text:
Writing Adolescent Fiction/Character names/Unisex

 * 1) Alex
 * 2) Alexis
 * 3) Arden
 * 4) Ariel
 * 5) Arlie
 * 6) Ashe
 * 7) Ashley
 * 8) Avery
 * 9) Avie
 * 10) Bailey
 * 11) Blair
 * 12) Bobby
 * 13) Brett
 * 14) Bronte
 * 15) Brooklyn
 * 16) Bryn
 * 17) Brynn
 * 18) Caden
 * 19) Cameron
 * 20) Casey
 * 21) Cassidy
 * 22) Cecil
 * 23) Colby
 * 24) Cole
 * 25) Corin
 * 26) Cory
 * 27) Courtney
 * 28) Dakota
 * 29) Daley
 * 30) Daly
 * 31) Dana
 * 32) Darby
 * 33) Darian
 * 34) Darnell
 * 35) Darrell
 * 36) Darryl
 * 37) Dayna
 * 38) Dell
 * 39) Devin
 * 40) Devon
 * 41) Dustin
 * 42) Elis
 * 43) Elliot
 * 44) Emerson
 * 45) Emery
 * 46) Evan
 * 47) Evelyn
 * 48) Fay
 * 49) Finley
 * 50) Finn
 * 51) Garnet
 * 52) Gay
 * 53) Gayle
 * 54) Ginkgo
 * 55) Hayden
 * 56) Hayley
 * 57) Heaven
 * 58) Hilary
 * 59) Hope
 * 60) Hyacinth
 * 61) Ivy
 * 62) Jack
 * 63) Jamie
 * 64) Jasper
 * 65) Jayden
 * Jo
 * 1) Johnnie
 * 2) Jordan
 * 3) Joyce
 * 4) Juniper
 * 5) Juno
 * 6) Kale
 * 7) Kasey
 * 8) Kelly
 * 9) Kelsey
 * 10) Kelsie
 * 11) Kennedy
 * 12) Kerry
 * 13) Kiernan
 * 14) Kim
 * 15) Kimberly
 * 16) Kit
 * 17) Kristen
 * 18) Kye
 * 19) Kyrie
 * 20) Lacey
 * 21) Lee
 * 22) Leigh
 * 23) Lennox
 * 24) Leo
 * 25) Lesley
 * 26) Leslie
 * 27) Lindsay
 * 28) Lindy
 * 29) Logan
 * 30) Loren
 * 31) Lucky
 * 32) Lyndsay
 * 33) Lynn
 * 34) Mackenzie
 * 35) Madison
 * 36) Mallory
 * 37) Marley
 * 38) Mason
 * 39) Mel
 * 40) Meredith
 * 41) Mica
 * 42) Monroe
 * 43) Morgan
 * 44) Noah
 * 45) Noel
 * 46) Odell
 * 47) Ollie
 * 48) Paige
 * 49) Paris
 * 50) Parker
 * 51) Parris
 * 52) Paxton
 * 53) Payton
 * 54) Pepper
 * 55) Quinn
 * 56) Randi/Randy
 * 57) Raven
 * 58) Rebel
 * 59) Ripley
 * 60) Robin
 * 61) Ronnie
 * 62) Rowan
 * 63) Sam
 * 64) Sandi
 * 65) Sandy
 * 66) Sasha
 * 67) Sawyer
 * 68) Schuyler
 * 69) Shannon
 * 70) Shirley
 * 71) Skyler
 * 72) Snow
 * 73) