=0$, $R>0$ implies a differential $\mathcal{L}_2$ gain bound. This differential dissipativity condition can  be generalized to determine the incremental stability of the corresponding nonlinear closed-loop system (e.g., the $\mathcal{L}_2$ gain from the disturbances to states), which is detailed below.
\begin{theorem}\label{thm:dif dis}
System \eqref{eq:sys_d},\eqref{eq:sys_d diff},\eqref{equ:control integral} is exponentially differentially (Q,S,R)-dissipative, provided a DCCM and differential feedback gain pair, $(M,K)$, are found satisfying
\begin{equation}\label{equ:diff diss sub}
\begin{split}
    &\delta_{x_{k}}^\top(A_k+B_{u,k} K_k)^\top M_{k+1}(A_k+B_{u,k} K_k)\delta_{x_{k}} - (1-\beta)\delta_{x_{k}}^\top M_{k} \delta_{x_{k}} \\
    &\quad + \delta^\top_{\nu_k} B^\top_{\nu_k}M_{k+1}B_{\nu_k}\delta_{\nu_k}  \leq \delta_{x_{k}}^\top Q \delta_{x_{k}} + 2 \delta_{\nu_{k}}^\top S \delta_{x_{k}} + \delta_{\nu_{k}}^\top R \delta_{\nu_{k}}.
\end{split}
\end{equation}
Furthermore, the closed-loop nonlinear system, \eqref{eq:sys_d} driven by \eqref{equ:control integral} with $Q<0$ and $R>0$ satisfying \eqref{equ:diff diss sub}, is incrementally exponentially stable, and exhibits a bounded disturbance response, for any feasible  reference $x_k^*$ and expected disturbance $\nu_k^*$ (typically zero for unknown disturbances). Over the finite interval $k \in [0,N]$,
\begin{equation}\label{eq:bounded dist response}
    \sum_{k=0}^N ||x_k - x_k^* ||_2 \leq \rho \sum_{k=0}^N||\nu_k - \nu_k^* ||_2,
\end{equation}
where $\rho$ is the incremental truncated $\mathcal{L}_2$ gain from $\nu$ to $x$.
\end{theorem}
\begin{proof}
Substituting \eqref{equ:differential feedback} into \eqref{eq:sys_d diff} yields, the closed-loop differential system
    \begin{equation}\label{eq:diffcl_d}
        \delta_{x_{k+1}} = \left(A(x_k) + B_u(x_k)K(x_k)\right)\delta_{x_k} + B_\nu(x_k)\delta_{\nu_k}.
    \end{equation}
    Given a differential $(Q,S,R)$-supply rate as in \eqref{eq:qsr def} and substituting the closed-loop differential dynamics \eqref{eq:diffcl_d} and generalised squared distance \eqref{equ:generalised distance} (as a differential storage function) into the exponential differential dissipativity condition of \eqref{eqe:Vdotcondition} provides the  exponential differential $(Q,S,R)$-dissipativity condition in \eqref{equ:diff diss sub}.
Exponential incremental stability for \eqref{eq:sys_d} when $Q<0$, is then straightforward from the proof of Theorem \ref{theorem:incremental stability} by integrating \eqref{equ:diff diss sub} along the geodesic $\gamma(x,x^*)$ \eqref{equ:geodesic}.

Moreover, to see the that the disturbance response of \eqref{eq:sys_d} is bounded on the finite interval $k\in[0,N]$, integrate the differential dissipativity condition \eqref{eq:diff_diss} (omitting the $\beta V_k$ term, since the rate of change in the differential storage function is not relevant to the incremental $\mathcal{L}_2$ gain) along the geodesic $\gamma(x,x^*)$ and sum over all $k \in [0,N]$ to obtain
\begin{equation}\label{eq:sum_int_bounded}
    \sum_{k=0}^N \int_0^1 (V_{k+1} - V_k )ds \leq \sum_{k=0}^N \int_0^1 \left( \delta_x^\top Q \delta_x + 2\delta_x^\top S \delta_\nu + \delta_\nu^\top R \delta_\nu \right)ds.
\end{equation}
By induction and definition for Riemannian energy, \eqref{eq:sum_int_bounded} yields
\begin{equation}
    E(\gamma_{N+1}) - E(\gamma_0) \leq \sum_{k=0}^N \int_0^1 \delta_x^\top Q \delta_x + 2\delta_x^\top S \delta_\nu + \delta_\nu^\top R \delta_\nu.
\end{equation}
Given an equivalent initial condition $E(\gamma_0) = 0$ (i.e., $x(0)=x^*(0)$ -- reasonable via definition of the initial time step $k=0$), and noting that by definition $E(\gamma) \geq 0$, gives
\begin{equation}
   \sum_{k=0}^N \int_0^1  \delta_x^\top Q \delta_x + 2\delta_x^\top S \delta_\nu + \delta_\nu^\top R \delta_\nu \, \,  ds \geq E(\gamma_{N+1}) \geq 0.
\end{equation}
Denoting $\hat{Q}  = -Q$ and completing the square gives
\begin{equation}
    \sum_{k=0}^N \int_0^1 \delta_\nu^\top (R+S^\top \hat{Q}^{-1}S)\delta_\nu - ||\hat{Q}^{\frac{1}{2}} \delta_x - \hat{Q}^{-\frac{1}{2}} S \delta_\nu ||^2_2 \,\,ds \geq 0.
\end{equation}
Evaluating the integral by the Cauchy-Schwarz inequality, parameterising the disturbance input as $\nu_k(s) = (1-s)\nu^*_k +s\nu_k$ and noting that $d_\gamma(x,x^*) \leq ||x-x^*||$
gives
\begin{equation}
  \sum_{k=0}^N (R+S^\top \hat{Q}^{-1}S) ||\nu_k - \nu_k^* ||^2_2 -  ||\hat{Q}^{\frac{1}{2}}(x_k - x_k^*) - \hat{Q}^{-\frac{1}{2}} S (\nu_k - \nu_k^*)||^2_2 \geq 0,
\end{equation}
 and hence \eqref{eq:bounded dist response},
where $\rho = ||\hat{Q}^{-\frac{1}{2}}||_2(||\hat{Q}^{-\frac{1}{2}} S||_2 + (R+S^\top \hat{Q}^{-1}S)^{\frac{1}{2}})$.
\end{proof}

Note that these dissipativity results can be further extended to output feedback systems, or tailored for performance measures that are governed by particular states or outputs, by considering a measured output, e.g., $y_k = C(x_k)x_k$, and suitably modifying the dissipativity condition in \eqref{eq:diff_diss} (with arguments $(y,\delta_y)$). The disturbance response with respect to output $y$, could then be shaped by a weighted $\mathcal{L}_2$ gain from $d$ to $y$.

Note that the $\beta V_k$ term is not relevant to the incremental $\mathcal{L}_2$ gain (see the proof of Theorem \ref{thm:dif dis}). However, this term ensures the exponential incremental stability when the disturbance is zero (hence the results of Theorem \ref{thm:dif dis} are permitted to collapse to those of Theorem \ref{theorem:incremental stability}).

It is also interesting to note that the dissipativity condition in \eqref{equ:diff diss sub} links the input to the contraction condition in \eqref{inequ: control affine contraction condition}, whereby satisfaction of \eqref{equ:diff diss sub} ensures additional closed-loop properties, by imposing additional conditions on the permitted controller. As an example, by choosing $Q=-I$, $S=0$ and $R =  \alpha^2 I$, when solving for the pair $(M,K)$ in \eqref{equ:diff diss sub}, yields an incremental truncated $\mathcal{L}_2$ gain equal to $\alpha$, providing an additional design variable (implicit to the contraction-based controller \eqref{equ:control integral}) with respect to disturbance rejection.

\section{Tractable Contraction Metric and Controller Conditions for Discrete-time Nonlinear Systems}\label{sec:tractable}

This section presents the transformation of contraction and differential dissipativity conditions (developed in Section \ref{sec:DCCM diss}) for discrete-time control-affine nonlinear systems into a tractable synthesis problem.
\subsection{Obtaining a Tractable Contraction Condition}
As characterised by Theorem \ref{theorem:incremental stability}, two conditions are needed to ensure contraction of discrete-time nonlinear systems -- the first is the discrete-time contraction condition \eqref{inequ: control affine contraction condition}, and the second is the positive definite property of the metric $M$. Inspired by \cite{manchester2017control}, an equivalent condition to \eqref{inequ: control affine contraction condition} is developed in the following Proposition as a computationally tractable means for handling the coupled terms in \eqref{inequ: control affine contraction condition}.

\begin{proposition}\label{thm:condition}
    Consider a differential feedback controller \eqref{equ:differential feedback} for the differential dynamics \eqref{equ:differential control affine} of a discrete-time control-affine nonlinear system \eqref{equ:control affine}. The discrete-time nonlinear system \eqref{equ:control affine} is contracting with respect to a DCCM, $M$, if a pair of matrix functions $(W,L)$ satisfies
    \begin{equation}\label{inequ:LMI}
        \begin{bmatrix}
            W_{k+1}              & A_kW_k+B_kL_k \\
            (A_kW_k+B_kL_k)^\top    & (1-\beta)W_k
        \end{bmatrix} > 0,
    \end{equation}
    where $A_k,B_k$ and $K_k$ are functions in \eqref{equ:control affine} and \eqref{equ:differential feedback} respectively, $W_{k}:=M_k^{-1}$, $W_{k+1}:= M_{k+1}^{-1}= M^{-1}(f(x_k)+B(x_k)u_k)$, $L_k := K_kW_k$ and $\beta \in (0,1]$.
\end{proposition}
\begin{proof}
    Condition \eqref{inequ: control affine contraction condition} is equivalent to
    \begin{equation}\label{Metric_Condi}
        (1-\beta)M_{k} - (A_k+B_kK_k)^\top M_{k+1}(A_k+B_kK_k) > 0.
    \end{equation}
    Applying Schur's complement \cite{boyd1994linear} to (\ref{Metric_Condi}) yields
    \begin{equation}
    \label{equ:schursCC}
        \begin{bmatrix}
            M_{k+1}^{-1} & (A_k+B_kK_k) \\
            (A_k+B_kK_k)^\top     & (1-\beta)M_k
        \end{bmatrix} > 0.
    \end{equation}
    Defining $W_k := M^{-1}(x_k)$ and $W_{k+1} := M^{-1}(x_{k+1}) = M^{-1}(f(x_k)+B(x_k)u_k)$, we then have
    \begin{equation}
    \label{equ:inequality_step}
        \begin{bmatrix}
            W_{k+1}     & (A_k+B_kK_k) \\
            (A_k+B_kK_k)^\top    & (1-\beta)W_k^{-1}
        \end{bmatrix} > 0.
    \end{equation}
    Left/right multiplying \eqref{equ:inequality_step} by an invertible positive definite matrix, $\diag\{I,W_k\}$ (and its transpose), yields

    \begin{equation}
        \begin{bmatrix}
            W_{k+1}          & (A_k+B_kK_k)W_k \\
            W_k^\top (A_k+B_kK_k)^\top     & (1-\beta)W_k
        \end{bmatrix} > 0.
    \end{equation}
    Finally, defining $L_k:=K_kW_k$, we have the condition \eqref{inequ:LMI}.
\end{proof}

\subsection{Exponential Differential Dissipativity Conditions}\label{sec:dissipativity}
As described in Theorem \ref{thm:dif dis}, dissipativity properties of the closed-loop can be formed by considering a differential storage function and supply rate, yet the resulting condition \eqref{equ:diff diss sub} is computationally complex. Even with the supply rate  provided, finding a DCCM and feedback gain satisfying \eqref{equ:diff diss sub} is at least as difficult (e.g., considering coupled terms) as the ``reduced'' inequality \eqref{inequ: control affine contraction condition} (forming the left-hand side of \eqref{equ:diff diss sub}).
Inspired by \cite{wang2017distributed}, an equivalent computationally tractable condition to \eqref{equ:diff diss sub} is developed in the following Proposition.
\begin{proposition}\label{prop:dif dis}
 Consider a differential feedback controller \eqref{equ:differential feedback} for the differential dynamics \eqref{eq:sys_d diff} of a discrete-time control-affine nonlinear system \eqref{eq:sys_d}. The system is exponentially incrementally differentially (Q,S,R)-dissipative (in the sense of Definition \ref{def:diff diss}), with respect to a DCCM, $M$, and differential (Q,S,R,)-supply rate \eqref{eq:qsr def}, if a pair of matrix functions $(W,L)$ satisfies
\begin{equation}\label{equ:LMI_dd}
\begin{bmatrix}
W_{k+1} & A_k W_k + B_{u,k} L_k & B_{\nu,k} & 0\\
(A_k W_k + B_{u,k} L_k)^\top & (1 - \beta)W_k & W_k S & W_k \\
B_{\nu,k}^\top & S^\top W_k & R & 0\\
0 & W_k & 0 & -Q^{-1}
\end{bmatrix}  \geq 0,
\end{equation}
where $A_k$, $B_{u,k}$, $B_{\nu,k}$ and $K_k$ are defined in \eqref{eq:sys_d diff} and \eqref{equ:differential feedback} respectively, $W_{k}:=M_k^{-1}$, $W_{k+1}:= M_{k+1}^{-1}= M^{-1}(f(x_k)+B(x_k)u_k)$, $L_k := K_kW_k$ and $\beta \in (0,1]$.
\end{proposition}
\begin{proof}
From \eqref{equ:diff diss sub}, we have
\begin{equation}
\begin{split}
\begin{bmatrix}\delta_{x}\\ \delta_\nu\end{bmatrix}^\top
\begin{bmatrix}A_{c,k} & B_{\nu,k} \end{bmatrix}^\top M_{k+1} \begin{bmatrix}A_{c,k} & B_{\nu,k} \end{bmatrix}
\begin{bmatrix}\delta_{x}\\ \delta_\nu\end{bmatrix}
-  (1 - \beta) \delta_x^\top M_k \delta_x
 \\ \leq
\begin{bmatrix}\delta_{x}\\ \delta_\nu\end{bmatrix}^\top \begin{bmatrix} Q & S\\S^\top & R \end{bmatrix}
\begin{bmatrix}\delta_{x}\\ \delta_\nu\end{bmatrix},
\end{split}
\end{equation}
where $B_{\nu,k} = A_k + B_{u,k} K_k$. This inequality can be equivalently written
\begin{equation}
\begin{bmatrix} Q + (1 - \beta)M_k & S\\S^\top & R \end{bmatrix} -
\begin{bmatrix}A_{c,k} & B_{\nu,k} \end{bmatrix}^\top M_{k+1} \begin{bmatrix}A_{c,k} & B_{\nu,k} \end{bmatrix}  \geq 0.
\end{equation}
By Schur's complement, this is equivalent to
\begin{equation}
\begin{bmatrix}
M_{k+1}^{-1} & A_{c,k} & B_{\nu,k} \\
A_{c,k}^\top & Q + (1 - \beta)M_k & S\\
B_{\nu,k}^\top & S^\top & R
\end{bmatrix}  \geq 0.
\end{equation}
Defining $W_k := M^{-1}_k$
and appropriately (left/right) multiplying by $\diag\{ I , W_k, I\}$ (and its transpose) gives
\begin{equation}
\begin{bmatrix}
W_{k+1}  & A_{c,k} W_k & B_{\nu,k} \\
W_k^\top A_{c,k}^\top & W_k^\top Q W_k + (1 - \beta)W_k^\top & W_k^\top S\\
B_{\nu,k}^\top & S^\top W_k & R
\end{bmatrix}  \geq 0.
\end{equation}
Applying Schur's complement, expanding $A_{c,k}$,
defining $L_k := K_k W_k$ and noting by symmetry $W_k^\top = W_k$ gives \eqref{equ:LMI_dd}.
\end{proof}

\section{DCCM Synthesis and Controller Implementation}\label{sec:synthesis}
Based on the contraction \eqref{inequ:LMI} and differential dissipativity \eqref{equ:LMI_dd} conditions presented in Section \ref{sec:tractable},  here we develop control synthesis approaches from these conditions using Sum of squares (SOS) programming  (see \cite{wei2021discrete} for an alternative neural network-based approach). Details for numerical implementation of a contraction-based controller are also provided.

\subsection{Overview of SOS Programming as a Synthesis Method}
SOS programming (see, e.g. \cite{boyd2004convex}) was proposed as a tractable method in \cite{manchester2017control} for computing CCMs for continuous-time control-affine nonlinear systems and is demonstrated in this work as an additionally tractable approach in the discrete-time setting. A polynomial $p(x)$, is an SOS polynomial, provided it satisfies
\begin{equation}\label{equ:SOS definition}
    p(x) = \sum_{i=1}^n{q_i(x)^2},
\end{equation} where $q_i(x)$ is a polynomial of $x$. Thus, it is easy to see that any SOS polynomial, $p$, is positive provided it can be expressed as in \eqref{equ:SOS definition}. Furthermore, in \cite{aylward2006algorithmic}, determining the SOS property expressed in \eqref{equ:SOS definition} is equivalent to finding a positive semi-definite $Q$ such that
\begin{equation}
    p(x)=\phi(x)^\top Q\phi(x) \in \Sigma (x),
\end{equation}where $\phi(x)$ is a vector of monomials that is less or equal to half of the degree of polynomial $p(x)$, $\Sigma$ is stated in Notation. An SOS programming problem is an optimisation problem to find the decision variable $p(x)$ such that $p(x)\in\Sigma (x)$, which can be relaxed (in terms of computational complexity) for a particular range (e.g., a region of interest). Suppose we want $p(x)\in\Sigma (x)$ for $x \in [x_{i_{min}},x_{i_{max}}]$, with  $i =1,\cdots,n$. The relaxed condition can be constructed as follows,
\begin{equation}\label{equ:sos constraint}
\begin{aligned}
    &q(x) = p(x) - \sum_i^n{-(x_i-x_{i_{min}})(x_i-x_{i_{max}})c_i(x)} \in \Sigma (x)\\
    &\forall i, c_i(x) \in \Sigma (x),
\end{aligned}
\end{equation}
where the term $-(x_i-x_{i_{min}})(x_i-x_{i_{max}})c_i(x)$ represents a function that is only positive within the range $[x_{i_{min}},x_{i_{max}}]$. This means $p(x)$ only needs to be positive inside the constraints and $c_i(x)$ can be selected to relax the problem. In~\cite{Parrilo00}, the non-negativity of a polynomial is determined by solving  a semi-definite programming (SDP) problem (e.g. SeDuMi~\cite{Sturm1999}).
\subsection{DCCM Synthesis via SOS Programming}
Here we introduce SOS programming with relaxations for Theorem \ref{theorem:incremental stability}, followed by some discussion and natural progression to SOS programming for Proposition \ref{thm:condition}, and hence extensions for Proposition \ref{prop:dif dis}. From Theorem \ref{theorem:incremental stability}, two conditions need to be satisfied: the contraction condition \eqref{inequ: control affine contraction condition} and positive definite property of the matrix function $M$. These conditions can be transformed into an SOS programming problem (see, e.g.,~\cite{Parrilo00,aylward2006algorithmic}) if we assume the functions are all polynomial functions or polynomial approximations (see, e.g.,~\cite{EbeAll06}), i.e.
\begin{equation}
    \begin{aligned}
                & \min_{k_c,m_c} \trace(M)\\
        s.t. ~ &\phi^\top  \Omega_1 \phi  \in \Sigma (x_k,u_k,\phi),\\
                &\phi^\top  M_k \phi \in \Sigma (x_k,\phi),
    \end{aligned}
\end{equation}where $\Omega_1 = -((A_k+B_kK_k)^\top M_{k+1}(A_k+B_kK_k) - (1-\beta)M_{k})$ represents the discrete-time contraction condition and $k_c,m_c$ are coefficients of polynomials for the controller gain, $K$ in \eqref{equ:differential feedback}, and metric, $M$, respectively (see the example in Section \ref{sec:example} for additional details). This programming problem is computationally difficult (if not intractable) due to the hard constraints imposed by the inequality \eqref{inequ: control affine contraction condition}. One possible improvement can be made by introducing relaxation parameters to soften the constraints (see, e.g. \cite{boyd2004convex}), i.e. introducing two small positive values, $r_1$ and $r_2$, as
\begin{equation}
    \begin{aligned}
        & \min_{k_c,m_c,r_1,r_2} r_1 + r_2 \\
        & s.t. ~ \phi^\top  \Omega_1 \phi - r_1I \in \Sigma (x_k,u_k,\phi)\\
        & \phi^\top  M_k \phi -r_2I \in \Sigma (x_k,\phi),r_1 \geq 0,r_2 \geq 0.
    \end{aligned}
\end{equation}
Note that the contraction condition holds if the two relaxation parameters $r_1$ and $r_2$ are some positive value, then we get a required DCCM as long as the relaxation parameters are positive. Although this relaxation reduces the programming problem difficulty, the problem remains infeasible, due to the terms coupled with unknowns, e.g. $B_kK_k^\top M_{k+1}B_kK_k$.

Naturally, substitution for the equivalent contraction condition \eqref{inequ:LMI} in Proposition \eqref{thm:condition}, solves this computational obstacle, and hence a tractable SOS programming problem can be formed as follows
\begin{equation}\label{min:sos}
    \begin{aligned}
        &\underset{l_c,w_c,r}{\min} r\\
        &s.t. \ \tilde{\phi}^\top  \Omega \tilde{\phi} -rI \in \Sigma(x_k,u_k,\tilde{\phi}), ~r \geq 0,
    \end{aligned}
\end{equation}
where $\Omega =
\begin{bmatrix}
    W_{k+1}              & A_kW_k+B_kL_k \\
    (A_kW_k+B_kL_k)^\top     & (1-\beta)W_k
\end{bmatrix}$ and $w_c,l_c$ are the polynomial coefficients of the metric-controller dual $(W_k,L_k)$ (see the example in Section \ref{sec:example} for additional details). Note that the inverse and coupling terms are not present in \eqref{min:sos} and that its SDP tractable solution yields the matrix function $W_k$ and $L_k$, as required for contraction analysis and control.

As an extension of this approach, synthesis of a DCCM and differential controller with desirable differential dissipativity properties, satisfying Proposition \ref{prop:dif dis} (and hence Theorem \ref{thm:dif dis}), can be expressed as the following SOS programming problem
\begin{equation}\label{min:sos dd}
    \begin{aligned}
        &\underset{l_c,w_c,r}{\min} r\\
        &s.t. \ \breve{\phi}^\top  \Phi \breve{\phi} -rI \in \Sigma(x_k,u_k,\nu_k,\breve{\phi}), ~r \geq 0,
    \end{aligned}
\end{equation}
where $\Phi = \begin{bmatrix}
W_{k+1} & A W_k + B_c L_k & B_\nu & 0\\
(A W_k + B_c L_k)^\top & (1 - \beta)W_k & W_k S & W_k \\
B_\nu^\top & S^\top W_k & R & 0\\
0 & W_k & 0 & -Q^{-1}
\end{bmatrix}$.

\subsection{Numerical Implementation}
Suppose that the optimisation problem \eqref{min:sos} (or \eqref{min:sos dd}) is solved for the pair $(W,L)$ and hence $(M,K)$ are obtained. The next step in implementing the contraction-based controller \eqref{equ:control integral} is to integrate \eqref{equ:differential feedback} along the geodesic, $\gamma$ \eqref{equ:geodesic}.
Subsequently, one method to numerically approximate the geodesic is shown. From \eqref{equ:Riemannian distance and energy} and \eqref{equ:geodesic}, we have the following expression for computing the geodesic,
\begin{equation}\label{equ:geo cal}
    \gamma(x,x^*) = \argmin_c \int_0^1{\frac{\partial c(s)}{\partial s}^\top  M(c(s)) \frac{\partial c(s)}{\partial s}ds}.
\end{equation}
where (see Section \ref{sec:discrete-time contraction analysis}) $c(s)$ is an $s$-parameterized smooth curve connecting $x$ ($s=0$) to $x^*$ ($s=1$). Since \eqref{equ:geo cal} is an infinite  dimensional problem over all smooth curves, without explicit analytical solution, the problem must be discretized to be numerically solved. Note that the integral can be approximated by discrete summation provided the discrete steps are sufficiently small. As a result, the geodesic \eqref{equ:geo cal} can be numerically calculated by solving the following optimization problem,
\begin{equation}\label{min:geodesic}
\begin{aligned}
    \bar \gamma(x,x^*) = \argmin_{\Delta \tilde{x}_{s}} &\sum_{i=1}^{N}{\Delta \tilde{x}_{s_i}^\top  M_{NN}(\tilde{x}_i) \Delta \tilde{x}_{s_i} \Delta s_i}\\
    s.t. \quad  & \tilde{x}_1 = x,~\tilde{x}_N = x^*,
\end{aligned}
\end{equation}where $\bar \gamma(x,x^*) \approx \gamma(x,x^*)$ represents the numerically approximated geodesic, $x$ and $x^*$ are the endpoints of the geodesic, $\tilde{x}_i$ represents $i$-th point on a discrete path in the state space, $\Delta \tilde{x}_{s_i} := \Delta {\tilde{x}_i} / \Delta {s_i} \approx {\partial c(s)}/{\partial s}$ can be interpreted as the displacement vector discretized with respect to the $s$ parameter, $\Delta \tilde{x}_s:=(\Delta \tilde{x}_{s_1},\cdots,\Delta\tilde{x}_{s_N})$ is the discretized path joining $x$ to $x^*$ (i.e., discretization of c(s) in~\eqref{equ:geo cal}), all $\Delta {s_i}$ are small positive scalar values chosen such that $\sum_{i=1}^N \Delta s_i = 1$, $N$ is the chosen number of discretization steps (of s), $\tilde{x}_i = \sum_{j=1}^i \Delta \tilde{x}_{s_j} \Delta s_j + x$ represents the numerical state evaluation along the geodesic.

Note that \eqref{min:geodesic} is the discretization of \eqref{equ:geo cal} with $\Delta \tilde{x}_{s_i}$ and $\Delta_{s_i}$ as the discretizations of  $\frac{\partial c(s)}{\partial s}$ and $\delta_s$ respectively, whereby the constraints in \eqref{min:geodesic} ensure that the discretized path connecting the start, $x$, and end, $x^*$, state values align with the continuous integral from $s=0$ to $s=1$. Hence, as $\Delta_{s_i}$ approaches 0, i.e., for an infinitesimally small discretization step size, the approximated discrete summation in \eqref{min:geodesic} converges to the smooth integral in \eqref{equ:geo cal}.

After the geodesic is numerically calculated using \eqref{min:geodesic}, the control law in \eqref{equ:control integral} can be analogously calculated using an equivalent discretisation as follows
\begin{equation}\label{equ:controller}
    u_k = u_k^* +  \sum_{i=1}^N K_k(\tilde{x}_i) \Delta \tilde{x}_{s_i} \Delta s_i.
\end{equation}
Substituting $K_k = L_kW_k^{-1}$ into \eqref{equ:controller}, we can then implement the control law as follows
\begin{equation}
\label{equ:disccontr}
    u_k = u_k^* +  \sum_{i=1}^N L_k(\tilde{x}_i)W_k^{-1}(\tilde{x}_i) \Delta \tilde{x}_{s_i} \Delta s_i,
\end{equation}
where $(x_k^*,u_k^*)$ are the state and control reference trajectories at time $k$. Note that this real-time control approach results in a setpoint-independent control structure, whereby the structure (imposed  by matrix functions $L$ and $W$) required to implement this control law are obtained offline via solution to \eqref{min:geodesic}. To see this, recall that $\tilde{x}_i$ represents the numerical state evaluation along the geodesic at discrete points and $K(\tilde{x}_i)$ is the evaluation of the matrix function $K=LW^{-1}$ at each of those points. Then, as the desired state value, $x_k^*$, changes, the feed-forward component, $u_k^*$, can be instantly updated, and the feedback component, $\sum L_k(\tilde{x}_i)W_k^{-1}(\tilde{x}_i) \Delta \tilde{x}_{s_i} \Delta s_i$ (i.e., $\int K(\gamma)\delta_\gamma ds$ in \eqref{equ:control integral}), can be automatically updated through online geodesic calculation.

