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Micronations/List of already existing micronations/Ladonia

Ladonia is a micronation, proclaimed in 1996 as the result of a years-long court battle between artist Lars Vilks and local authorities over two sculptures. The claimed territory is part of the natural reserve of Kullaberg in southern Sweden.

History
In 1980, artist Lars Vilks began construction of two sculptures, Nimis (Latin for "too much", a structure made of 75 tonnes of driftwood) and Arx (Latin for "fortress", a structure made of stone), in the Kullaberg nature reserve in north-west Skåne, Sweden. The location of the sculptures is difficult to reach, and as a consequence they were not discovered for two years, at which point the local council declared the sculptures to be buildings, the construction of which was forbidden on the nature reserve, and demanded that they should be dismantled and removed. Despite the confrontations with the local council, a large percentage of the local community supports the sculptures, especially people working in the tourism industry. Vilks appealed the decision of the council, but lost. He appealed repeatedly, and finally the case was settled, in the council's favour, by the Swedish government. However, in the meantime Nimis had been bought from Vilks by the artists Christo and Jeanne-Claude after the death of Joseph Beuys, who bought it in 1984. On 2 June 1996, in protest of the local council, Vilks declared the area surrounding Nimis an independent, sovereign country named Ladonia.

In 1999, another sculpture, Omphalos (named after Omphalos, a small sculpture in the temple at Delphi, "marking the centre of the world"), was created. It was made of stone and concrete, 1.61 metres high and weighing a tonne. The Gyllenstiernska Krapperup Foundation, formed to promote art and culture, accused Vilks of building this sculpture and complained to the police, and in August 1999 the district court ordered its removal. The Foundation had also demanded the removal of Nimis and Arx, but the court ruled against it. The Foundation appealed this decision to the Supreme Court, who eventually ruled against it. The police were unable to positively identify Vilks as the sculptor, but the district court held that he was.

The removal of Omphalos was itself controversial. Vilks was ordered to find an acceptable way to remove the sculpture. He proposed blowing it up on 10 December 2001, Nobel Day and the 100th anniversary of the Nobel Prize, and applied to the county council for permission to do so. The county council made a decision on 7 December, but kept it secret until 10 December. By that time, another artist, Ernst Billgren, had bought Omphalos from Vilks, and had requested that it not be damaged. In the early hours of 9 December, a crane boat was sent (by DYKMA, under contract from the Enforcement Administration) to the site and removed the sculpture (at a cost of SEK 92,500, billed to Vilks). Despite the new owner's request, the sculpture was damaged by handling. In response to this, the Enforcement Administration was satirically declared to be "Performance Artist of the Year" in 2002.

Afterwards, Vilks applied to the county council again, this time for permission to erect a memorial in the place that Omphalos had stood. Permission was granted by the council to erect a monument no greater than 8 centimetres high. This was duly done, and the monument was inaugurated on 27 February 2002.

The ensemble Neurobash performed in Ladonia during the 10th Anniversary celebrations in 2006.

Government
The government of Ladonia is jointly led by a Queen and a President. The President and Vice-President are elected tri-annually, while the Queen, once crowned, reigns for life. the position of President is held by Interim-President The President's Old Shoes, and the Vice-President position is vacant, pending new elections in 2019. Vilks, the State Secretary, performs or oversees many of the day-to-day operations of the microstate, including processing new citizenship applications and posting photos and news items to the Ladonian online "newspaper". The Cabinet Ministers are the legislative body in Ladonia, and they participate in debates and vote on proposals via the Internet. Many Ladonian ministries have artistic connotations and whimsical names. As of 2018 the Cabinet of Ladonia has voted to amend the constitution to create the position of Prime Minister. The PM candidates will be nominated by and selected from current members of the Cabinet. The amendment abolishes the positions of President and Vice-President. This move was made as a way to prevent the election fraud that occurred in the previous election.

Demographics
When it was created, Ladonia had no population. As of 2016, there are 17,752 Ladonian citizens from more than 50 countries. None of the citizens resides within Ladonia's borders, although there has been at least one resident citizen in the past. Ladonia coexists as both a physical territory and as a large and active online community that resembles a third place, distinguishing itself from other micronations, which are either active online communities or claim small physical territories.

A comparison can be drawn between Ladonia accepting online citizenship applications and the E-Residency of Estonia. Virtual residency program shows how established nations may also adopt similar strategies to micronation, providing an example of what citizenship might look like in postnationalism.

Nimis
Nimis is a series of wooden sculptures situated along the coast in the Kullaberg Nature Reserve, Höganäs Municipality, in the northern part of Skåne County, Sweden. They are a massive, wooden labyrinthine structure connected by several wooden towers, and are said to be mostly constructed from driftwood.

They were begun by the artist Lars Vilks in 1980 and have been the subject of a long-running legal dispute between the Swedish authorities and the artist. As no permission was given to build on the site within the nature reserve, the County Administrative Board in Skåne has sought to have Nimis demolished, despite the fact that it has become a popular tourist attraction.

As Nimis's existence is not sanctioned by Sweden, it is difficult to find – there are no official sign posts in Sweden, nor is it marked on maps. However, it can be found on Google Maps. It lies a few kilometres northwest of the town of Arild and somewhat farther from the town of Mölle, and can only be reached on foot following a well-worn path with yellow Ns painted on trees and fences. The path begins as an easy stroll past Himmelstorp, a well-preserved eighteenth-century farmstead, but quickly becomes a steep and rocky climb down to the coast. Reportedly, thousands of local and international tourists visit Nimis each year, especially during summer.


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Micronations/List of already existing micronations/Empire of Atlantium

The Empire of Atlantium is a micronation and secular, pluralist progressive lobby group based in New South Wales, Australia precisely at -34,1924000, 149,0335000.

“Micronations: The Lonely Planet Guide to Home-Made Nations” described Atlantium as "a refreshing antidote to the reactionary self-aggrandisement of so many micronations", and "an extremely sophisticated nation-state experiment, as well as an entirely serious claimant to legitimate statehood". The book's entry on Atlantium notes its espousal of "progressive, liberal policies" and characterises it as a "secular humanist utopia".

Among the causes Atlantium supports were the right to unrestricted international freedom of movement, the right to abortion, the right to assisted suicide and decimal calendar reform.

It had almost 3,000 "citizens", most of whom signed up online from over 100 countries and have never been to Atlantium.

History
Atlantium was established in 1981 by three Sydney teenagers – George Francis Cruickshank, Geoffrey John Duggan and Claire Marie Coulter (née Duggan). The three claimed a 10 m2 Provisional Territory in the southern suburb of Narwee as Atlantium's first capital, and declared Cruickshank to be Sovereign Head of State, with the title "Emperor George II". Geoffrey Duggan was elected as Prime Minister in 1982, serving until 1986. Damian Scott held the position from 1986 to 1988 and Kevin Fanucchi became Prime Minster in 1988, but by 1990, when the original group members had graduated from university and moved to different locations, the group ceased to be active.

In 1999 Cruickshank purchased an apartment in the inner Sydney suburb of Potts Point, and soon after revived Atlantium, launching a website, which was instrumental in attracting new members. The 61 m2 apartment, known as the Imperium Proper, became the second capital of Atlantium. Concordia became the third capital of Atlantium on 12 January 2008, when the rural 0.76 km2 Province of Aurora, approximately 350 km southwest of Sydney, was created. The Atlantium website describes Aurora as Atlantium's "global administrative capital, ceremonial focal point and spiritual homeland".

Status and operations
The Atlantium website used several different self-descriptions, including "self-declared state", "aspirant microstate" and "global sovereign state". In line with its claim to be a "primarily non-territorial" state Atlantium does not maintain any formal territorial claims; however, it does promote the idea that Cruickshank's apartment, and the Province of Aurora, have extraterritorial status; in practice these properties remain under Australian jurisdiction.

No established nation had recognised Atlantium's sovereignty claims, and it had no reciprocal diplomatic relations. Atlantium had appointed "unaccredited diplomatic representatives" called "Imperial Legates" in the United States, Pakistan, Poland, Brazil, India, Italy, Iran, Singapore, Serbia and Switzerland. The group had awarded "Imperial Honours" to various recipients, generally in recognition of political activism or for service to local communities.

Stamps, coins and banknotes
Stamps, coins and banknotes were sold by Atlantium, which used a decimal currency system of 100 imperial centi to the imperial solidus. Atlantium's website claimed that the profits from those sales are used for "the Empire's ongoing operations" as well as charitable causes.

The earliest documented media report referring to Atlantium is a 1984 philatelic magazine article about its cinderella stamp releases. There were 12 issues of Atlantian stamps.

Atlantium minted coins, though the levels of Atlantian economic activity remained low. The first coin was the 20th anniversary 10 solidi coin, which has a likeness of George Cruickshank as the obverse and the imperial eagle as the reverse. In 2011, a 30 solidi coin was issued to commemorate the 30th anniversary of Atlantium. The coin had the likeness of George Cruickshank on the obverse and the wedge-tailed eagle (Aquila audax) on the reverse and was struck in cupronickel and plated with 9 carat gold. Banknotes of Atlantium were denominated in imperial solidi (10, 25, 50 and 100 imperial solidi) and are currently divided into a 2006 series and a 2007 series.

Citizenship
, Atlantium had almost 3,000 citizens from (and still therein residing due to the small area of the Empire of Atlantium) over 100 countries. The website names individuals holding such functions as minister, director, magister and imperial legate. Atlantium said its citizenship does not supersede existing citizenships. Atlantians contend that they are all dual citizens, and that Atlantium actively encourages its citizens to participate in the political processes of their resident countries.

In late 2016, Atlantium stopped accepting citizenship requests according to a reply posted on the official Facebook page. On 1 January 2018, citizen application process was re-opened with a US$25 fee.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


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LMIs in Control/pages/LMI for Feasibility Problem

LMI for Feasibility Problem in Optimization

The feasibility problem is to find any feasible solutions for an optimization problem without regard to the objective value. This problem can be considered as a special case of an optimization problem where the objective value is the same for all the feasible solutions. Many optimization problems have to start from a feasible point in the range of all possible solutions. One way is to add a slack variable to the problem in order to relax the feasibility condition. By adding the slack variable the problem any start point would be a feasible solution. Then, the optimization problem is converted to find the minimum value for the slack variable until the feasibility is satisfied.

The System
Assume that we have two matrices as follows:

$$ \begin{align} A(x) = A_{0}+A_{1}x_{1}+ ... + A_{n}x_{n} \quad i=1, 2, ..., n \end{align}$$

$$ \begin{align} B(x) = B_{0}+B_{1}x_{1}+ ... + B_{n}x_{n} \quad i=1, 2, ..., n \end{align}$$

which are matrix functions of variables $$ x = [x_{1} \quad x_{2} \quad ... \quad x_{n}]^{\text{T}} \in \mathbb{R}^{n} $$.

The Data
Suppose that the matrices $$ \begin{align} A_{0}, A_{1}, ..., A_{n} \quad \end{align}$$ and $$ \begin{align} B_{0}, B_{1}, ..., B_{n} \quad \end{align}$$ are given.

The Optimization Problem
The optimization problem is to find variables $$\begin{align} x = [x_{1} \quad x_{2} ... x_{n}] \end{align}$$ such that the following constraint is satisfied:

$$\begin{align} A(x) < B(x) \end{align}$$

The LMI: LMI for Feasibility Problem
This optimization problem can be converted to a standard LMI problem by adding a slack variable, $$t$$.

The mathematical description for this problem is to minimize $$ t $$ in the following form of the LMI formulation:

$$ \begin{align} & \text{min} \quad t\\ & \text{s.t.} \quad A(x) < B(x) + tI \\ \end{align}$$

Conclusion:
In this problem, $$ x $$ and $$ t $$ are decision variables of the LMI problem.

As a result, these variables are determined in the optimization problem such that the minimum value of $$t$$ is found while the inequality constraint is satisfied.

Implementation
A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Feasibility-Problem-of-Convex-Optimization

Related LMIs
LMI for Linear Programming

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LMIs in Control/pages/LMI for Linear Programming

LMI for Linear Programming

Linear programming has been known as a technique for the optimization of a linear objective function subject to linear equality or inequality constraints. The feasible region for this problem is a convex polytope. This region is defined as a set of the intersection of many finite half-spaces which are created by the inequality constraints. The solution for this problem is to find a point in the polytope of existing solutions where the objective function has its extremum (minimum or maximum) value.

The System
We define the objective function as:

$$ c_{1}x_{1} + c_{2}x_{2} + ... c_{n}x_{n} $$

and constraints of the problem as:

$$ a_{11}x_{1} + a_{12}x_{2} + ... + a_{1n}x_{n} < b_{1} $$

$$ a_{21}x_{1} + a_{22}x_{2} + ... + a_{2n}x_{n} < b_{2} $$

.

.

.

$$ a_{m1}x_{1} + a_{m2}x_{2} + ... + a_{mn}x_{n} < b_{m} $$

The Data
Suppose that $$ c_{j} \in \mathbb{R}^{n} $$, $$ a_{ij} \in \mathbb{R}^{n}$$, and $$ b_{i} \in \mathbb{R}^{m} $$ are given parameters where $$ i =1, 2, ..., m $$ and $$ j =1, 2, ... ,n $$. Moreover, $$ x = [x_{1} \quad x_{2} \quad ... \quad x_{n}]^{\text{T}} $$ is an $$ n\times 1 $$ vector of positive variables.

The Optimization Problem
The optimization problem is to minimize the objective function, $$ c^{\text{T}}x $$ when the aforementioned linear constraints are satisfied.

The LMI: LMI for linear programming
The mathematical description of the optimization problem can be readily written in the following LMI formulation:

$$ \begin{align} &\text{min} \quad c^{\text{T}}x&\\ &\text{s.t.} \quad a_{i}^{\text{T}}x \leq b_{i}x \\ \end{align}$$

Conclusion:
Solving this problem results in the values of variables $$x$$ which minimize the objective function. It is also worthwhile to note that if $$ m \geq n $$, the computational cost for solving this problem would be $$ mn^{2} $$.

There does not exist an analytical formulation to solve a general linear programming problem. Nonetheless, there are some efficient algorithms, like the Simplex algorithm, for solving a linear programming problem.

Implementation
A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Linear-Programming

Related LMIs
LMI for Feasibility Problem

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LMIs in Control/pages/LMI for Attitude Control of Nonrotating Missiles

LMI for Attitude Control of Nonrotating Missles, Pitch Channel

The dynamic model of a missile is very complicated and a simplified model is used. To do so, we consider a simplified attitude system model for the pitch channel in the system. We aim to achieve a non-rotating motion of missiles. It is worthwhile to note that the attitude control design for the pitch channel and the yaw/roll channel can be solved exactly in the same way while representing matrices of the system are different.

The System
The state-space representation for the pitch channel can be written as follows:

$$ \begin{align} \dot{x}(t) &= A(t)x(t) + B_{1}(t)u(t) + B_{2}(t)d(t) \\ y(t) &= C(t)x(t) + D_{1}(t)u(t) + D_{2}(t)d(t) \end{align} $$

where $$ x = [\alpha \quad w_{z} \quad \delta_{z}]^{\text{T}}$$, $$ u = \delta_{zc} $$, $$ y = [\alpha \quad n_{y}]^{\text{T}}$$, and $$ d = [\beta \quad w_{y}]^{\text{T}}$$ are the state variable, control input, output, and disturbance vectors, respectively. The paprameters $$ \alpha $$, $$ w_{z} $$, $$ \delta_{z} $$, $$ \delta_{zc} $$, $$ n_{y} $$, $$ \beta $$, and $$ w_{y} $$ stand for the attack angle, pitch angular velocity, the elevator deflection, the input actuator deflection, the overload on the side direction, the sideslip angle, and the yaw angular velocity, respectively.

The Data
In the aforementioned pitch channel system, the matrices $$ A(t), B_{1}(t), B_{2}(t), C(t), D_{1}(t),$$ and $$D_{2}(t)$$ are given as:

$$ \begin{align} A(t) = \begin{bmatrix} -a_{4}(t) & 1 & -a_{5}(t) \\ -\acute{a}_{1}(t)a_{4}(t) - a_{2}(t) & \acute{a}_{1}(t) - a_{1}(t) & \acute{a}_{1}(t)a_{5}(t) - a_{3}(t) \\ 0 & 0 & -1/\tau_{z} \end{bmatrix} \end{align} $$

$$ \begin{align} B_{1}(t) = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \quad B_{2}(t) = \frac{w_{x}}{57.3} \begin{bmatrix} -1 & 0 \\ -\acute{a}_{1}(t) & \frac{J_{x}-J_{y}}{J_{z}} \\ 0 & 0 \end{bmatrix} \end{align} $$

$$ \begin{align} C(t) = \frac{w_{x}}{57.3} \begin{bmatrix} 57.3g & 0 & 0 \\ V(t)a_{4}(t) & 0 & V(t)a_{5}(t) \end{bmatrix} \end{align} $$

$$ \begin{align} D_{1}(t) = 0, \quad D_{2}(t) = \frac{1}{57.3g} \begin{bmatrix} 0 & 0 \\ V(t)b_{7}(t) & 0 \end{bmatrix} \end{align} $$

where $$ a_{1}(t) \sim a_{6}(t), \quad b_{1}(t) \sim b_{7}(t), \acute{a}_{1}(t), \acute{b}_{1}(t) $$ and $$ c_{1}(t) \sim c_{4}(t) $$ are the system parameters. Moreover, $$ V $$ is the speed of the missle and $$J_{x}$$, $$J_{y}$$, and $$J_{z}$$ are the rotary inertia of the missle corresponding to the body coordinates.

The Optimization Problem
The optimization problem is to find a state feedback control law $$ u= Kx$$ such that:

1. The closed-loop system:

$$ \begin{align} \dot{x} &= (A+B_{1}K)x + B_{2}d \\ z &= (C + D_{1}K)x + D_{2}d \end{align} $$

is stable.

2. The $$ H_{\infty} $$ norm of the transfer function:

$$G_{zd}(s) = (C + D_{1}K)(sI-(A+B_{1}K))^{-1}B_{2} + D_{2}$$

is less than a positive scalar value, $$ \gamma $$. Thus:

$$ ||G_{zd}(s)||_{\infty} < \gamma $$

The LMI: LMI for non-rotating missle attitude control
Using Theorem 8.1 in [1], the problem can be equivalently expressed in the following form:

$$ \begin{align} &\text{min} \quad \gamma \\ &\text{s.t.} \quad X > 0 \\ &\begin{bmatrix} (AX+B_{1}W)^{T} + AX+B_{1}W & B_{2} & (CX+D_{1}W)^{T} \\ B_{2}^{T} & -\gamma I & D_{2}^{T} \\ CX + D_{1}W & D_{2} & -\gamma I \end{bmatrix} < 0 \end{align}$$

Conclusion:
As mentioned, the aim is to attenuate the disturbance on the performance of the missile. The parameter $$ \gamma $$ is the disturbance attenuation level. When the matrices $$ W $$ and $$ X $$ are determined in the optimization problem, the controller gain matrix can be computed by:

$$ K = WX^{-1} $$

Implementation
A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Non-rotating-Missle-Attitude-Control

Related LMIs
LMI for Attitude Control of Nonrotating Missles, Yaw/Roll Channel

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LMIs in Control/pages/LMI for convex minimization problem

The System
Let $$ A(x) $$ and $$ B(x) $$ are matrix functions of appropriate dimensions.

Moreove, the matrix functions $$ A(x) $$ and $$ B(x) $$ are linear in vector $$ x $$ where $$ x $$ is an $$m \times 1$$ vector.

The Data
A convex function $$ f(x) $$ is given and $$ x $$ is the vector of decision variables.

The Optimization Problem
The optimization problem is to minimize the convex function $$ f(x) $$ such that a contraint between two convex functions are satisfied for the decision variables.

The LMI: LMI for convex minimization problem
$$ \begin{align} &\text{min} \; \quad f(x)&\\ &A(x) < B(x) \end{align} $$

Implementation
A link to Matlab codes for this problem in the Github repository:


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LMIs in Control/pages/LMI for Generalized eigenvalue problem

LMI for Generalized Eigenvalue Problem

Technically, the generalized eigenvalue problem considers two matrices, like $$A$$ and $$B$$, to find the generalized eigenvector, $$x$$, and eigenvalues, $$\lambda $$, that satisfies $$ Ax = \lambda Bx $$. If the matrix $$ B $$ is an identity matrix with the proper dimension, the generalized eigenvalue problem is reduced to the eigenvalue problem.

The System
Assume that we have three matrice functions which are functions of variables $$ x = [x_{1} \quad x_{2} \quad ... \quad x_{n}]^{\text{T}} \in \mathbb{R}^{n} $$ as follows:

$$ A(x) = A_{0}+A_{1}x_{1}+ ... + A_{n}x_{n} $$

$$ B(x) = B_{0}+B_{1}x_{1}+ ... + B_{n}x_{n} $$

$$ C(x) = C_{0}+C_{1}x_{1}+ ... + C_{n}x_{n} $$

where are $$A_{i}$$, $$B_{i}$$, and $$C_{i}$$ ($$i=1, 2, ..., n$$) are the coefficient matrices.

The Data
The $$A(x)$$, $$B(x)$$, and $$C(x)$$ are matrix functions of appropriate dimensions which are all linear in the variable $$x$$ and $$A_{i}$$, $$B_{i}$$, $$C_{i}$$ are given matrix coefficients.

The Optimization Problem
The problem is to find $$\begin{align} x = [x_{1} \quad x_{2} ... x_{n}] \end{align}$$ such that:

$$A(x) < \lambda B(x) $$, $$ B(x) >0 $$, and $$ C(x) <0 $$ are satisfied and $$\lambda$$ is a scalar variable.

The LMI: LMI for Schur stabilization
A mathematical description of the LMI formulation for the generalized eigenvalue problem can be written as follows:

$$ \begin{align} &\text{min} \quad \lambda \\ & \text{s.t.} \quad A(x) < \lambda B(x) \\ & \quad \quad B(x) > 0 \\ & \quad \quad C(x) <0 \end{align} $$

Conclusion:
The solution for this LMI problem is the values of variables $$x$$ such that the scalar parameter, $$\lambda$$, is minimized. In practical applications, many problems involving LMIs can be expressed in the aforementioned form. In those cases, the objective is to minimize a scalar parameter that is involved in the constraints of the problem.

Implementation
A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Schur-Stability

Related LMIs
LMI for Generalized Eigenvalue Problem

LMI for Matrix Norm Minimization

LMI for Maximum Singular Value of a Complex Matrix

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LMIs in Control/pages/LMI for Mixed H2 Hinf Output Feedback Control

The System
The objective is to minimize the combined $$ H_{2} $$ and $$ H_{\infty} $$ norms of the closed-loop system.

The LMI: LMI for mixed $$H_{2}/H_{\infty}$$
$$ \begin{align} \text{min} \; \quad t:&\\ \end{align} $$
 * S(P,K) ||_{H_{2}}^{2} + || S(P,K) ||_{H_{\infty}}^{2}

Implementation
A link to Matlab codes for this problem in the Github repository:


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LMIs in Control/pages/LMI for Decentralized Feedback Control

LMI for Decentralized Feedback Control

In large-scale systems like a multi-agent robotic system, national economies, or chemical refineries, an actuator should act based on local information, which necessitates a decentralized or distributed control strategy. In a decentralized control framework, the controllers are distributed and each controller has only access to a subset of local measurements. We describe LMI formulations for a general decentralized control framework and then provide an illustrative example of a decentralized control design.

The System
In a decentralized controller design, the state feedback controller $$ u = Kx$$ can be divided into $$n$$ sub-controllers $$u_{i} = K_{i}x_{i}, \quad i=1,2, ..., n$$.