\section{Illustrative Example}\label{sec:example}
In this section, we present a case study to illustrate the synthesis and implementation of a contraction-based controller using the proposed approach. \sloppy Consider a well mixed, nonisothermal CSTR where 3 parallel irreversible elementary exothermic reactions take place of the form $A \to B$, $A \to C$, $A \to D$ \cite{christofides2011networked}, whereby the process can be modelled as follows,
\begin{equation}\label{equ:sim sys}
\begin{aligned}
    x_{1_{k+1}} &= x_{1_k} + \Delta_t \left(\frac{F}{V_r}(C_{A0} -x_{1_k}) + \nu_k +  \sum_{i=1}^3 \kappa_{i0} e^{\frac{-E_i}{R x_{2_k}}} x_{1_k}\right) \\
    x_{2_{k+1}} &= x_{2_k} + \Delta_t \left(\frac{F}{V_r}(T_{A0}-x_{2_k}) - \sum_{i=1}^3 \frac{\Delta H_i}{\sigma c_p} k_{i0} e^{\frac{-E_i}{R x_{2_k}}}x_{1_k} + u_k \right),
\end{aligned}
\end{equation}
where the feed to the reactor consists of reactant $A$ with molar concentration $C_{A0}$ at flow rate $F$ and temperature $T_{A0}$. This process model consists of two states, where $x_1 = C_A$ denotes the concentration of reactant $A$, and $x_2 = T$ denotes the temperature of the reactor. The states $x_1$ and $x_2$ are controlled by manipulating  $u = \frac{Q}{\sigma C_p V_r}$, where $Q$ represents the rate of heat input/removal. The process disturbance $\nu$ is caused by the variation in the feed concentration $\Delta C_{A0}$ with $\nu =\frac{F}{V_r} \Delta C_{A0}$. For the remaining process variables, $V_r$ denotes the volume of the reactor, $\Delta H_i,\kappa_{i0},E_i$ denotes the enthalpies, preexponential constants and activation energies of the three reactions, respectively, and $c_p$ and $\sigma$ denote the heat capacity and density of the fluid in the reactor, respectively. For simulation purposes, the system model is normalized in the range of operation with a sampling period of $\Delta_t = 0.05\ h$. All parameters are shown in Table \ref{tab:process parameters}.
\begin{table}[]
    \centering
    \begin{tabular}{cc|cc}
        $F$          & $4.998[m^3/h]$               & $\kappa_{10}$ & $3\times 10^6[h^{-1}]$\\
        $V_r$        & $1[m^3]$                     & $\kappa_{20}$ & $3\times 10^5[h^{-1}]$\\
        $R$          & $8.314[KJ/kmol K]$           & $\kappa_{30}$ & $3\times 10^5[h^{-1}]$\\
        $T_{A0}$     & $300[K]$                     & $E_1$    & $5\times 10^4[KJ/kmol]$    \\
        $C_{A0}$     & $4[kmol/m^3]$                & $E_2$    & $7.53\times 10^4[KJ/kmol]$ \\
        $\Delta H_1$ & $-5.0 \times 10^4[KJ/kmol]$  & $E_3$    & $7.53\times 10^4[KJ/kmol]$ \\
        $\Delta H_2$ & $-5.2 \times 10^4[KJ/kmol]$  & $\sigma$ & $1000[kg/m^3]$             \\
        $\Delta H_3$ & $-5.4 \times 10^4[KJ/kmol]$  & $c_p$    & $0.231[KJ/kg K]$
    \end{tabular}
    \caption{Process Parameters}
    \label{tab:process parameters}
\end{table}

To synthesise a metric and controller pair $(M,K)$, the exponential functions in \eqref{equ:sim sys} are first approximated using second order polynomial functions, such that the system description is amenable to SOS programming. Then, the Jacobian matrices, $A_{k},B_{c,k},B_{\nu,k}$, are calculated to obtain the corresponding differential system model \eqref{eq:sys_d diff}. As required for the differential dissipativity-based SOS problem \eqref{min:sos dd}, a $(Q,S,R)-$supply rate, with $Q=I$, $S=0$, $R=0.81I$, is selected to achieve a truncated incremental $\mathcal{L}_2$ gain of $0.9$ (from the disturbance to state). The respective contraction metric and feedback gain duals, $W_k$ and $L_k$, are matrices of polynomial functions in the following forms, i.e.
\begin{equation}
\label{equ:WkLk}
    W_k =
    \begin{bmatrix}
        W_{{11}_k} & W_{{12}_k} \\
        W_{{12}_k} & W_{{22}_k}
    \end{bmatrix}, \qquad
    L_k =
    \begin{bmatrix}
        L_{1_k} \\
        L_{2_k}
    \end{bmatrix},
\end{equation}
where $W_{{\cdot \cdot}_k}=w_{{\cdot \cdot c}}v(x_k)$ and $w_{{\cdot \cdot c}}$ is a row vector of unknown coefficients, and similarly for $L_k$. As required to solve the SOS problem in \eqref{min:sos} and \eqref{min:sos dd}, the functions $W_{{\cdot \cdot k}}$  need to be polynomial functions and as such are expressed using the common monomial vector, $\phi(x_k)$, defined as
\begin{equation}
    \phi(x_k) =
    \begin{bmatrix}
        x_{1_k}^4 & x_{1_k}^3x_{2_k} & \cdots & x_{2_k}^4
    \end{bmatrix}^\top ,
\end{equation}
where the polynomial order is chosen to be 4.
Additionally, $W_{k+1}$, is constructed as a matrix of polynomials
\begin{equation}
    W_{k+1} =
    \begin{bmatrix}
        W_{{11}_{k+1}} & W_{{12}_{k+1}} \\
        W_{{12}_{k+1}} & W_{{22}_{k+1}}
    \end{bmatrix},
\end{equation}
where the elements are defined as $W_{{\cdot \cdot}_{k+1}}=w_{{\cdot\cdot c}}\phi(x_{k+1})$ and $w_{{\cdot \cdot c}}$ is the same coefficient vector for $W_k$ in \eqref{equ:WkLk}. Two design scenarios are then considered: i) without dissipativity, i.e. solving the optimisation problem in \eqref{min:sos}, and; ii) with dissipativity, i.e., solving the optimisation problem in \eqref{min:sos dd}. Both SOS problems were then solved for the contraction rate, $\beta=0.9$, and by considering the normalised range of operation $x_{min} = 0.01,x_{max} = 1$, with the resulting metric and controller matrix coefficients shown in \ref{Appen}.

A time-varying reference, corresponding to the variations in product specifications and energy cost, was considered as the sequence of setpoints \sloppy $(x_1^*,x_2^*,u^*)$=$(0.750,0.680,2.241)$, $(0.850,0.716,2.264)$, $(0.950,0.748,2.269)$ on the respective $k \Delta_t$ intervals of  $[0,10.0)$, $[10.0,19.9)$, $[19.9,29.8]$ $h$. Such feasible setpoints can be generated using an RTO layer -- a befittingly popular method for real-time process control. It is worth noting that for contraction-based control the optimal or even \textit{complete} trajectories (e.g., between the current state and target setpoint) are not needed, only the desired feasible setpoints are required. Suppose that a feasible complete set of reference trajectories was in fact available (typically a non-trivial task), then the same contraction based controller could be additionally used to drive the system to such references, and without structural redesign (simply update the geodesic information in \eqref{equ:control integral}). The contraction-based control methods does not require redesigning the control algorithm as the reference changes (setpoint or otherwise), unlike the Lyapunov-based control designs.

The CSTR \eqref{equ:sim sys} was then simulated with the discrete-time contraction-based controller \eqref{equ:disccontr} under two scenarios: i) without disturbance, see Figure \ref{fig:cstr sim}, and; ii) with feed concentration disturbance, see Figure \ref{fig:cstr sim d}. The results shown in Figure \ref{fig:cstr sim} demonstrate that the proposed  contraction-based controller is capable of driving the CSTR error-free to the desired time-varying product specifications in the absence of disturbances (as per Theorem \ref{theorem:incremental stability}). Figure \ref{fig:cstr sim d} shows that the contraction-based design which incorporates a differential dissipativity condition additionally achieves disturbance attenuation (as per Theorem \ref{thm:dif dis}).

\begin{figure}
    \begin{center}
        \includegraphics[width=\linewidth]{./pics/CSTR_SIM_NO_DIS.eps}
        \caption{Contraction-based control of the CSTR \eqref{equ:sim sys} without disturbance}
        \label{fig:cstr sim}
    \end{center}
\end{figure}

\begin{figure}
    \begin{center}
        \includegraphics[width=\linewidth]{./pics/sim.eps}
        \caption{Contraction-based control of the CSTR \eqref{equ:sim sys} with disturbance attenuation}
        \label{fig:cstr sim d}
    \end{center}
\end{figure}

\section{Conclusion}\label{sec:conclusion}
A systematic approach to the implementation of contraction-based control for discrete-time nonlinear processes was developed. Through the differential system framework, contraction and dissipativity conditions were derived, to ensure stability of the resulting closed-loop control system, i.e., exponential convergence to feasible time-varying references and bounded disturbance response. Computationally tractable equivalent conditions were then derived and additionally transformed into SOS programming problems, such that joint synthesis of a discrete-time control contraction metric and stabilising feedback controller could be completed with desired closed-loop properties. Complete controller implementation details were provided and the overall approach was effectively demonstrated through a numerical CSTR case study.

\bibliographystyle{elsarticle-num}

redpajama_set_name: RedPajamaArXiv


Featuring a garden and a terrace, Helena's place provides accommodations in Pale with free WiFi and mountain views. This apartment has accommodations with a balcony.
Offering free WiFi and mountain views, Biljana Sweet Home is a property located in Pale. Guests have a private patio. This apartment has 2 bedrooms, a cable TV, and a kitchen with an oven.
Holiday home Srdic is located in Pale and offers barbecue facilities and a garden. The property has mountain views.
Offering free WiFi, Bergkranc Hotel & Resort is located in Pale. Featuring free shuttle service, this property also welcomes guests with a restaurant and a terrace. The hotel has ski storage space.
Apartman Pale is located in Pale. This bed and breakfast also has free WiFi. If you would like to discover the area, skiing is possible nearby.
Located in Pale, Apartments Kokotovic has accommodations with a flat-screen TV. Free WiFi is featured.
Located in Pale, APARTMAN KNEZ offers BBQ area, free WiFi, room service, and a tour desk. Guests staying at this apartment can use the terrace.


text:
An Old Time Halloween
We recently came across the statement "Classic children's literature teaches audiences of all ages the truth about the world and human nature." Do you think this is true? Let's take a look at the Octobers of long ago through illustrations from four of Tasha's books; Pumpkin Moonshine (1938), The New England Butt'ry Shelf Cookbook (1968), Tasha Tudor's Five Senses (1978) and Around the Year (1957.)
Pumpkin Moonshine is available in hardcover, board book and paperback in our online shop.
A Bit of New England Halloween
Pumpkin Moonshine contemplates the excitement of autumn, pumpkins, adventure, and how children experience tradition. It was originally published as a very small book, and was one of a series of five referred to as the 'calico' books. Due to the diminutive dimensions of the volumes, bookstores experienced heightened incidents of the books departing from the premises without payment. Later editions were much larger!
Illustration from The New England Butt'ry Shelf Cookbook
The New England Butt'ry Shelf Cookbook possesses recipes and observations valuable to anyone involved in the process of day to day cooking and baking. This illustration documents two young children carving a "pumpkin moonshine" Tasha's name for what many people call Jack-o'-lanterns. Notice the wooden hayrake, the wooden water barrel, the basket full of apples.
What would this scene look like today? Would the boy be wearing shoes? Would the dog have a collar and dog tag? Most children I know love to carve pumpkins as much as the two children pictured showing a universal truth. This book is now out-of-print but used copies can be found for aficionados of New England recipes and lore.
Mary Mason Campbell, the author of this book, writes "Hallowe'en is many autumn things, none of them very spooky. Not that we have no ghosts. A ghost lives in our house who wouldn't scare a youngster if he could help it. Once or twice, when he has turned himself into a bat, he has given some of the grown-ups a few bad moments. He only does this when the butt'ry window has been left open at night and it's his way of showing how careless we can be."
"Hallowe'en is the last of the flowers in the garden, as the chrysanthemums and calendulas bravely outlast all the tender blossoms. It is gathering in of the pumpkins and squashes and putting the gardens to bed. It is frosty nights with brilliant starts and sometimes a cold north wind to remind us to put on storm windows. It is raking leaves and filling the woodbox. It is building a fire in the cookstove to warm the kitchen in the morning. It is making gingerbread and baking apples and planning the mincemeat."
"It is time to fill the butt'ry cooky and candy jars and to make the sweet cider. It is time to decorate the barn with cornshocks and with scooped-out pumpkins and funny-shaped squashes with cutout faces of leering goblins and with black cats and witches on broomsticks. It is the time to frost pumpkin tarts with whipped cream and to caramel apples on a stick and to fill the wash tub with water for bobbing Macs. It is the time to get into the attic trunks for costumes and to start the record player for square dancing and choose your partner, line up here for the Hallowe'en party."
The Five Senses by Tasha Tudor
In The Five Senses, we see a young girl putting a candle in her pumpkin moonshine. Notice how she's finished scooping the seeds and pulp out and has left that in a pile on the ground. Will she leave it there so that the many seeds can grow into plants and create more pumpkins to be carved into pumpkin moonshines? Notice how the corn is tied into shocks, common at the time but not something you see since the advent of harvest machinery and larger farms. One wonders if the child is alone, but of course she isn't, her corgi is there to keep her company. She's wearing warmer clothes in the fall chill, very similar to those of the child in the second illustration, though they were made years apart.
Like the child in the Five Senses illustration above, Tasha always wore a scarf over her hair plus a second around her neck. We offer these for sale in our online shop.
Around the Year is available in hardcover and paperback in our online shop.
Around the Year by Tasha Tudor
The fourth book, Around the Year is another favorite of young and old alike in that adults love the illustrations and seasonal celebrations, and children love being read to. This book is a treasure trove of the traditions of both historic 19th century life and the very similar life Tasha and her family lived in the mid-20th century. This image shows a fall town fair with vegetables and flowers on display for judging.
The book has retained its original size over the many decades it has been in print. The main difference between recent print editions and earlier is the former has but one large image on the front dust jacket, while the latter has four separate images depicting the respective New England seasons. As Tasha drew from the immediate world around her, it seems reasonable to believe the beautiful watercolors in this book are of scenes and circumstances she encountered while creating it. To this day, a great many of the objects in drawings Tasha did eighty years ago are yet present in her house in Vermont.
It is interesting to consider that Tasha made these books at different stages of her life. Pumpkin Moonshine when she was in her early twenties, and The New England Butt'ry Self Cookbook when she was older, and a mother. One of the great delights of Tasha's artwork and writing is seeing both her technique and what she observed evolve through the years and decades of her long and fulfilling career.
This portrait of Tasha Tudor by Richard Brown is available in our online shop.
One of Tasha's delights was creating pencil or charcoal drawings. She was especially effective in this medium, and those books in which such illustrations are present deserve particular attention. Many of her earlier books were created when her four children, all born in the 1940s, were quite young. In some illustrations, one wonders if Tasha baked the food seen in the oval wooden bowls, made the cider seen in the glass bottles, and collected the apples in the baskets. Always busy, it is probable that Tasha planted the apple trees from which she later picked the apples!

redpajama_set_name: RedPajamaCommonCrawl


Please select an option 10 11 12 5 6 7 8 9 Please select a ringsize.
Known to boost productivity, garnet helps to transmute negative energies into positive and sharpen your perceptions. A mystic crystal that helps to widen knowledge and ward off evil vibes, Garnet is also the stone of wisdom.


text:
Nelle sue opere ha colto con suggestivo lirismo personaggi e ambienti della provincia francese: L'uomo della segheria (L'homme de la sciene, 1943), Il paese dove non si arriva mai (Le pays où l'on n'arrive jamais, 1955, con cui ottenne il Prix Femina), Il cielo del sobborgo (Le ciel du faubourg, 1957).

Biografia

André Dhôtel trascorre i primi sei anni della vita nel paese natale di Attigny, nelle Ardenne. Scriverà i ricordi di questo periodo in Un jour viendra, nel 1972. Nel 1907, il padre è nominato banditore d'asta a Autun e si trasferisce con la famiglia. Ma André torna sempre dai nonni nelle Ardenne e non si lega a Autun. Dopo i primi studi, va a frequentare il liceo Sainte-Barbe a Parigi.

Nominato sorvegliante da novembre 1918 a Sainte-Barbe, in compagnia del futuro chansonnier Raymond Souplex, prepara nello stesso tempo una licenza di filosofia. Effettua il servizio militare dal 1920 al 1923, con gli scrittori Georges Limbour, Roger Vitrac e Marcel Arland. Con quest'ultimo fonda nel 1921 la rivista Aventure e pubblica l'unico numero della rivista Dès.

Le lacune di effettivi, dovute alla prima guerra mondiale, portano il giovane ad essere nominato professore all' Istituto superiore di studi francesi di Atene nel 1924; scopre così la Grecia e le sue isole durante i suoi quattro anni all' Istituto francese di Atene. Nei suoi romanzi posteriori non è raro trovare personaggi greci o espressioni filoelleniche. Al mondo ellenico ha dedicato quattro romanzi: Ce lieu deshérité, 1949, Ma chère âme, 1961, L'île de la Croix d'Or, 1979, Lorsque tu reviendras, 1986.

Rientrato in Francia, prende servizio al collegio di Béthune nel 1928 e pubblica i primi testi poetici. Nel 1930 esce Campements, suo primo romanzo, e si sposa nel 1932 con Suzanne Laurent. Il loro figlio François nasce nel 1933. Tuttavia la sua carriera letteraria ristagna. Dal 1930 al 1940, lo scrittore è rifiutato dai principali editori. A ciò si uniscono altri dispiaceri inerenti l'insegnamento, sicché André Dhôtel si ammala di depressione. Si ristabilisce tuttavia, e nel 1940, allo scoppio della guerra, è per qualche tempo mobilitato.

Finalmente, grazie a Jean Paulhan, torna a pubblicare, nonché a un insegnamento più tranquillo. Paulhan impiega la sua influenza per farlo nominare professore a Coulommiers nel 1943 (dove rimarrà fino al suo ritiro nel 1961) e fa accettare il romanzo Le village pathétique alla Nouvelle Revue Française. Qui Dhôtel pubblica un romanzo quasi ogni anno. Fonda anche la rivista 84, insieme a Marcel Bisiaux, Jacques Brenner, Alfred Kern, Armen Lubin e Henri Thomas. Il suo romanzo David, del 1947 alle Éditions de Minuit, riceve il Prix Sainte-Beuve.

Nell'estate 1947, Jean Dubuffet ne dipinge il ritratto, che intitola Dhôtel nuancé d'abricot. Riceve il Prix Femina nel 1955 per Le Pays où l'on n'arrive jamais, il Grand Prix de littérature pour les Jeunes nel 1960, il Grand Prix de littérature de l'Académie française nel 1974 e il Grand Prix national des lettres per il romanzo Les Disparus nel 1975. Dopo il ritiro dall'insegnamento, va finalmente a vivere a Parigi.

Muore nel 1991, circa un anno dopo la moglie, ed è sepolto nel cimitero di Provins.