The Data
A general state space representation of a linear time-invariant system is as follows:

$$ \begin{align} &\dot{x} = Ax + Bu \\ &y = Cx + Du \end{align} $$

where $$x$$ is a $$n\times n$$ vector of state variables, $$B$$ is the input matrix, $$C$$ is the output matrix, and $$D$$ is called the feedforward matrix. We assume that all the four matrices, $$A$$, $$B$$, $$C$$, and $$D$$ are given.

The Optimization Problem
We aim to solve the $$H_{\infty}$$-optimal full-state feedback control problem using a controller $$ u = Kx$$.

In a decentralized fashion, the control input $$u$$ can be divided into sub-controllers $$ u_{1}, u_{2}, ..., u_{j} $$ and can be written as $$ u = [u_{1} \quad u_{2} \quad ... \quad u_{j}]_{1\times n}^{\text{T}}$$.

For instance, let $$ j = 3$$ and $$n=6$$. Thus, there are three control inputs $$ u_{1}$$, $$u_{2}$$, and $$u_{3}$$. We also assume that u_{1} only depends on the first and the second states, while $$u_{2}$$ and $$ u_{3} $$ only depend on thrid to sixth states. For this example, the controller gain matrix can be described by:

$$ K = \begin{bmatrix} k_{1} & k_{2} & 0 & 0 & 0 & 0 \\ 0    & 0     & k_{3} & k_{4} & k_{5} & k_{6} \\ 0    & 0     & k_{7} & k_{8} & k_{9} & k_{10} \\ \end{bmatrix} $$

Thus, the decentralized controller gain consists of sub-matrices of gains.

The LMI: LMI for decentralized feedback controller
The mathematical description of the LMI formulation for a decentralised optimal full-state feedback controller can be described by:

$$ \begin{align} & \text{min} \quad \gamma\\ & \begin{bmatrix} YA^{\text{T}} + AY + Z^{\text{T}}B_{2}^{\text{T}} + B_{2}Z & *^{T} & *^{T} \\ B_{1}^{T} & -\gamma I & *^{T} \\ YC_{1}^{T} + Z^{T}D_{12} & D_{11} & -\gamma I \end{bmatrix} \end{align} $$

where $$Y > 0$$ is a positive definite matrix and $$ Z $$ such that the aforemtntioned constraints in LMIs are satisfied.

Conclusion:
The controller gain matrix is defined as:

$$ K = \begin{bmatrix} 0 & 0 \\ 0 & F \end{bmatrix} $$

where $$F$$ can be found after solving the LMIs and obtaining the variables matrices $$Y$$ and $$Z$$. Thus,

$$ F = ZY^{-1}$$.

Implementation
A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI_for_decentralized_feedback_controller/tree/master

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The Story of Rhodesia

This book is intended to provide information about Rhodesia. It will give information about the history and legacy of this storied nation.

Part 1: Pre-Colonial History

 * 1) /Kingdom of Mapungubwe/ Southern Rhodesia (1909) (14804694073).jpg
 * 2) /Kingdom of Zimbabwe/ Smith Dinner.jpg
 * 3) /Kingdom of Mutapa/ British South Africa Police.jpg
 * 4) /Kingdom of Batua/
 * 5) /Rozwi Empire/ Coat of arms of Rhodesia.svg
 * 6) /Matabeleland (Ndebele Kingdom)/

Part 2: Company Rule in Rhodesia

 * 1) /Rhodes’s Dream/ Flag map of Rhodesia (1968–1979).svg
 * 2) /Rudd Concession & Royal Charter/
 * 3) /Territorial Expansion/
 * 4) /Matabele Wars/
 * 5) /Administration & the Name Rhodesia/
 * 6) /Development/
 * 7) /Military & Involvement in the second Boers War, WWI/

Part 3: Colonial History of Southern Rhodesia

 * 1) /1922 Southern Rhodesian government referendum & End of Company Rule/
 * 2) /Colony of Southern Rhodesia before WWII/
 * 3) /Involvement in WWII/
 * 4) /Colony of Southern Rhodesia Post WWII/
 * 5) /Central African Federation & Return to “Rhodesia”/

Part 4: Rhodesia Under U.D.I.

 * 1) /Rhodesia's Unilateral Declaration of Independence/
 * 2) /Post-U.D.I. Rhodesia/
 * 3) /Zimbabwe-Rhodesia/
 * 4) /Lancaster House Agreement/
 * 5) /British Dependency & End of Rhodesia/
 * 6) /Legacy of Rhodesia/

End Notes

 * 1) /Authors and Contributors/
 * 2) /Important Notes/
 * 3) /More resources about Rhodesia/

title: The Story of Rhodesia


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Visual Basic for Applications/Array Data To WorkSheet (1D or 2D)

Summary
This MS Excel VBA code listing transfers data from a one or two dimensional array to a worksheet.

Code Notes

 * Arr1Dor2DtoWorksheet transfers data from an array to a specified worksheet, and at a specified location within it. It takes one-dimensional or two-dimensional arrays, and is able to distinguish between them, prior to the transfer.   Non-array inputs are detected as are arrays that are not allocated. One-dimensional arrays are transferred into a sheet row in all cases.   Two-dimensional arrays are displayed in the same row and column shape as in the array.   There are no facilities in the procedure to transpose data, but procedures exist elsewhere in this series for that purpose.

The VBA Module

 * Copy the entire code listing into a VBA standard module and run the top procedure. Save the worksheet as type .xlsm.  Comment and de-comment lines in the top procedure and adjust parameters to test the main procedure.


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The Story of Rhodesia/Company Rule in Rhodesia

The British South Africa Company's administration of what became Rhodesia was chartered in 1889 by Queen Victoria of the United Kingdom, and began with the Pioneer Column's march north-east to Mashonaland in 1890. Empowered by its charter to acquire, govern and develop the area north of the Transvaal in southern Africa, the Company, headed by Cecil Rhodes, raised its own armed forces and carved out a huge bloc of territory through treaties, concessions and occasional military action, most prominently overcoming the Matabele army in the First and Second Matabele Wars of the 1890s. By the turn of the century, Rhodes's Company held a vast, land-locked country, bisected by the Zambezi river. It officially named this land Rhodesia in 1895, and ran it until the early 1920s. The area south of the Zambezi became Southern Rhodesia, while that to the north became North-Western and North-Eastern Rhodesia, which were joined in 1911 to form Northern Rhodesia. Within Northern Rhodesia, there was a separate Kingdom called Barotseland which later became a British protectorate alongside other territories under the British sphere of influence. Each territory was administered separately, with an administrator heading each territorial legislature. In Southern Rhodesia, which attracted the most white immigrants and developed fastest, a legislative council was established in 1898. This comprised a blend of Company-nominated officials and elected members, with the numbers of each fluctuating over time. Partially motivated by Rhodes's dream of a Cape to Cairo Railway, railway and telegraph lines were laid across previously barren Rhodesia with great speed, linking South Africa to the Belgian Congo's southern Katanga province by 1910. The British South Africa Police, responsible for law enforcement in Southern Rhodesia, was established in 1896. A number of police forces north of the river amalgamated to form the Northern Rhodesia Police in 1911. Northern and Southern Rhodesians fought alongside the British in the Second Boer War and the First World War; about 40% of Southern Rhodesian white men fought in the latter, mostly on the Western Front in Europe. Black soldiers served in East Africa with the Rhodesia Native Regiment. As the number of elected members in the Legislative Council rose, power in Southern Rhodesia gradually transferred from complete Company rule to effective self-government by the growing number of white settlers. In a 1922 referendum, Southern Rhodesians chose responsible government within the British Empire over incorporation into the Union of South Africa. The Company's charter was duly revoked by Whitehall in 1923, and Southern Rhodesia became a self-governing colony of Britain in October that year. Northern Rhodesia became a directly-run British protectorate in April 1924.

Rhodes's dream
Amid the Scramble for Africa during the 1880s, the South African-based businessman and politician Cecil Rhodes envisioned the annexation to the British Empire of a bloc of territory connecting the Cape of Good Hope and Cairo—respectively at the southern and northern tips of Africa—and the concurrent construction of a line of rail linking the two. On geopolitical maps, British territories were generally marked in red or pink, so this concept became known as the "Cape to Cairo red line". In the immediate vicinity of the Cape, this ambition was challenged by the presence of independent states to the north-east of Britain's Cape Colony: there were several Boer Republics, and to the north of these was the Kingdom of Matabeleland, ruled by Lobengula. Having secured the Rudd Concession on mining rights from Lobengula in October 1888, Rhodes and his British South Africa Company were granted a Royal Charter by Queen Victoria in October 1889. The Company was empowered under this charter to trade with local rulers, form banks, own and manage land, and raise and run a police force. In return for these rights, the Company would govern and develop any territory it acquired, while respecting laws enacted by extant African rulers, and upholding free trade within its borders.

North to the Zambezi; territorial rivalry with Portugal
The projected Company sphere was initially Matabeleland and its immediate neighbours between the Limpopo River and the Zambezi. Portugal's colonies in Angola and Mozambique, coastal territories respectively to the west and east of this general area, were over three centuries old, and Lisbon's alliance with Britain formally dated back to the 1386 Treaty of Windsor. However, the exceedingly lethargic pace of local Portuguese colonisation and development was such that even in the 1880s, Portugal's dominions in Mozambique comprised only a few scattered ports, harbours and plantations, all of which were administered from the island of Mozambique, just north of the Mozambique Channel. Angola differed little, with gigantic tracts of hinterland coming under the largely nominal purview of Portugal's modest colony on the coast.

Rhodes quietly planned to annex some of Mozambique into the Company domain so he could establish a major port at the mouth of the Pungwe River. He thought this might make an ideal sea outlet for his proposed settlement in Mashonaland, the area directly to Matabeleland's north-east where Lobengula held dominion over many Mashona chiefs. Rhodes believed that the Portuguese claim to Mozambique was tenuous enough that he could win much of it without provoking major ire: "the occupation of the Portuguese even along the coast line is in most places merely a paper one," he wrote to Whitehall in late 1889, "and if this has not been recognised by international agreement I think it might be left open." But contrary to Rhodes's opinion, general consensus at the Berlin Conference of 1884–85 had made Portugal's hold over the Mozambican coastline very secure. The Portuguese had expanded inland during the late 1880s, creating Manicaland in the eastern Mashona country. They founded Beira, a port on Rhodes's proposed Pungwe site, in 1890. Portugal issued the so-called "Pink Map" around this time, laying claim to the very corridor of land between Angola and Mozambique that Rhodes desired. The British government issued a firm ultimatum against the Portuguese claims in January 1890; Lisbon swiftly acquiesced and left the area open for the Company's drive north.

Pioneer Column
The Pioneer Column, initially comprising about 100 volunteers referred to as "pioneers", was raised by the Company during 1890. Led by Major Frank Johnson, a 23-year-old adventurer, the column was designed by the Company to be the instrument by which it would not only acquire Mashonaland, but also begin its development. Men from a wide variety of backgrounds therefore filled its ranks; according to one member, "prospectors predominate, but nearly every trade and profession under the sun is represented ... one troop is called the gentlemanly troop because the majority in it are brokers". Most of the pioneers self-identified as South African rather than British, and many of them were Afrikaners. At Rhodes's insistence, several sons of the Cape Colony's leading families were also included. Each pioneer was promised 3000 acre of land and 15 mining claims in return for his service.

Lobengula gave his approval to the ostensibly non-military expedition, but many of his izinDuna (advisors) were fiercely against it, seeing it as an appropriation of Matabele territory. Wary that one or more of these izinDuna might turn rogue and attack the pioneers, the Company gradually enlarged the escorting detachment of British South Africa Company's Police until it numbered 500 men, headed by Lieutenant-Colonel Edward Pennefather, an officer seconded from the British Army. To Johnson's chagrin, the Imperial officer was also given ultimate command of the column.