Opere

Romanzi
 Campements, Parigi, Gallimard, 1930 ; ried. 1987.
 Le Village pathétique, Parigi, Gallimard, 1943; ried. coll. «Folio», 1975.
 Nulle part, Parigi, Gallimard, 1943; ried. Parigi, Éditions Pierre Horay, 1956, 1977; Rombaldi, 1979.
 Les Rues dans l'aurore, Parigi, Gallimard, 1945.
 Le Plateau de Mazagran, Parigi, Éditions de Minuit, 1947; ried. Parigi, Guilde du livre, 1960; Parigi, Marabout, 1977.
 David, Parigi, Éditions de Minuit, 1948; ried. Parigi, Marabout, 1979 (scritto nel 1930).
 Ce lieu déshérité, Parigi, Gallimard, 1949; ried. 1997.
 Les Chemins du long voyage, Parigi, Gallimard, 1949; ried. coll. «Folio», 1984.
 L'Homme de la scierie, Parigi, Gallimard, 1950.
 Bernard le paresseux, Parigi, Gallimard, 1952; ried. coll. «L'Imaginaire», 1984.
 Les Premiers Temps, Parigi, Gallimard 1953; ried. 1987.
 Le Maître de pension, Parigi, Grasset, 1954.
 Mémoires de Sébastien, Parigi, Grasset, coll. «Les Cahiers verts», 1955.
Le memorie di Sébastien, S.A.I.E., Torino 1956
 Le Pays où l'on n'arrive jamais, Parigi, P. Horay, 1955; ried. Parigi, J'ai lu , 1960; Parigi, Gallimard, 1975.
Il paese dove non si arriva mai, Ist. geografico De Agostini, Novara 1968
 Le Ciel du Faubourg, Parigi, Grasset 1956; ried. Parigi, Grasset, coll. Les Cahiers rouges, 1984.
 Dans la vallée du chemin de fer, Parigi, P. Horay, 1957.
 Le Neveu de Parencloud, Parigi, Grasset, 1960.
 Ma chère âme, Parigi, Gallimard, 1961; ried. Parigi, Phébus, coll. «Libretto», 2003.
 Les Mystères de Charlieu-sur-Bar, Parigi, Gallimard, 1962; ried. Rombaldi, 1979.
 La Tribu Bécaille, Parigi, Gallimard, 1963; ried. 1977.
 Le Mont Damion, Parigi, Gallimard, 1964.
 Pays natal, Parigi, Gallimard, 1966; ried. Parigi, Phébus, coll. «Libretto», 2003.
 Lumineux rentre chez lui, Parigi, Gallimard, 1967; ried. Phébus, coll. «Libretto», 2003.
 L'Azur, Parigi, Gallimard, 1968; ried. coll. «Folio», 2003.
 L'Enfant qui disait n'importe quoi, Parigi, Gallimard, 1968; ried. coll. «Folio Junior», 1978.
 Un jour viendra, Parigi, Gallimard, 1970; ried. Parigi, Phébus, Libretto, 2003.
 La Maison du bout du monde, Parigi, P. Horay, 1970.
 L'Honorable Monsieur Jacques, Parigi, Gallimard, 1972.
 Le Soleil du désert, Parigi, Gallimard, 1973; ried. 1997.
 Le Couvent des pinsons, Parigi, Gallimard, 1974
 Le Train du matin, Parigi, Gallimard, 1975.
 Les Disparus, Parigi, Gallimard, 1976.
 Bonne nuit Barbara, Gallimard, 1978.
 L'Île de la croix d'or, Parigi, Gallimard, 1000 soleils, 1978.
 La Route inconnue, Parigi, Phébus, 1980.
 Des trottoirs et des fleurs, Parigi, Gallimard, 1981.
 Je ne suis pas d'ici, Parigi, Gallimard, 1982.
 Histoire d'un fonctionnaire, Parigi, Gallimard, 1984.
 Vaux étranges, Parigi, Gallimard, 1986.
 Lorsque tu reviendras, Parigi, Phébus, 1986.

Novelle, storie e racconti
 Du Pirée à Rhodes, Le Rouge et le noir, 1928;
 Ce jour-là, Parigi, Gallimard, 1947.
 La Chronique fabuleuse, Parigi, Éditions de Minuit, 1955; ried. Parigi, Mercure de France (ampliata), 1960;
 L'Île aux oiseaux de fer, Parigi, Éditions Fasquelle, coll. «Libelles», 1956;
 Nulle part, Parigi, P. Horay, 1957.
 Les Voyages fantastiques de Julien Grainebis, Parigi, P. Horay, 1957;
 Idylles, Parigi, Gallimard, 1961;
 La Plus belle main du monde, Parigi, Casterman, coll. «Plaisir des contes», 1962.
 Le Robinson de la rivière, Parigi, Casterman, coll. «Plaisir des contes», 1962.
 Les Lumières de la forêt, Parigi, Fernand Nathan, 1964.
 Un soir, Parigi, Gallimard, 1977.
 La Merveilleuse Bille de verre, Parigi, Robert Laffont, 1980
 Comment Fabien regarda l'aurore, Parigi, Clancier-Guénaud, Les Premiers temps, 1982.
 La Princesse et la lune rouge, Parigi, Casterman, 1982.
 Le Bois enchanté et autres contes, Parigi, Hachette, 1983.
 Rhétorique fabuleuse, Parigi, Garnier, 1983; ried. Cognac, Le Temps qu'il fait, 1990.
 La Nouvelle Chronique fabuleuse, Parigi, P. Horay, 1984.
 Pierre Marceau (Mesures, N. 2, 15 aprile 1940), Parigi, Mont Analogue, 1993
 Un adieu, mille adieux, Parigi, La bibliothèque Gallimard , 2003
 Beauté, Parigi, Phébus, coll. «Libretto», 2004
 D'un monde inconnu, ill. di Daniel Nadaud, Saint-Clément-de-Rivière, Fata Morgana, 2012.

Poemi
 Le Petit Livre clair, Arras, Le Rouge et le noir, 1928;
 La Chronique fabuleuse, Parigi, Mercure de France, 1960;
 La Vie passagère, Parigi, Phébus, 1978.
 Poèmes comme ça, Cognac, Le Temps qu'il fait, 2000 (prefazione di Jean-Claude Pirotte).

Saggi
 L'œuvre logique de Rimbaud, Mézières, Éditions de la Société des écrivains ardennais, 1933.
 Rimbaud et la révolte moderne, Parigi, nrf Gallimard, 1951.
 Saint Benoit-Joseph Labre, Parigi, Éditions Plon, 1957.
 Le Roman de Jean-Jacques, Parigi, Éditions du Sud, 1962.
 La vie de Rimbaud, Parigi, Éditions du Sud, 1965.
 Nord-Flandre Artiois-Picardie, testo di André Dhôtel, fotografie di Jacques Fronval, Christian de Rudder, Alain Perceval, Jacques Verroust, série Tourisme en France nr 17, Parigi, Éditions SUN, 1971.
 Jean Follain, série Poètes d'aujourd'hui nr 49, Parigi, Éditions Pierre Seghers, 1972.

Interviste
 L'École buissonnière, interviste con Jérôme Garcin, Parigi, P. Horay, 1984.

Prefazioni e postfazioni
 Edmond Dauchot, Ardenne bien aimée, prefazione d'André Dhôtel, Parigi-Gembloux, J. Duculot, 1976.
 René Hénoumont, Edmond Dauchot: le photographe de l'Ardenne d'autrefois, introduzione di Georges Vercheval, con una testimonianza di René Henoumont e una postfazione d'André Dhôtel, Tournai, La Renaissance du Livre, 2000,

Premi letterari
 1944 - Prix de l'Académie française
 1955 - Prix Femina
 1974 - Gran premio di letteratura dell'Accademia francese per l'insieme delle sue opere letterarie;
 1975 - Grand Prix national des lettres.
 1988 - Grand prix de Littérature de la Société des Gens de Lettres

Note

Altri progetti

Collegamenti esterni

 Association des amis d'André Dhôtel

 Le fonds André Dhôtel de la bibliothèque universitaire d'Angers

redpajama_set_name: RedPajamaWikipedia


Bentonite Performance Minerals Llc USDOT 2401343 - Bentonite Performance Minerals Llc is a licensed and bonded freight shipping and trucking company running freight hauling business from Colony, Wyoming.
foundwel minerals e auction bentonite, mobile crusher, dmg machine with mineral casting foundwel minerals e auction bentonite - Newest Crusher >Minerals, bentonite crushing mill manufacturer lt germany; .
foundwel minerals e auction bentonite Bentonite Stone Images foundwel minerals e auction bentonite equipment bentonite germany application of barite, bentonite . bentonite stone images Application Of Barite Bentonite And Lime Stone Powders. foundwel minerals e auction bentonite.
Overview ResponsibilitiesWood Mining amp Minerals business is currently seeking a dynamic and innovative individual to fill the role of Mechanical Plant Layout .
foundwel minerals e auction bentonite foundwel minerals e auction bentonite Bentonite Stone Images foundwel minerals e auction bentonite equipment bentonite germany application of barite, read more.
arumpo bentonite mine - z.
arumpo bentonite mine - Newest Crusher, …. Arumpo Bentonite - Mine Site .Arumpo Bentonite Pty Ltd is a world-class bentonite mining site, the largest in Australia with indicated reserves of 70 million tonnes.
Rock Crusher: Mining Equipment | eBay. Find great deals on eBay for Rock Crusher in Mining, Rock Crusher 1hp Elec Motor-Gold Ore-14, Impact mill grinder 48" crusher rock gold silver mining jaw .
foundwel minerals e auction bentonite Bentonite Stone Images foundwel minerals e auction bentonite equipment bentonite germany application of barite, bentonite .
Mineral Resources Malaysian Minerals Malaysia have abundant resources of aggregates loed in the states of Perak, Selangor, Johore, Sabah and Sarawak. Production of aggregates in 2016 decreased by 16 per cent to 133,072,882 tonnes from 158,744,150 tonnes produced in the previous year.