The column was to move roughly east from Macloutsie, a small camp near the border of Matabeleland and Bechuanaland, and then march north to its destination. It would build a road as it went, founding minor forts along the way, and establish a major town in Mashonaland, whereupon the pioneers would be released to farm, prospect and trade. Frederick Courteney Selous, a famed hunter with intimate knowledge of Mashonaland, was made the column's "intelligence officer". He chose as its intended destination an open patch of veld he had discovered during his travels, which he called Mount Hampden. The proposed site was about 650 km to the north-east of Macloutsie. The column departed on 28 June 1890, and on 11 July crossed the Tuli River into Matabeleland. Its first settlement, Fort Tuli, was inaugurated near the riverbank. Though Johnson was nominally in command of the pioneers, he was generally seen as untried and green when contrasted with the experienced, respected authority of Selous. According to most contemporary accounts, Selous was effectively in control. The officers were outwardly harmonious, but Johnson was privately troubled by pangs of jealousy.

The column was initially accompanied by about 200 Ngwato provided by the Tswana chief Khama, who had firmly aligned his country with Britain. The Ngwato provided much assistance in building the new road, but animosity soon developed between them and the whites, principally because the latter were not used to treating blacks as equals. By mutual consent, the Ngwato returned home. As the column continued its march north, Selous split off with a small section and headed east to challenge the Portuguese in Manicaland. Pennefather and Johnson continued at the head of the main force and founded Fort Victoria, Fort Charter, and, on 12 September, Fort Salisbury.

The site of Salisbury was a naturally flat and marshy meadow, bounded by a rough kopje. The pioneers were about 15 km short of Mount Hampden, but Pennefather climbed the kopje, surveyed the open veld and insisted that it was "magnificent", so they need go no further. He reported back to Rhodes in triumphant tones: "Site selected ... All well. Magnificent country. Natives pleased to see us". On the morning of 13 September 1890, about 10:00, the officers and men of the Pioneer Column paraded atop the kopje before an improvised flagstaff. With the column standing to attention, Lieutenant Edward Tyndale-Biscoe hoisted the Union Jack, a 21-gun salute was fired, and three cheers were given for the Queen. Work then began on the fort, which was completed by the end of September. The Pioneer Column was then disbanded.

Eastern skirmishes with Portugal
Instructed by Rhodes to hurry east, Selous met with the Manica chief, Mtassa, on 14 September 1890, and agreed with him a concession whereby Mtassa promised not to ally with any other foreign power, and granted the Company exclusive rights to mine within his territory, as well as to build railways, bridges, canals and other projects typical of colonial settlement. In return, the Company gave Mtassa rifles and other equipment (worth £100 in total), and a promise of protection against attacks by the Portuguese or the neighbouring Shangaan (or Tsonga) people. Portugal despatched a small force to militarily overwhelm Mtassa and reclaim the area in early November 1890.

Captain Patrick Forbes rode to Mtassa's aid from Salisbury, quickly routed the Portuguese, and thereupon advanced all the way to Beira, securing further concessions from local chiefs along the way to build a railroad. Tense negotiations between Britain and Portugal followed, finally concluding with a treaty signed in Lisbon on 11 June 1891: prominent among the numerous territorial revisions was the integration of Manicaland into the Company domain as part of Mashonaland. Britain concurrently recognised Portugal's authority over the entire Mozambican coast, putting an end to Rhodes's designs for a Company port on the Mozambique Channel.

North to Katanga
Representatives of the Company crossed the Zambezi to venture even further north. The Shire Highlands of Nyasaland, far to the north-east on the banks of Lake Nyasa, had been settled by a modest number of British missionaries for about a decade, and in Barotseland, to the north-west, King Lewanika hosted François Coillard of the Paris Evangelical Missionary Society. Rhodes sent Elliot Lochner north to negotiate with Lewanika in late 1889, and in June 1890 the king signed the Lochner Concession, which gave the Company rights to mine, trade and build railways in Barotseland in return for British protection over his domain from external threats, and a British resident in Lewandika's court at Lealui. The British government thereupon chartered the Company to defend Barotseland, as well as all country to the east as far as Nyasaland, and to the north as far as Lake Tanganyika and Katanga.

A country where resources were scarce and tropical disease was rampant, Barotseland offered comparatively few economic opportunities for the Company and little incentive for white immigration. The main objective of Lochner's expedition was all along to clear a path towards Katanga, a mineral-rich area further north, where Msiri ruled the Yeke Kingdom. Katanga was also coveted by the owner of the Congo Free State, King Leopold II of the Belgians, whose representatives Rhodes hoped to beat there. "I want you to get Msiri's," Rhodes told one of his agents, Joseph Thomson; "I mean Katanga ... You must go and get Katanga."

The efforts of Thomson and Alfred Sharpe to secure a Company concession over the area were furiously rebuffed by Msiri in late 1890, and ultimately foiled by the 1891–92 Stairs Expedition—a multinational force in Leopold's service, led by a Canadian British Army officer, Captain William Grant Stairs—which violently clashed with the obstreperous Msiri, and eventually shot him dead when an attempt to arrest him turned into a firefight. Msiri had been in the habit of displaying the heads of his enemies atop poles outside his boma (enclosure), and the expedition's men hoisted his own head alongside them in an attempt to strike fear into the locals. The country promptly capitulated to the Free State, ending the Company's expansion north.

The Company did little to fulfil Britain's obligations to Lewandika; having failed in Katanga, its hierarchy saw Barotseland as an unworkable inconvenience that might later be exchanged with Portugal. Whitehall, by contrast, regarded Lewandika's domain as an important buffer against further Portuguese claims inland. Neither the Company nor the British government proved eager to take practical responsibility for the Barotse; in 1894, while informing Britain of his willingness to administer on Whitehall's behalf north of the Zambezi, Rhodes stressed that he would not take Barotseland. The promised British resident at Lealui remained conspicuously absent, despite Lewandika's repeated enquiries, until the appointment by Rhodes of Robert Thorne Coryndon in 1897.

Matabele Wars; the First Chimurenga

Though the Company made good on most of the pledges it had made to local leaders in Matabeleland, the assent of Lobengula and other less prominent figures, particularly regarding mining rights, was often evaded, misrepresented or simply ignored. Company officials also demanded that Lobengula cease the habitual raids on Mashona villages by Matabele impis (regiments). Enraged by what he perceived as slights against his authority, Lobengula made war on Mashonaland in 1893. Matabele warriors began the wholesale slaughter of Mashonas near Fort Victoria in July that year. The Company organised an indaba (tribal conference) to try to end the conflict, but this failed. The First Matabele War had begun.

The Company moved on Lobengula during October and early November 1893, and used the inexorable firepower of its Maxim machine guns to crush attacks by the far larger Matabele army as it rode south-west. As the whites approached his royal town at Bulawayo, Lobengula fled, torching it as he went. Company troops were despatched to bring him back, and the resultant pursuit north ended with the ambush and annihilation of the 34-man Shangani Patrol by the remnants of Lobengula's army on 4 December 1893. The king died from smallpox while on his way north in January 1894, and his izinDuna made peace with the Company soon after. Bulawayo was rebuilt as a Company-run city atop the ruins of the former Matabele capital. Rhodes subsequently funded education for three of Lobengula's sons.

The Matabele rose again in 1896 at the behest of Mlimo, a spiritualist leader who was revered as a god by much of the local populace. The botched Jameson Raid on the Transvaal at the end of 1895 had severely depleted the Company's garrison in Matabeleland, and the settlers in Bulawayo had little to defend themselves with. Mlimo convinced his followers that the white man was responsible for all their ills—hut tax, forced work, locusts, rinderpest, drought and so on—and that he and other tribal prophets could ensure the success of a massed rebellion by turning the settlers' bullets into water. This uprising, called the Second Matabele War or the First Chimurenga (liberation war), began in March 1896. Over the following three months the Matabele killed hundreds of isolated settlers and their families, but Bulawayo itself held out. When the Company mustered reinforcements from South Africa, the Matabele retreated into the Matopos Hills; here Frederick Russell Burnham, an American scout long in Company service, discovered and killed Mlimo in June 1896.

Starting in August 1896, Rhodes personally played a key role in ending the Matabele insurgency. With one of the widows of Mzilikazi (Lobengula's father) acting as a go-between, the Company and the rebel izinDuna arranged an indaba for 21 August: the izinDuna agreed to meet Rhodes and three companions in the Matopos Hills. At this meeting, the insurgents vehemently protested against their prior treatment under Company rule, prompting Rhodes to walk away from the other whites and to sit among the Matabele instead, apparently intending to symbolically demonstrate empathy and a spirit of reconciliation. He told the Matabele that he was on their side, and that he would personally ensure the non-recurrence of any abuses. The izinDuna would be fully restored to the status they had held under Lobengula, he said, and there would be no retribution against those who had taken part in the Chimurenga. After four hours, it was agreed to continue the talks. Bitterness endured among some of the rebels, but three further indabas progressed well, and the Matabele rising ended amicably in October 1896.

Around the same time, Mashona svikiro (spiritualist prophets), most prominently Mukwati, Kaguvi and Nehanda Nyakasikana, instigated their own Chimurenga in Mashonaland. The Company forcibly put down this uprising during 1897, and afterwards took significant steps to demilitarise the tribal population and improve relations with the local chiefs. Small pockets of Mashona unrest continued sporadically until 1903, but peace endured in Matabeleland. Including both theatres, the Chimurenga has been estimated to have taken around 8,450 lives; roughly 8,000 blacks died, and about 450 whites, of whom 372 were locally based settlers. The rest were soldiers in Company or British service from outside Rhodesia.

The name "Rhodesia"
The Company initially referred to each territory it acquired by its respective name—Mashonaland, Matabeleland and so on—but there was no official term for them collectively. Rhodes preferred the name "Zambesia" while Leander Starr Jameson proposed "Charterland". Many of the first settlers instead called their new home "Rhodesia", after Rhodes; this was common enough usage by 1891 for it to be used in newspapers. In 1892 it was used in the name of Salisbury's first newspaper, The Rhodesia Herald. The Company officially adopted the name Rhodesia in 1895, and three years later the UK government followed suit. "It is not clear why the name should have been pronounced with the emphasis on the second rather than the first syllable," the historian Robert Blake comments, "but this appears to have been the custom from the beginning and it never changed."

Administrative divisions and centres
Matabeleland and Mashonaland, both of which lay south of the Zambezi, were officially referred to collectively as "Southern Rhodesia" from 1898, and formally united under that name in 1901. Meanwhile, the areas to the river's north became North-Western and North-Eastern Rhodesia, which were governed separately, and amalgamated in 1911 to form Northern Rhodesia.

The overall centre of Company administration was Salisbury, which was also the Southern Rhodesian capital. The administrative centre in North-Eastern Rhodesia was Fort Jameson, while in North-Western Rhodesia the capital was Kalomo initially, and Livingstone from 1907. Livingstone became the capital of Northern Rhodesia when the two northern territories joined in 1911, and remained so at the end of Company rule.

Administrative posts, politics and legislature
The head of government in each territory under Company rule was in effect a regional administrator appointed by the Company. In Southern Rhodesia, a ten-man Legislative Council first sat in 1899, originally made up of the administrator himself, five other members nominated by the Company, and four elected by registered voters. The number of elected members rose gradually under Company rule until they numbered 13 in 1920, sitting alongside the administrator and six other Company officials in the 20-member Legislative Council. The Company's Royal Charter was originally due to run out in October 1914, but it was renewed for a further ten years in 1915.

In Northern Rhodesia, administration was entirely undertaken by the Company until 1917, when an Advisory Council was introduced, comprising five elected members. This council did little to lighten the Company's administrative burden north of the river, but endured until the end of Company rule.

Railways and the telegraph
Chief among the endeavours pursued by the Company during its early years were the construction of railroads and telegraph wires across the territory it governed. These respective arteries of transport and communication, vital both for the successful development of the new country and for the realisation of Rhodes's Cape to Cairo dream, were laid across the previously bare Rhodesian landscape with great speed. Strategically planned, the railways were not intended or expected to turn a profit during their early years; their construction was largely subsidised by the Company. The telegraph line from Mafeking in South Africa reached Salisbury—one third of the way from Cape Town to Cairo—in February 1892. Just under six years later, in December 1897, the Bechuanaland railway from Vryburg reached Bulawayo, making it possible to travel between the Cape and Rhodesia by train.

A narrow gauge railway towards Salisbury from the Mozambican port of Beira was begun by the Portuguese in 1892, and reached the Rhodesian border at Umtali after six years. Umtali and Salisbury were linked in 1899, on a different track gauge; the gauges between Beira and Salisbury were regularised the following year. The Second Boer War then restricted the further extension of the line from Vryburg, but the completion of the Beira–Salisbury railway allowed the importation of materials. Salisbury was connected to Bulawayo and the Cape in 1902. The Vryburg–Bulawayo railway was meanwhile extended up to the Zambezi, and across when the Victoria Falls Bridge opened in 1905. Continuing through North-Western Rhodesia, the railway reached Élisabethville in Katanga—by this time part of the Belgian Congo—in 1910.

Agriculture and land distribution: the rise of Rhodesian tobacco

The Company originally hoped that gold prospecting between the Limpopo and Zambezi rivers would reveal mineral deposits comparable to those of the South African Rand, and indeed acquired its charter in part because its founders convinced Whitehall that a "second Rand" would be found and exploited in what would become Southern Rhodesia, thereby providing more than enough capital to develop the territory without help from London. Though much gold was discovered during the 1890s, these grand expectations were not met. The Company resolved after about a decade that it could not financially sustain its domain through gold mining alone, and therefore shifted its priority to the development of white agriculture.

To maximise the potential of new, white-run farms, the Company launched a wide-scale land settlement programme for white settlers. As part of this drive it reorganised the geographical distribution of native reserve areas, moving the reserves and often reducing them in size where the land was of particularly high quality. To ensure that the white farmers would retain the reliable access to markets that the nascent railway network provided, tribal reserve boundaries in various relevant places were redrawn by the Company to place the railway lines outside. The new hut taxes concurrently compelled black peasants to find paid work, which could be found in the new agricultural industry, though most tribesmen were reluctant to abandon their traditional lifestyles in favour of the capitalist labour market. Managers at farms and mines often had great trouble sourcing sufficient manpower.

Tobacco, initially just one of several crops earmarked for wholesale production, soon emerged as Southern Rhodesia's most prominent agricultural product, though its early development was far from stable: aside from the climactic uncertainties of the unfamiliar country and the mercurial quality of the product, the early industry was cursed by a debilitating boom and bust cycle that continued well into the 1920s. All the same, tobacco endured as the territory's staple crop, while the growers came to dominate Southern Rhodesian politics, holding a majority in the Legislative Council from 1911. Holding considerable political and economic power up to the end of Company rule in 1923, the Southern Rhodesian tobacco industry retained its prominent position for decades afterwards.

Immigration and economic performance
White immigration to the Company realm was initially modest, but intensified during the 1900s and early-1910s, particularly south of the Zambezi. The economic slump in the Cape following the Second Boer War motivated many White South Africans to move to Southern Rhodesia, and from about 1907 the Company's land settlement programme encouraged more immigrants to stay for good. The Southern Rhodesian mining and farming industries advanced considerably during this period; Southern Rhodesia's annual gold output grew in worth from £610,389 in 1901 to £2,526,007 in 1908. The territory first balanced revenue and expenditure in 1912. There were 12,586 w Whites in Southern Rhodesia in 1904, and 23,606 in 1911; in 1927, four years after the end of Company rule, the Black and White populations in Southern Rhodesia were respectively 922,000 and 38,200.

The White population north of the river was far smaller, with only about 3,000 settlers spread across the of Northern Rhodesia. In the same area, there were roughly 1,000,000 Black people. The Whites in Northern Rhodesia were primarily concentrated in the far west, along the railway line between Bulawayo and Élisabethville in the Belgian Congo. A community of about 250 lived in the vicinity of Fort Jameson near the eastern border. In between were vast swathes of largely uninhabited bush, which lacked roads, railways and telegraph lines, making communication between the two White communities very difficult. The amalgamation of North-Western and North-Eastern Rhodesia in 1911 did little to improve the situation. Northern Rhodesia suffered as the result of its artificial nature—the country was not homogeneous in terms of population, history or geography—and unlike Southern Rhodesia, it never turned a profit or became self-supporting. During 1921 alone, the Company's administration of Northern Rhodesia lost it more than £1,250,000.

Evolution of the Southern Rhodesian police
In line with the terms of its royal charter, the Company formed the British South Africa Company's Police (BSACP) in late 1889. A paramilitary, mounted infantry force, the BSACP initially boasted 650 men, but it proved so expensive to maintain that it was reduced to only 40 in 1892. This rump force was renamed the Mashonaland Mounted Police. With its size regularly fluctuating, it and other more irregular units—prominently the Bulawayo Field Force, including figures such as Selous and Burnham as commanders—proceeded to play a central role in the two Matabele Wars of the 1890s.

Following the formation of the Matabeleland Mounted Police in 1895 with 150 members, it and the Mashonaland force were collectively referred to as the Rhodesia Mounted Police. This was run directly by the Company until 1896, when it was reorganised into an independent entity called the British South Africa Police (BSAP). The word "Rhodesia" was omitted at the insistence of the British Colonial Secretary, Joseph Chamberlain, because Britain did not yet formally consider that the country's name, despite the Company's official adoption of it the year before. This anomaly was resolved in 1898 but the BSAP name remained.

Police forces in Southern Rhodesia were initially all-white, but this changed over time: the Native Police Force, first raised in May 1895, was made up entirely of Matabele non-commissioned officers and men, many of whom were veterans of Lobengula's impis. Its 200 members, of whom 50 were posted to Mashonaland, were trained in the Western manner, drilling and learning marksmanship. They were held in high regard by their white officers for their formidable soldiering ability, but they became hugely unpopular among the black civilian population for their perceived arrogance and abuse of the law they were supposed to uphold. At the 1896 indaba with Rhodes in the Matopos Hills, the Matabele chief Somabhulana complained at length about the native police, saying they did not respect the traditional tribal structure and generally oppressed the populace, reportedly raping women on a regular basis. The parties agreed to abolish the native police in Matabeleland, and Rhodes promised not to reintroduce it.

On its reconstitution in 1896, the BSAP was authorised to recruit 600 officers and men in Matabeleland—all of whom were white because of Rhodes's promise at the indaba—and 680 in Mashonaland, of whom 100 should be black. In practice, the "Native Contingent" in Mashonaland numbered 120. The BSAP thereafter operated alongside the Southern Rhodesian Constabulary (SRC), a town police force covering Salisbury, Bulawayo, Fort Victoria, Gwelo and Umtali. The constabulary was far smaller than the BSAP—in 1898 it included only 156 officers and men, black and white—and it was run by local magistrates, as opposed to the paramilitary BSAP, which had a military-style structure.

The commissioned ranks in the BSAP were entirely white, but the number of black constables in its ranks gradually rose, with many being recruited abroad. This kind of recruitment was not uncommon in colonial Africa, as many white officials of the day believed that blacks who policed their own communities were easily corruptible, and often not inclined to properly ensure the payment of colonial institutions such as hut tax. In Southern Rhodesia, many constables came from Barotseland, Zululand and Zanzibar. Locally sourced black policemen were officially reintroduced to Matabeleland in 1904; that year the force nominally contained 550 whites and 500 blacks. The SRC was merged into the BSAP in 1909, putting law enforcement in Southern Rhodesia into the hands of a single authority for the first time. Following the end of Company rule in 1923, the BSAP endured as Southern Rhodesia's police force.

Policing north of the Zambezi
North-Eastern Rhodesia was initially policed by locally recruited rank-and-filers, led by white officers from south of the river; the first force was raised in 1896. During its early years it busied itself eliminating the slave trade, in which foreign traders, mostly Arabs, captured villagers for sale as slaves overseas. A more regular police force was then introduced by the Company in each of the northern territories. Because there were so few white immigrants to North-Eastern Rhodesia—and because most of them were men of the church or of business rather than potential recruits—the North-Eastern Rhodesia Constabulary was almost exclusively black, including all of its non-commissioned officers.

North-Western Rhodesia attracted more white immigrants than its north-eastern counterpart, and its police force initially comprised an all-white detachment of Company police seconded from Southern Rhodesia. The unit proved expensive to maintain, however, and many of its constables fell victim to the unfamiliar tropical diseases of Barotseland. Local black constables were introduced in 1900 after the Company unsuccessfully attempted to recruit more whites. In 1902, the Barotse Native Police was formed, with Bemba, Ngoni and Ila recruits making up most of the ranks. Minor forces of white policemen were formed in the towns north of the Zambezi.

After North-Western and North-Eastern Rhodesia merged into Northern Rhodesia in 1911, the police forces were amalgamated as the Northern Rhodesia Police (NRP). Like the BSAP, the NRP was effectively a paramilitary rather than civil organisation, with its armed constables receiving martial training under military command. Because they were not trained in the civil manner considered normal in a more developed country, most of them were illiterate. The main purpose of the force during the early 1910s was not to police Northern Rhodesia's towns, but rather to prevent and combat potential uprisings. The constables were also considered suitable for use as soldiers in the bush. It was not a large force; just before the outbreak of the World War I in 1914, it had only 800 personnel.

Second Boer War (1899–1902)
The BSAP served in the Second Boer War of 1899–1902 in its paramilitary capacity, with the newly formed Rhodesia Regiment also taking part, drawing most of its men from the Southern Rhodesia Volunteers. Rhodesia contributed approximately 1,000 men in all, about 20% of the white male population at the time. Rhodesia contributed part of the British garrison at the Battle of the Elands River in August 1900, during which a 500-man force made up principally of Australians and Rhodesians held off attacks from a far larger Boer army under General Koos de la Rey, and repeatedly refused offers of safe passage in return for surrender. Captain "Sandy" Butters, the Rhodesian commanding officer, encouraged his men with shouts towards the Boers that "Rhodesians never surrender!" The Rhodesia Regiment was disbanded later that year, shortly after the relief of Mafeking.

World War I (1914–18)
With its fledgling White population largely characterised by youth, hardiness and Imperial patriotism, Southern Rhodesia proved a bountiful source of volunteers during the First World War, in which about 40% of Southern Rhodesian White males of service age fought. The majority of Southern Rhodesian personnel served with British, South African and other regiments on the Western Front (in Belgium and France). The Company raised exclusively Rhodesian units for African service.

Following the start of the war in August 1914, the Rhodesia Regiment was reformed in October, initially comprising 20 officers and 500 men, mostly Southern Rhodesians. This force, which became called the 1st Rhodesia Regiment, was sent to the Cape to fight alongside the South Africans in South-West Africa. The 2nd Rhodesia Regiment, raised a month later, was sent to the East African Front. Following the end of the South-West African Campaign in 1915, the 1st Rhodesia Regiment was dissolved; most of its men travelled to England to volunteer for the Western Front, while others joined the 2nd in East Africa. Boasting an effective strength of about 800 for the rest of its tour of duty, the 2nd Rhodesia Regiment returned home in April 1917, and disbanded in October.

Influenced by South Africa's reluctance to use Black soldiers in what was widely considered a "White man's war", Southern Rhodesia did not recruit Blacks in large numbers until 1916, when the number of potential White volunteers not already in uniform became too small to merit further drafts. The Rhodesia Native Regiment (RNR) was formed in that year to join the 2nd Rhodesia Regiment in East Africa, and included 2,507 men by 1918. Organisers expected that most Black volunteers would come from the Matabele people, famous for its martial tradition, and therefore originally named the unit the "Matabele Regiment"; however, when the ranks proved to be ethnically diverse, the name was changed. Led by White officers, the Black soldiers served with distinction in East Africa, soon becoming regarded as formidable bush fighters. Pitted against the German Generalmajor Paul von Lettow-Vorbeck—who was mounting a successful guerrilla campaign against the far larger Allied forces—they remained in East Africa for the rest of the war, returning home only in December 1918, soon after von Lettow-Vorbeck surrendered at Abercorn in Northern Rhodesia on 25 November. The RNR was thereupon dissolved.