Southern England. Dorset, DT1 1HU.
Georgian townhouse and secluded courtyard annexe combine to provide a relaxed haven located centrally in Dorchester.
Стоимость за ночь: от £60 до £160.
This Georgian residence with its concealed courtyard annexe combines to provide a comfortable and relaxed haven behind the bustling street scene of this pleasant and historic county town. Located centrally in Dorchester and close to many restaurants and cafés and convenient for visitors to the town centre. All rooms have direct dial telephone, colour television, hair dryer, tea and coffee making facilities and en-suite bath or shower rooms. You may prefer the Georgian style of our four poster bedroom, or the intimacy of a peaceful attic room, or even the contemporary style of a ground floor room with its own private patio. The Casterbridge has level access from the street and to all public areas. Bedrooms are situated on the ground, first and second floors. if you wish to reserve a particular room for any reason, or require a room with wheelchair access, please request this in the Additional Information section of the booking form. The family room sleeps two people in the main room and and a further two in bunk beds in the internally connecting room with an additional television and refrigerator. Children of all ages are welcome but we regret no dogs except guide-dogs.
Although the Casterbridge has been caring for guests since the 1930's it was not until 1981, when the third generation of the Turner family undertook an extensive modernisation programme, that it became recognised in the Good Hotel Guide, quickly followed by WHICH Hotel Guide, Michelin, Lonely Planet, Frommer, Berlitz and Charming Small Hotels. A generous breakfast can be taken in the dining room or conservatory, choosing from a selection of healthy fresh fruits, yogurts, porridge, mixed grill and all the traditional favourites. Alternatively, home made marmalade, croissants, muffins and assorted juices can afford a leisurely start from the privacy of your room. Vegetarian options are available and with a little notice most dietary requests can be met. The comfortable drawing room provides seclusion and there is a wealth of local literature in the residents' bar/library which may be studied over a quiet drink. Staff are available throughout the day to ensure a plentiful supply of freshly brewed tea or coffee and biscuits. As we do not serve lunch or dinner, information on nearby restaurants, pubs and takeaways can be found with a selection of menus in our bar and we are happy to make reservations on your behalf. Free wireless broadband is available in the public rooms and most bedrooms. Please specify at the time of reservation if it is important that you are allocated one of these rooms.
Check In: From 2pm Check Out: By 11am Dorchester is easily accessible by most forms of transport. Please be aware that there are TWO towns called Dorchester in England. This is the Dorchester in Dorset. By Road: Those wishing to navigate by GPS should enter "DT1 1HU" which will bring you within a few yards. If you are driving to Dorset from the north, leave the M5 at junction 25 and follow the signs for Yeovil, then Weymouth. From the east, follow the M3 and M27 to Ringwood, then the A31 to Dorchester. The A31 and A35 cross the county, giving access to the west country. By Bus and Coach: Services run to London, Bournemouth, Bristol, Exeter, Plymouth, Salisbury and all local towns. By Rail: Dorchester South Station is on the main line from London Waterloo to Weymouth. There are hourly trains to London and Southampton (and Southampton Airport). Change at Poole or Southampton for services to the east and north of Britain. Dorchester West Station provides direct services to Bristol, Cardiff and Weymouth. Both stations are approximately a 15 minute walk from the hotel and taxis are usually waiting in the ranks.


HARRISBURG, Pa. (CIRCA via STORYFUL) -- A girl and a grocery store cashier in Harrisburg, Pennsylvania, formed a special bond on February 24 when the pair bagged groceries together.
Lisa Teach shared a video of her 9-year-old daughter, Lilly, bagging groceries with the help of Giant Food Store cashier, Issac. According to Teach, Lilly loves to pack groceries at the store.
"We've never met Isaac before, but as he watched Lilly approaching the bagging area, she started to get the first bag ready," she wrote, "Isaac, in a moment I'll never forget, asked Lilly if she would like to help him and bag the groceries." The duo can be seen working together as Issac hands Lilly each item and she places them in bags, putting them into her mom's cart.
When they finish bagging, Lilly says "I love you" to Issac. "She doesn't say this to everyone, but in the moment, I believe this was her way of saying thank you for seeing me and for letting me help, I love you," Teach said.


Registration for Catechism classes will be held on Wednesday, August 29, 2018, from 3:30 PM to 5:00 PM at Our Lady of Fatima School. Please bring a photocopy of your child's Baptism certificate at the time of registration, if you have not previously done so. There is a registration fee of $20 per child for grades K-2 and 4-7 to help cover the cost of books. The fee for 3rd grade is $30 per child because of the additional textbooks needed for First Reconciliation and First Communion.


In summary HRW says "The British government has an international legal obligation not to deport people who have a credible fear of torture upon return. Convincing reports of arbitrary arrests and torture demand that the UK government suspend returns of rejected Tamil asylum seekers to Sri Lanka until it can fairly and thoroughly assess their individual claims based on up-to-date human rights information on Sri Lanka."
TAG argued that deportations should be halted based on new evidence collated in a UKBA bulletin dated the 30th November 2011. This bulletin makes reference to reports submitted to the UN Committee against Torture (UNCAT) that concluded its investigation in respect of reports concerning torture on the 25th November 2011. The UNCAT published its concluding observations which amounted to a damning indictment of the Sri Lankan authorities record on torture.
TAG argued that UKBA's decision to remove failed asylum seekers to Sri Lanka by way of Charter flight on the 15thDecember 2011, is unlawful. This is because it breaches a legitimate expectation held by the returnee that the policy on removal will be reviewed, in the light of 'credible and relevant evidence' [see ministerial statement, 28th November 2011 set out below] concerning safety. This 'credible and relevant' evidence has only recently existed in collated form and constitutes the basis of the UKBA bulletin entitled 'Recent reports on torture and ill treatment'.
Mittel J. ruled that TAG could not bring a 'generic' application against the Secretary of State for the Home Department (SSHD), currently Theresa May, and the arguments used in the above grounds should be brought individually in the context of individual appeals.
We recommended that solicitors incorporate the generic TAG grounds with the specific grounds of their clients. We understand that many who did so, succeeded in December.
The case Tamils Against Genocide v United Kingdom on the above grounds is now before the European Court of Human Rights (ECTHR): application number 77045/11, TAG being represented by Birnberg Peirce and Partners. It will take some months to be heard and it will not prevent deportations in the mean time.
If the ECTHR ultimately rules the deportation unlawful UKBA may be forced to grant asylum to the deportees. In the event that UKBA contributed to the torture by for example handing over confidential documents belonging to the returnee to GOSL a damages claim may arise.
The HRW statement of 25th February provides fresh evidence to support the grounds raised in TAG's case. We are aware the other NGOs also hold unpublished fresh evidence.
TAG will be supporting fresh grounds incorporating the HRW statement and other recent developments including the Swiss and New Zealand rulings (see our previous articles) - as before these are generic grounds. TAG will sponsor a lead case based on these fresh grounds. Vasuki solicitors are instructed on this lead case.
Option 1: they may request a copy of the new grounds from TAG by emailing us at tagdesk@tamilsagainstgenocide.org and the incorporate these points in their individual applications - tailoring them as appropriate based on whether they are applying for Judicial Review or to the Court of Appeal.
Option 2: If they have exhausted individual remedies, they can instruct Vasuki solicitors to add their names to the lead case that is being run by them.
Option 3: They may be able to do both option 1 and 2 but will need to take advice on this based on their situation.
The fresh general grounds incorporating the new evidence of torture on return (published by HRW on saturday the 25th) will be available from TAG by sunday the 26th. Please email tagdesk@tamilsagainstgenocide.org prior for more information, leaving your name, contact number, solicitor firm name and solicitors name if applicable.


text:
Q: Very simple application for managing a list of tasks and associating iCal events I have a list of tasks in a plan text file and I would like to create some very simple mac application where I can assign each task to an iCal event with an associated date, and the possibility to change later that date or delete associated event. Everything will be in my computer.
I do not know where can I start from or even if AppleScript would be useful here, any help?

redpajama_set_name: RedPajamaStackExchange


In describing and interpreting the results of her investigation of the Goldkrest Site near Albany, New York, archaeologist Lucianne Lavin (2004) referred to the radiocarbon dating of burned soil patches with associated charcoal as evidence of forest clearing through the use of fire. This evidence was found near the transition between soil strata that indicated a stabilizing landscape about AD 1000. The implication of this information is that as floods became less frequent and less violent due to changes in the Hudson River channel; this section of the Hudson River floodplain became more attractive for gardening to the ancestors of the Mohican Indians.


text:
Elena Chudinova: Telling the Truth
By Eugene Girin
In the autumn of 2005, I moved to New York City, breaking out of the green confines of bucolic and insufferably boring upstate New York to continue college. I wandered into one of the numerous Russian bookstores on Brighton Beach—a noisy, dirty, and delicious corner of the Soviet Union, preserved on the southernmost tip of Brooklyn. A book's glossy cover jumped out at me from the shelves. It showed a turbaned, bearded Mohammedan, mouth wide open in the call to prayer, against a dystopian landscape of Paris, full of onion-domed mosques, with the Eiffel Tower crowned by a Muslim crescent. In the coming days, I devoured the book on the subway under the curious gazes of Brooklyn's babushkas. The novel's title was Mechet Parizhskoi Bogomateri (The Notre Dame de Paris Mosque) by Russian writer, publicist, and poet Elena Chudinova.
Chudinova's magnum opus invites the readers to a Paris of a.d. 2048. This is a city ruled by the likes of sharia judge Malik, a Wahhabist, and Imam Abdulwahid. Here, Malik's burka-clad wife, Zeinab, crosses the Pont de Emirates bridge over the Seine on her way to the Champs-Élysées and interrupts her shopping spree to watch an elderly Catholic get stoned to death for making wine to celebrate Mass. In this Paris, the Christians are driven into ghettos, with the final massacre looming...