1922 Southern Rhodesian government referendum

In 1917, the Responsible Government Association (RGA) was formed. This party sought self-government for Southern Rhodesia within the Empire, just as Britain had previously granted "responsible government" to its colonies in Australia, Canada, New Zealand and South Africa as a precursor to full dominion status. Sir Charles Coghlan, a lawyer based in Bulawayo, led the RGA from 1919. The RGA opposed the proposed integration of Southern Rhodesia into the Union of South Africa, which had been formed in 1910 by the South Africa Act 1909, Section 150 of which explicitly provisioned for the accession of territories governed by the British South Africa Company. The Company originally stood against Southern Rhodesia's addition, fearing the territory might become dominated by Afrikaners, but abruptly changed its stance when, in 1918, the Privy Council in London ruled that unalienated land in the Rhodesias belonged to the British Crown rather than to the Company. This removed the longstanding stream of Company revenue created by the sale of land.

The loss of this source of income hampered the Company's ability to pay dividends to its shareholders, and caused its development of the Rhodesias to slow. The Company now backed Southern Rhodesia's incorporation into South Africa, hoping its membership in the union could help solve both problems. However, this prospect proved largely unpopular in Southern Rhodesia, where most of the settlers wanted self-government rather than rule from Pretoria, and came to vote for the RGA in large numbers. In the 1920 Legislative Council election, the RGA won ten of the 13 seats contested. A referendum on the colony's future was held on 27 October 1922—at the suggestion of Winston Churchill, then Britain's Colonial Secretary, continuing the initiative of his preprocessor Lord Milner—and responsible government won the day. Just under 60% of voters backed responsible government from a turnout of 18,810; Marandellas was the only district to favour the union option, doing so by 443 votes to 433.

Self-government vs. direct rule
Southern Rhodesia was duly annexed by the Empire on 12 September 1923, and granted full self-government on 1 October the same year. The new Southern Rhodesian government immediately purchased the land from the British Treasury for £2 million. The Company retained mineral rights in the country until 1933, when they were bought by the colonial government, also for £2 million.

The future administration of Northern Rhodesia, a proposition of little economic viability without its southern counterpart, was a burden the Company now endeavoured to rid itself of. Negotiations between the Company and the British government produced a settlement whereby the territory would become a protectorate under Whitehall, with government transferred to the Colonial Office in London, which would henceforth appoint a local governor. The Company would concurrently keep the country's mineral rights, extensive tracts of freehold property, and half the proceeds from future sales of land in what had been North-Western Rhodesia. Northern Rhodesia duly became an Imperial protectorate on 1 April 1924, with Sir Herbert Stanley installed as the inaugural governor. British South Africa Company rule in Rhodesia was thereby ended.

Notes and references

 * Notes

 * References

 * Online sources

 * Newspaper and journal articles

 * Bibliography


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Cambodian Language

 * The Cambodian or Khmer language (in Cambodian, ភាសាខ្មែរ /pʰāsākʰmær/) is one of the main Austro-Asian languages. Sanskrit and Pali have considerable influence on the language through Buddhism and Hinduism. As a result of its geographical proximity, the Cambodian in turn influenced Thai and Lao, and vice versa.

 * The Cambodian, also known as Khmer, is the official language of the Kingdom of Cambodia and almost all Cambodians speak it. Khmer is also understood by people in many bordering countries such as Thailand (in the eastern provinces of Buriram, Surin and Srisket in northern Thailand), in the Mekong Delta region in South Vietnam and in southern Laos.

 * The Cambodian is the main modern representative of the Mon-Khmer language family, which includes hundreds of related dialects dispersed throughout most of Southeast Asia. As for foreign influences in the language, the Khmer language has borrowed many words from Sankrit. With the advent of Theravada Buddhism in the early fifteenth century, Khmer began to borrow Pali's words, and continue to use Pali as an important source of neologisms today (Huffman 1970). During the period of French domination, many French words were borrowed from the language and have become part of the colloquial language, as well as in medical and technical terms. There are also a handful of loans in Chinese and Vietnamese in colloquial form.

 * Unlike Thai, Vietnamese and Lao, Khmer is not tonal and has a high percentage of disyllabic words that are derived from monosyllabic bases by prefixing and infixing.

Lessons

 * 1) /Writing System/
 * 2) /Grammar/
 * 3) /Conversation/
 * 4) /Vocabulary/
 * 5) /Reading/

title: Cambodian Language


text:
Cambodian Language/Introduction

 * The Cambodian, also known as Khmer, is the official language of the Kingdom of Cambodia and almost all Cambodians speak it. Khmer is also understood by people in many bordering countries such as Thailand (in the eastern provinces of Buriram, Surin and Srisket in northern Thailand), in the Mekong Delta region in South Vietnam and in southern Laos.

 * The Cambodian is the main modern representative of the Mon-Khmer language family, which includes hundreds of related dialects dispersed throughout most of Southeast Asia. As for foreign influences in the language, the Khmer language has borrowed many words from Sankrit. With the advent of Theravada Buddhism in the early fifteenth century, Khmer began to borrow Pali's words, and continue to use Pali as an important source of neologisms today (Huffman 1970). During the period of French domination, many French words were borrowed from the language and have become part of the colloquial language, as well as in medical and technical terms. There are also a handful of loans in Chinese and Vietnamese in colloquial form.

 * Unlike Thai, Vietnamese and Lao, Khmer is not tonal and has a high percentage of disyllabic words that are derived from monosyllabic bases by prefixing and infixing.

title: Cambodian Language/Introduction


text:
Cambodian Language/Writing System

Introduction
The Cambodian script (called Khmer letters) are all probably derived from various forms of the ancient Brahmi script of South India. The Cambodian script has symbols for thirty-three consonants, twenty-four dependent vowels, twelve independent vowels, and several diacritic symbols. Most consonants have reduced or modified forms, called sub-consonants, when they occur as the second member of a consonant cluster. Vowels may be written before, after, over, or under a consonant symbol.

Some efforts to standardize Khmer spelling have been attempted, but inconsistencies persist, and many words have more than one accepted spelling. A two-volume dictionary prepared under the direction of the Venerable Chuon Nath of the Buddhist Institute in Phnom Penh is the standard work on Khmer lexicography.

Khmer is divided into three historical stages: Old Khmer (seventh to twelfth century A.D.), Middle Khmer (twelfth to seventeenth century A.D.), and Modern Khmer (seventeenth century to the present). It is likely that Old Khmer was the language of Chenla. The language of Funan was most probably a Mon-Khmer language. The earliest inscription in Khmer, found at Angkor Borei in Takev Province south of Phnom Penh, dates from A.D. 611.

Consonants and Sub-consonants
There are thirty-three letters in the Cambodian writing system. They are arranged in five groups according to the position of the articulation, proceeding from the back to the front of the mouth, and a sixth group labeled as miscellaneous. The consonants in modern Cambodian are also divided into two categories or series. The first series is voiceless and the second series is voiced (sometimes are called light voiced and heavy voiced).

When two consonants are pronounced consecutively within a word, the second consonant's symbol is written in a special sub-consonant form which is placed below the first consonant. The sub-consonant always follows the consonant in the pronunciation. The form of the sub-consonant is in most cases a smaller version of its consonant version but some look completely different from the superscript.

The table below provides the symbol of all consonants sounds in IPA symbol with the first and second series sound, and the corresponding sub-consonant form.

Vowels
The Cambodian vowel may consist of one or a combination of elements written before, above, below, or after the initial consonant. There are 24 vowels in Khmer. Since the abstract vowel (AA) is embedded in a consonant, there are only 23 vowels shown in the table below. The pronunciation of a vowel in Khmer is determined by the series of the initial consonant that it accompanies.

The pronunciation of a vowel in Cambodian is determined by the two series of consonants (first and second series). The table below provides the symbol of all vowel sounds in IPA symbol with the first and second series sound.

Independent Vowels
Independent vowels are known as /Sra? phñ tue/ (complete vowel) because they incorporate both an initial consonant and a vowel. In the table below, independent vowel from 1 to 5 and 10 to 11 include an initial /q-/ and are listed in the official dictionary along with other words that are spelled with an initial G and the equivalent vowel. Independent vowel 6 and 7 include an initial /r/ and are listed in the official dictionary along with an initial r and the equivalent vowel. Independent vowel 8 and 9 includes an initial /l/ and are listed in the official dictionary along with an initial consonant l.

Bantaq ( ់ )
Diacritic Bantaq occurs on the top of the final consonant of a syllable and it is used to shorten the vowel of that syllable. All Khmer consonant there is an inherent vowel. The inherent vowel for first series consonant is /AA/ and second consonant is /OO/. In a syllable with invisible vowe, bantaq signalls the short inherent vowel. For example, vowel /AA/ change to /A/ after a first series consonant, and  /OO/ change to  /u/ or /u(e/ after the second series consonant. The vowel ( - a ) /aa/ followed by the Bantaq (  ' ) will pronounced shorter /a/ after the first series consonant and after the second series consonant, vowel ( -a ) /aa/ is pronounced /o(e/ or /e(e/.

Treysap ( ៊ )
Treysap is used to convert four of the first series consonants s h b and G which have no second series counterpart to the second series consonant s~ h~ b~ and G~.

Mousekatoan ( ៉ )
Mousekatoan is used to convert six of the second series consonants g Baj m y r and v which has no second series counterparts to first series consonants g" Baj"" m" y" r" and v" . Mousekatoan is used to convert a first series consonant b to b" and from which b"  has the conterpart in the second series B.

Sanyok-sanha ( ័ )
Sanyok-sanha has the same value as the (- a ). It is used in a certain words which borrowed from Pali or Sakrit. Usually, if the final consonant is silent, the words can be spelled with different way. If the sanyok-sanha plus a final y, it is pronounced (ai) in the first series and (ei) in the second series. If the s sanyok-sanha plus a final ( r ) /r/, it is pronounced ( oa).

Robaat ( ៌ )
Robaat is the reflex of an origanal /r/ in Sanskrit words. In most words, when the (robaat) occurs over a final consonant, both the consonant and the robaat are not pronounced. In some cases, the effect of the robaat is to change the vowel/OO/ to /Oe/. When the robaat appears over a medial consoant, the robaat is pronounced.

Khan ( ។ )
This is Cambodian full stop. It occurs less frequently than the full stop in English. It can be at the end of a single sentence or several sentences dealing with a single topic.

bāriyaosān ( ៕ )
Bāriyaosān is a full stop that marks the the entire end of a chapter or an entire text.

Laq (។ល។ )
Laq is used to indicate "etcetera".

Leiktoo ( ៗ )
Leiktoo is used to indicate that the word or phrase after which it occurs is to be repeated.

Ordinals
Cambodian numeral system based on five. From 6 to 9 is form by 5 + 1, 5 + 2, 5 + 3, and 5 + 4.

Number 7 is pronounced (prampii) in reading or formal speech; in mormal speech it pronounced (pramphl).

10,000 (muey mehn ) also can be counted as (dAp po).

100,000 (muey saen ) can be counted as ( dAp mehn ) or (muey rOOy po)

From 11-19 the numbers are pronounced in two different forms, the written and a colloquial.

In Cambodian numerals involving four or more symbols, a period is placed after every three symbols, counting from the right.

Cardinals
The particle ទី (tii) is used to place in front of ordinal number in order to change the ordinal number to Khmer cardinal number.

Currency
Cambodian has its own currency, called រៀល /riel/. There are the following types of bank notes or ក្រដាសប្រាក់ /kratāsbrāk/:

 * 100 riel = ១០០ រៀល
 * 200 riel = ២០០ រៀល
 * 500 riel = ៥០០ រៀល
 * 1000 riel = ១.០០០ រៀល
 * 2000 riel = ២.០០០ រៀល
 * 5000 riel = ៥.០០០ រៀល
 * 10000 riel = ១០.០០០ រៀល
 * 20000 riel = ២០.០០០ រៀល
 * 50000 riel = ៥០.០០០ រៀល
 * 100000 riel = ១០០.០០០ រៀល

title: Cambodian Language/Writing System


text:
Cambodian Language/Grammar

Personal Pronouns
One of the most difficult problems in Khmer for a foreign student is the choice of appropriate pronouns; there is a much larger number of pronouns than in English because they are context oriented, since they reflect gender, relative age, and social status of the speakers to the addressee. Thus there are no general pronouns such as "you" which are appropriate for all situation. The tables below provide the commonest Khmer personal pronoun and their explanation:

First Person

 *  ខ្ញុំ ទៅសាលារៀន
 * I go to school
 * ខ្ញុំ : khnyom
 * ទៅ : dau
 * សាលារៀន : sālārian
 *  គ្នា ដើរទៅផ្សារ  សុខចង់ទៅផ្សារជាមួយ គ្នា  ?
 * I walk to the market, Do you want to go with me Sok?
 *  យើង ចង់រៀនខ្មែរ។
 * We want to study Khmer.
 *  យើងខ្ញុំ ជានិសិត្
 * We are students.
 *  ពួកយើង រៀនខ្មែរជាមួយលោកគ្រូសំ
 * We study Khmer with teacher Sam.

Second Person

 * លោកចង់ពិសារគុយទាវទេ ? 	= Do you want to eat noodle soup?
 * លោកស្រីស្នាក់នៅឯណា ? 	= Where do you stay?
 * អ្នកឈ្មោះអី ?		= What is your name?
 * អស់លោកមកពីប្រទេសណា ? 	= Where are you all from?
 * អស់អ្នកស្នាក់នៅឯណា ? 	= Where do you all stay?

Third Person

 * ក្មេងនេះវាមកពីណា ? 		= Where is this kid come from?
 * គាត់រៀនខ្មែរនៅសាលាណា ? 		= At what school does he or she study Khmer?
 * គេចង់ទៅណា ? 			= Where is he or she want to go?
 * នាងនោះធើ្វការអីនៅស្រុកខ្មែរ ? 		= What does she do for a living in Cambodia?
 * ថ្ងៃនេះមានក្មេងច្រើនណាស់មកលេង សួននេះ។ ពួកវាមកពីណា ?	= Today, there are many children in this park. Where are they all from?
 * ពួកគាត់រៀនខ្មែរនៅសាលាណា ? 				= At what school do they study Khmer?
 * ពួកគេចង់ទៅណា ? 					= Where do they want to go?
 * ពួកនាងធើ្វការអីនៅស្រុកខ្មែរ ? 				= What do they do for a living in Cambodia?

Statements
Statements are normally accompanied by a falling contour on the last words of the sentence, for example:
 * ញុំ សុខសប្បាយជាទេ = I am quite well.

Rising contour may also occur at the end of statements. In this case, the statement seems to be obvious, for example:
 * ខ្ញុំ សុខសប្បាយជាទេ = I am fine [of course]

Questions
Questions are formed in three ways in Cambodian, question requesting specific information, question requiring a yes or no answer and the question that occur without question particle or question words.

Questions are usually accompanied by rising contour on the last words of the sentence. The rising contour question reflects deference, politeness and formality, for example:
 * លោកអញ្ជើញទៅណា ? = Where are you going, Sir?

The Question Particle តើ /tæ/
The question particle (តើ) are always occurs at the beginning of an interrogative clause in formal questions. When the question particle (តើ) is used, it signal the fact that a question follows and the rising and falling intonation of question is not necessary. For example:
 * លោកអញ្ជើញទៅណា ? = Where are you going, Sir?
 * តើលោកអញ្ជើញទៅណា ? = Where are you going, Sir?

Questions Requesting Specific Information
Questions requesting specific information require a question words for example: Question words: Where ណា (nā),  What អី (aī) / អី្វ (aīv)

 * ទៅណាហ្នឹង សុខ ?	 ខ្ញុំទៅណោះ មួយភ្លែត។ = Where are you going? I go there for a moment.
 * លោកអញ្ជើញទៅណា ? ខ្ញុំទៅសាលា។ = Where are you going? I go to school.
 * លោកមានការអីដែរ ។ ខ្ញុំចង់ជួបលោក សំណាង។ = What are you doing? I want to meet Samnang.
 * អ្នកឈ្មោះអី្វ ? ខ្ញុំឈោ្មះ ផល។ = What is your name? I'm Bhal.

Questions Requiring Yes or No Answer
Question requiring yes- or-no answer require a final question particle ទេ (de). The full form of particle (de) is ឬទេ (ȳde).

 * លោកសុខសប្បាយ ទេ ? បាទ ខ្ញុំសុខសប្បាយទេ។ = How are you? I'm fine.
 * លោកស្រីសុខសប្បាយ ឬទេ ? ចាស ខ្ញុំសុខសប្បាយទេ។ = How are you doing? I'm fine.

Response Particles: Yes បាទ /bād/, ចាស /cās/
The function of the response particles បាទ (a respond by men) and ចាស (a respond by women) is a polite acknowledgment of a previous utterance speech by other speakers, whether the previous utterance was a statement or a question. Following a question, the occurrence of a response particle as the sole constituent of the response implied affirmation.

A negative reply is indicated by a response particle followed by the negative final particle ទេ  or by ទេ alone.

In exchange between equals, response particles tend to be discontinued after the initial exchanges.

The exchange between a superior and an inferior, response particles tend to be used through the exchange by the inferior, but may be discontinued or not used at all by the superior party in the exchange.

Response particles usually occur with a falling intonation contour.

 * លោកសុខសប្បាយទេ ? = Are you fine? -> ចាស (women) / បាទ (men)
 * លោកស្រីទៅរៀនទេ ? = Is she going to school? -> ចាទេ (women) / បាទទេ (men)

Word ណា (nā)
ណា (nā) is used in many forms:

Interrogative forms
Pronoun: Adjective:
 * លោក មក ពី ណា ? = Where have you been ?
 * លោក ទៅ ណា ? = Where are you going ?
 * លោក ស្នាក់នៅ កន្លែង ណា ? = Which place do you stay ?
 * លោក ទៅ កន្លែង ណា ? = Which place are you going to ?
 * លោក មកពី ប្រទេស ណា ? = Which country are you from ?

Indefinite forms (somewhere/anywhere/wherever)
Pronoun: Adjective: Can also occur with the elements muey (one) and klah (some) as compound indefinite pronouns and adjectives:
 * លោកស្រី ទៅ ណា ទេ ? = Are you going somewhere or anywhere ?
 * ខ្ញុំ មិន ទៅ ណា ទេ។ = I am not going anywhere.
 * លោកស្រី ចង់ ទៅ ណា ទេ ? = Do you want to go somewhere or anywhere ?
 * ខ្ញុំ មិន ចង់ ទៅ ណា ទេ។ = I don't want to go anywhere.
 * លោក ចង់ ទៅ ណា ក៏ បានដែរ។ = You can go anywhere or wherever (you wish).
 * បើលោកស្រី ចង់ ទៅ ណា ខ្ញុំ អាច ជូន ទៅ បាន។ = If you want to go somewhere or anywhere I will take you. Or: Wherever you want to go, I will take you.
 * លោកស្រី ចង់ ទៅ កន្លែង ណា ? = Do you want to go some or any place ?
 * ខ្ញុំ មិន ចង់ ទៅ កន្លែង ណា ទេ។ = I don't want to go to any place.
 * លោក ចង់ ទៅ កន្លែង ណា ក៏ បាន ដែរ។ = You can go any place (anywhere) you wish.
 * បើ លោក ចង់ ទៅ ផ្សារ ណា ខ្ញុំ ជូនទៅ។ = If you want to go to any market I will take you. Or: Whatever / whichever market you want to go, I will take you.
 * បើ លោក ចង់ បាន អាវ ណា មួយ ប្រាប់ ខ្ញុំ មក ។ = if you want (any shirt), just tell me.
 * លោក ចង់ ទៅ កន្លែង ណា ខ្លះ។ = Wherever you would like to go, I will take you.

កិរិយាស័ព្ទ ស្គាល់ / Know a person, place or thing
The verb (ស្គាល់) here is use to express, be acquainted with person, to know and to know of.

Question: Subject + ស្គាល់ + person/place/thing + ទេ / ឬទេ ?

Answer: Response   Female 	Male Positive   ចាស ស្គាល់   បាទ ស្គាល់ Negative   អត់ទេ      អត់ ស្គាល់ ទេ

 * សូមទោស លោក ស្គាល់ គ្រូភាសាខ្មែរ ទេ ? = Excuse me, do you know a Khmer teacher?
 * សូមទោស លោក ស្គាល់ ភ្នំពេញ ទេ ? = Excuse me, do you know where Phnom Penh is?
 * សូមទោស លោក ស្គាល់ ផ្ទះលោក សារ៉េត ទេ ? = Excuse me, do you know Saret's house?

កិរិយាស័ព្ទ ចេះ / Know how to do something
The verb (ចេះ) here is use to express the ability of knowing to do things.

Question: Subject + ចេះ + do something + ទេ / ឬទេ ?

Answer: Response  Female 	Male Positive  ចាស ចេះ 	បាទ ចេះ Negative  អត់ទេ       អត់ ចេះ ទេ

 * លោក ចេះ និយាយ ភាសា ខ្មែរ ទេ ? = Do you know how to speak Khmer or Do you speak Khmer?.
 * អ្នក ចេះ ច្រៀង ទេ ? = Do you know how to sing?
 * អ្នកចេះ លេងភ្លេង ខ្មែរ ទេ ? = Do you know how to play Khmer music?

Word អី្វ /ʔaīv/ "What"
អី្វ /ʔaīv/ is used in many forms:

Interrogative forms
Pronoun: Adjective:
 * លោក ទិញ អី្វ ? ខ្ញុំ ទិញ សៀវភៅ។ = What do you buy ? I buy a book.
 * អ្នក លក់ អី្វ ? ខ្ញុំ លក់ ខោអាវ។ = What do you sell ? I sell clothing.
 * លោក ទិញ សៀវភៅ អី្វ ? ខ្ញុំ ទិញ សៀវភៅ សរសេរ។ = Which book are you buying ? I buy a notebook.
 * អ្នក លក់ ខោអាវ អី្វ ? ខ្ញុំ លក់ ខោអាវ បុរស។ = What clothing are you selling ? I sell men clothes.

Indefinite forms (somewhat/anywhat/whatever)
Pronoun: Adjective: Compound:
 * គាត់ ទៅ ទិញ អី្វ នៅ ផ្សារ។ = He goes to buy something at the market.
 * ខ្ញុំ លក់ អី្វ នៅ ផ្សារ។ = I sell something at the market.
 * ខ្ញុំ មិន ចង់ ញុំា អី្វ ទេ។ = I don't want to eat anything.
 * បើ លោក ចង់ បាន អ្វី ខ្ញុំ ទិញ អោយ។ = If you want anything I will buy it for you. or: whatever you want I will buy it for you.
 * អ្នក ចង់ ទិញ ខោអាវ អី្វ ទេ ? ខ្ញុំ មិន ចង់ ទិញ ខោអាវ អី្វ ទេ។ = Do you want to buy any clothes? I don't want to buy any clothes.
 * បើ លោក ចង់ បាន ខោអាវ អ្វី ខ្ញុំ ទិញ អោយ។ ? លោក ទិញ ខោអាវ អ្វី អោយ ខ្ញុំ ក៏បាន ដែរ។ = If you want any clothes I will buy it for you. You can buy any clothes for me.
 * បើ អ្នក ចង់ ទិញ អី្វខ្លះ ប្រាប់មក ខ្ញុំ អាច ទិញ អោយ បាន។ = If you want me to buy something, just let me know, I will buy it for you.
 * បើ អ្នក ចង់ ពិសារ អី្វមួយ ប្រាប់មក ខ្ញុំ ទិញ អោយ។ = If you want to eat anything just let me know I will buy it for you.
 * បើ អ្នក ចង់ ទិញ សៀវភៅ អី្វមួយ ពី ស្រុក អាមេរិក ប្រាប់មក ខ្ញុំ អាច ទិញ អោយ បាន។  = If you want me to buy any books from America, just let me know, I will buy it for you.