The chapel is considered to be the largest fishing chapel. It was built in 1750 but was expanded in the 1890s and now has 250 seats.
In a small bay furthest out towards the Sea of Bothnia is Berghamns fishing village with guest harbor, boat houses and service facilities. As late as the 1940s went a dozen fishing boats from the harbor, and here you can still buy fresh salmon, among other things.


From getting infant furniture to toys, there are a lot of things which feature on the purchase listing of moms and dads. Infant bedding is an important part of the nursery and makes sure outright comfort and convenience for the infant. Today, the market is stuffed with a lot of options when it comes to infant bedding.
Keep in mind when you are deciding on your purposes of whether you mean to include in storage area under the bottom bunk. This can be a good way to obtain included drawer area for your kids. If you decide this is an attribute you desire, ensure that you select plans with all the storage area consisted of in the plans. Picking the right bunkbed plans is in fact severe company. The hardest sound you may hear in the middle of the evening would be the bed collapsing, and your kids being hurt. Get the right plans and abide by them to the letter, and you make sure to have a secure, tough set of bunk beds.
When discussing platform beds it needs understanding precisely what one is and how it works. Merely put a platform bed is a bed that makes use of either a slatted or solid foundation in lieu of a box springtime foundation that is developed into or is a component of the complete bed. There are many variations of platform beds but this definition seems to sum up well exactly what this group of bedroom furniture is. One might ask however, "What about the box springtimes, won't I wreck my mattress?" In all truthfully North America seems to be among the few places worldwide that really make use of mattress box springtime mixes. Odd as that may seem it's real. Places such as Asia, Europe and Africa make use of a mattress on a platform and it seems to have worked quite well for them approximately this point. The key is ensuring that the mattress is well supported. You do not have to purchase a box springtime for a platform design bed.
Your bedroom speaks volumes about your taste and character, so choose bedroom essentials that match your choice. If you are preparing to spruce-up your bedroom with bedroom sets, below are some ideas that can assist you get started. Bedroom sets contribute a terrific offer in making the bedroom look cozy and welcoming while emphasizing the appeal of the room.


The scientists who first responded to my work on true bugs from the fallout areas of the Chernobyl radioactive cloud [ 1 ] asserted that the licensed, admitted radioactivity leaving the chimneys of the Swiss nuclear power plants was less than the Chernobyl fallout and therefore even not harmful. They claimed that the morphological disturbances I had found in Heteroptera could not possibly be a result of the radioactivity emitted by nuclear power plants.
My logical conclusion was: All the true bugs around the Swiss nuclear power plants should be in good health. In the summer of 1988 I started collecting Heteroptera true bugs and other insects around the Swiss nuclear power plants Gösgen and Leibstadt, in the canton of Aargau.


Review of Dearly Departed | Books, Movies, Reviews. Oh my!
Fanny and Amy Abel, the dynamic mother-and-daughter owners of a NYC travel agency, have just booked their biggest trip yet. But with danger in the air, the itinerary may include murder... Paisley MacGregor, a maid to the rich, made a dying request to send all of her wealthy employers on a first-class wake to spread her ashes around the world. Amy has her suspicions about these "mourners," especially when one has a life-threatening "accident" at the first stop in Paris. And when a mysterious American stranger tagging along with the group has his ticket punched in the shadow of the Taj Mahal, Amy knows she may have a killer on her tour. Who was this stranger, and what's the connection to someone in her group? Digging for clues while continuing on with the trip is a lot for Amy to manage, especially when another mourner has a possibly fatal encounter with a Hawaiian volcano. Back in the States, Fanny and Amy start to piece together a secret worth killing for, but someone is hot on their trail, and ready to send them on a one-way trip—to the morgue!
At the sound of the electronic buzz, Amy Abel glanced up and let out a little moan.
Though I thought the ending sort of dragged out a little I must say that I enjoyed this one much more than the first book in this series.
Amy's travel agency isn't doing very well. Having a murder happen on your murder mystery tour doesn't help. Though some people are morbid enough to want to go on murder mystery tour after hearing they actually had a murder she just can't make herself book another one.
Peter who also has a travel agency and is crushing on Amy invites her to help him with his newest assignment. He is hoping spending time together she would see that she might actually like him. One of his old maids Paisley has died and he has been assigned to make sure that her ashes travel to various destinations and that several of her old bosses go along to spread them.
Things were going fine at first but then of course it doesn't last. They don't actually have a murder in the tour but there is a murder and one of the tourist could be the murderer. Amy just can't find the connection between the victim and her group.
Her mother and her boyfriend Marcus get in on the hunt for the killer back at home by trying to get into Paisley's home which doesn't go well. It does get Detective Rawlings attention as the man was from New York. They don't find the suspect while on the tour and it ends and they are all back home.
The hunt for the killer continues on american soil and could prove to be very dangerous.
The mystery was done well as I had several suspects, but the hunt for the killer and catching him just sort of drug on a bit in my opinion.
I like Amy and her mother Fanny, they are fun characters. I really love her mother she is so funny but if I was Amy I might want to smack her a few times…lol. I don't like her boyfriend Marcus, if I had to choose I would rather she be with Peter. Marcus is a conman more or less, he lost his job and then got his new job by being sketchy. I just don't like him.
I think because this author wrote some of the MONK books that his cozies sort of play out a bit more like a clean detective book but with an amateur sleuth. If you like that sort of book then you will enjoy this series. It does have some funny moments and overall I did like it.
I do like a good cozy mystery, and the NYC setting for the travel agency is unique- most of the cozies I read are set in quaint little towns. I might have to check this series out! My brother loves the Monk books, but I haven't read any of them yet.
I have the first one in the series (which is why I wanted you to read this one!) but haven't read it yet. This does sound fun but not like a must read. I definitely don't like the sound of Marcus. I'm glad this one was better then the first!
I don't read Cozy's but I love reading your reviews about them. Sorry the end was so slow but I am glad overall you enjoyed it.
I am glad you enjoyed this even if it it had some issues. Great review!
I haven't read a cozy in a while, but one where the ending drags on probably isn't the one for me to jump in with. How cool that he wrote some of the MONK series though! I loved that show (though I only watched it in bits and pieces).
I'm glad your TBR pick was a good one, even if it wasn't perfect.
Glad you liked it and I like when you can't guess the killer right off the bat. I like cozies, but I am like you and I want them to wrap up without dragging on.
It's nice if a series picks up like that and the second book is better than the first one. I like the sound of the main character having her own travel agency! Although I don't like it when the mystery drags on too long, but it is nice when the mystery keeps you guessing. Great review!
I love the cover, not sure the mystery has enough of a hook for me though, I like cozies, but usually they have to be a theme that I'm really interested in, ghosts, archaeology, something like that. Glad you enjoyed it, it's nice when you're able to connect with the main characters, although in mysteries it does seem like often they miss obvious things, because they have to!
I sometimes enjoy a cozy that's a bit more mystery than cozy. This sounds like a good one and I like the author.


Come out and join us May 13, 2019 for a Golf Tournament Sponsored by Bill & Kristi Jenkins at NAPA Auto Parts.
Camp Creation is a 5-week summer program held in Foley, Al at Liberty Church. Camp begins on June 24th and ends on July 26th. Please pre-register to save your camper's spot by May 1, 2019. The cost is $125 for the whole summer session. Scholarships are available for those with financial hardship. Transportation is not currently available.
Camp begins at 8:30am and ends at 12:30pm. Lunch and snacks are provided daily. We have many volunteers visiting on a weekly basis from local churches and youth groups.The schedule includes lessons in reading, math, character, arts and crafts, PE, and praise and worship.
Campers should come prepared to play outside with summer clothes and tennis shoes. The classrooms are air conditioned so a jacket may be needed indoors.


The Vietnamese food processing industry is driven by strong domestic consumption and strong exports, and demand keeps growing, - but fierce competition forces the industry to continue investing to improve efficiency and meet global food safety standards.
As a result Vietnam's food processing companies adopted low-cost strategies and invest large sums of money for equipment upgrades, which is mainly sourced from overseas - which opens great opportunities for international machinery manufacturers. In 2017 Vietnam imported over $425.5 million USD worth of F&B machinery.
With new FTAs coming into effect and old one starting to generate traffic, comparing to 2016, import of F&B machinery increased drastically: packaging/wrapping machinery +49%, bakery machinery for pasta +59%, cleaning & drying of bottles +184%, machinery for the industrial preperation of fruits, nuts & vegetables +154% and for manufacture of confectionery, cocoa or chocolate 145%.
Vietnam has the largest beer market among the 10 members of the Association of Southeast Asian Nations, with annual production of more than 4 million kilo-liters, according to Kirin Holdings research.
Revenue in the Food & Beverages segment amounts to US$148m in 2018.
Revenue is expected to show an annual growth rate (CAGR 2018-2023) of 21.9%, resulting in a market volume of US$400m by 2023.
User penetration is 15.9% in 2018 and is expected to hit 20% by 2023.
The average revenue per user (ARPU) currently amounts to US$9.88.
The Business Monitor International (BMI) forecast that the growth rate of the sector would be at 10.9 per cent in the 2017-19 period, thanks to people's increasing income and a trend of using high value products.
Join over 180 delegates and 20 manufacturing professionals and expert speakers. Learn about the latest packaging technologies and how to apply newest production techniques to increase your profitability. Network with industry professionals like you and explore business opportunities.
Field Application Engineer at Chang Woen Machinery Co., Ltd.