Used forms

 * លោក ត្រូវការ ស្ករ ប៉ុនា្មន គីឡូ ? ខ្ញុំ ត្រូវការ ស្ករ ២ គីឡូ។ = How many kilos of sugar do you want? I want 2 kilos of sugar.
 * អ្នក ទិញ ទឹកសុទ្ធ ប៉ុនា្មន ដប ? ខ្ញុំ ទិញ ទឹកសុទ្ធ ១២ ដប។ = How many bottle of pure water do you want to buy? I buy pure water 12 bottles.
 * ម៉ោង ប៉ុនា្មន ហើយ ? ម៉ោង ៨ ហើយ។ = What time is it now? It is 8 o'clock.
 * គាត់ ត្រូវការ ប៉ុនា្មន គីឡូ ? 	គាត់ ត្រូវការ ៦ គីឡូ = How many kilos does he want? He wants 6 kilos
 * រូបគំនូរ នេះ ថ្លៃ ប៉ុនា្មន ? រូបគំនូរ នេះ ថ្លៃ ១៨ ដុល្លា។ = How much is this painting? It is 18 dollars.
 * លោក ត្រូវការ ប៉ុនា្មន ? ខ្ញុំ ត្រូវការ ២០ ។ = How much do you need? I need 20.
 * លោក ចង់បាន ប៉ុនា្មន ? គាត់ ចង់បាន ១៥ = How much do you want? I want 15.
 * ពួកគេ មាន ប្រាក់ ប៉ុនា្មន ? ពួកគេ មាន ប្រាក់ ៥០០ រៀល។ = How much money do they have? They have 500 riels.

Affirmative sentences
Unlike English, Khmer verbs do not take inflection in the present tense.

Simple Sentence with Intransitive Verb: Subject + Verb (intransitive)
 * គាត់ យំ ។ 	= He cries.
 * សិស្ស សើច។ 	= Student laughs.
 * ពួកគេ ដើរ ។ 	= They walk.
 * ខ្ញុំ ដេក ។ 	= I sleep.

Simple Sentence with Intransitive Verb and Adverb: Subject + Verb + Adverb
 * ក្មេងៗ ដើរ តាមដងផ្លូវ។ 	= Children walk on the street.
 * សំ រស់នៅ ក្នុងទីក្រុង ។ 	= Sam lives in the city.

Simple Sentence with Transitive Verb and follow by Object: Subject + Verb + Object
 * ខ្មុំ ស្នាក់នៅ សណ្ឋាគារអាស៊ី ។ 	= I stay in Asie Hotel.
 * ខ្មុំ ជួប គាត់ ។ 	               = I meet him.
 * ក្មេងៗ ចូលចិត្ដ ផ្លែឈើ ។ 	       = Children like fruit.
 * យើង ស្រឡាញ់ សន្ដិភាព ។ 	       = We love peace.

Negative sentences
The negative sentence in Khmer is formed by inserting of a negative word ( មិន ) or ( អត់ ) between the subject and the main verb of a sentence and ( ទេ ) at the end of that sentence. Particle ( ទេ ) can be deleted from the negative sentence, but it doesn't sound natural. Subject + មិន or អត់ + ( Verb [intransitive] / Verb + Adverb / Verb + Object ) + ទេ

 * គាត់ មិន/អត់ យំ ទេ។                     = He doesn't cry.
 * សិស្ស មិន/អត់ សើច ទេ។                   = Student doesn't laugh.
 * ពួកគេ មិន/អត់ ដើរ ទេ។                   = They don't walk.
 * ខ្ញុំ មិន/អត់ ដេក ទេ។ ។ 	                = I don't sleep.
 * ក្មេងៗ មិន/អត់ ដើរ តាមដងផ្លូវ ទេ។          = Children don't walk on the street.
 * សំ មិន/អត់ រស់នៅ ក្នុងទីក្រុង ទេ។            = Sam deosn't live in the city
 * ខ្មុំ មិន/អត់ ស្នាក់នៅ សណ្ឋាគារអាស៊ី ទេ។       = I don't stay in Asie Hotel.
 * ខ្មុំ មិន/អត់ ជួប គាត់ ទេ។                 = I don't meet him.
 * ក្មេងៗ មិន/អត់ ចូលចិត្ដ ផ្លែឈើ ទេ។         = Children don't like fruit.
 * យើង មិន/អត់ ស្រឡាញ់ សន្ដិភាព ទេ។        = We don't love peace.

Interrogative sentences
The question particle (តើ) always occours at the beginning of an interrogative clause in formal questions. When the question particle (តើ) is used, it signal the fact that a question follows and the rising and falling intonation of question is not necessary.

Question requiring yes- or-no answer require a final question particle ទេ (de)). The full form of particle (de) is ឬទេ (yde). តើ + Subject + ( Verb [intransitive] / Verb + Adverb / Verb + Object ) + ឬទេ ?

 * តើ គាត់ យំ ឬទេ/ទេ ? = Does he cry?
 * តើ សិស្ស សើច ឬទេ/ទេ ? = Does the student laugh?
 * តើ ពួកគេ ដើរ ឬទេ/ទេ ? = Do they walk?
 * តើ ខ្ញុំ ដេក ឬទេ/ទេ ? = Do I sleep?
 * តើ ក្មេងៗ ដើរ តាមដងផ្លូវ ឬទេ/ទេ ? = Do children walk on the street?
 * តើ សំ រស់នៅ ក្នុងទីក្រុង ឬទេ/ទេ ? = Do Sam live in the city?
 * តើ លោក ស្នាក់នៅ សណ្ឋាគារអាស៊ី ឬទេ/ទេ ? = Do you stay in Asie Hotel?
 * តើ លោក ជួប គាត់ ឬទេ/ទេ ? = Do you meet him?
 * តើ ក្មេងៗ ចូលចិត្ដ ផ្លែឈើ ឬទេ/ទេ ? = Do children like fruit?
 * តើ យើង ស្រឡាញ់ សន្ដិភាព ឬទេ/ទេ ? = Do we love peace?

Present Tense with Connecting Verb ជា /jā/
Note: The connecting verb (ជា ) can be dropped in from the affirmative, negative, or interogative sentence in the spoken form and the sentence still carries the same meaning.

Affirmative Sentence
The meaning of connecting verb (ជា) is (be, is) as in the sentence: Subject + Connecting verb (ជា ) + Noun

 * ខ្ញ្ញុំ ជា ជនជាតិ អាមេរិក ។ 	= I am an American.
 * គាត់ ជា សាស្រ្ដាចារ្យ ។ 	= He is a professor.
 * ខ្ញ្ញុំ ជា គ្រូបង្រៀន ។ 	= 	I am a teacher.
 * ពួកគេ ជា កសិករ ។ 	= They are farmers.
 * អាគារនេះ ជា បណ្ណាល័យ ។ 	= This building is a library.

Negative Sentence
The negative sentence of simple sentence, with connecting verb ( ជា ), is form by inserting the negative word ( មិនមែន ) or ( អត់មែន ) between the subject and the main verb of a sentence and ( ទេ ) at the end of that sentence. Particle ( ទេ ) can be deleted from the negative sentence, but it doesn't sound natural. Subject + មិនមែន or អត់មែន + Connecting verb (ជា ) + Noun + ទេ

 * ខ្ញ្ញុំ មិនមែន/អត់មែន ជា ជនជាតិ អាមេរិក ទេ។ 		= I am not an American
 * គាត់ មិនមែន/អត់មែន ជា សាស្រ្ដាចារ្យ ទេ។ 	 	= He is not a professor.
 * ខ្ញ្ញុំ មិនមែន/អត់មែន ជា គ្រូបង្រៀន ទេ។ 	 	= I am not a teacher.
 * ពួកគេ មិនមែន/អត់មែន ជា កសិករ ទេ។ 	 	= They are not farmers.
 * អាគារនេះ មិនមែន/អត់មែន ជា បណ្ណាល័យ ទេ។ 	 	= This building is not a library.

Interrogative Sentence
The interrogative clause, with connecting verb (ជា ), is form by adding particle (តើ) at the beginning of the clause and follows by ( មែនទេ ) at the end of the clause. តើ + Subject +	Connecting verb (ជា ) + Noun + មែនទេ ?

 * តើ លោក ជា ជនជាតិ អាមេរិក មែនទេ ?   	= 	Are you an American?
 * តើ គាត់ ជា សាស្រ្ដាចារ្យ មែនទេ ? 	= 	Is he a professor?
 * តើ នាង ជា គ្រូបង្រៀន មែនទេ ? 	= 	Is she a teacher?
 * តើ ពួកគេ ជា កសិករ មែនទេ ? 	=	Are they famers?
 * តើ អាគារនេះ ជា បណ្ណាល័យ មែនទេ ? 	= 	Is this building a library?

title: Cambodian Language/Grammar


text:
LMIs in Control/pages/Continuous time Quadratic stability

To study stability of a LTI system, we first ask whether all trajectories of system converge to zero as $$ t \rightarrow \infty $$. A sufficient condition for this is the existence of a quadratic function $$ V(\xi)=\xi^T P\xi $$, $$ P > 0 $$ that decreases along every nonzero trajectory of system. If there exists such a P, we say the system is quadratically stable and we call $$ V $$ a quadratic Lyapunov function.

The System

\begin{align} \dot x(t)&=A(\delta(t))x(t)\\

\end{align}$$

The Data
The system coefficient matrix takes the form of

\begin{align} \dot x(t)&=A_0 + \Delta A(\delta(t))x(t)\\ \end{align}$$

where $$ A_0 \in \mathbb{R} $$is a known matrix, which represents the nominal system matrix, while $$ \begin{align} \Delta A(\delta(t))x(t) = \delta_1(t))A_1 + \delta_2(t))A_2 +...+\delta_k(t))A_k \end{align}$$ is the system matrix perturbation, where

 * $$ A_i \in \mathbb{R}^{n \times n}, i = 1,2,..,k, $$ are known matrices, which represent the perturbation matrices.
 * $$ \delta_i(t), i =1,2,...,k, $$ which represent the uncertain parameters in the system.
 * $$ \delta(t) = [\delta_1 (t) \delta_2(t)... \delta_k(t)]^T $$ is the uncertain parameter vector, which is often assumed to be within a certain compact and convex set : : $$ \Delta $$ that is

\delta(t) = [\delta_1 (t) \delta_2(t)... \delta]^T \in \Delta $$

The LMI: Continuous-Time Quadratic Stability
The uncertain system is quadratically stable if and only if there exists $$ P \in \mathbb{S}^n $$, where $$ P > 0, $$ such that

\begin{align} (A_0 + \Delta A(\delta(t))x(t))^T + P(A_0 + \Delta A(\delta(t))x(t)) <0   \delta(t) \in \Delta \end{align}$$

The following statements can be made for particular sets of perturbations.

Case 1: Regular Polyhedron
Consider the case where the set of perturbation parameters is defined by a regular polyhedron as

\begin{align} \Delta = { \delta(t) = [\delta_1 (t) \delta_2(t)... \delta_k(t)] \in \mathbb{R}^k \mid \delta_i(t) ,\underline{\delta_i}(t),\overline{\delta_i}(t),\underline{\delta_i}\leq \delta_i(t) \leq \overline{\delta_i)} } \end{align}$$

The uncertain system is quadratically stable if and only if there exists $$ P \in \mathbb{S}^n $$, where $$ P > 0, $$ such that

\begin{align} (A_0 + \Delta A(\delta(t))x(t))^T + P(A_0 + \Delta A(\delta(t))x(t)) <0  \delta_i(t) \in {\underline{\delta_i},\overline{\delta_i}}, i = 1,2,...k. \end{align}$$

Case 2: Polytope
Consider the case where the set of perturbation parameters is defined by a polytope as

\begin{align} \Delta = { \delta(t) = [\delta_1 (t) \delta_2(t)... \delta_k(t)] \in \mathbb{R}^k \mid \delta_i(t) \in \mathbb{R}_{\geq0}}, \sum_{i=1}^k \delta_i(t)=1 \end{align}$$

The uncertain system is quadratically stable if and only if there exists $$ P \in \mathbb{S}^n $$, where $$ P > 0, $$ such that

\begin{align} (A_0 + A_i)^T P+ P(A_0+A_i) <0, i=1,2...,k. \end{align}$$

Conclusion:
If feasible, System is Quadratically stable for any $$ x \in \mathbb{R}^n$$

Implementation
https://github.com/Ricky-10/coding107/blob/master/PolytopicUncertainities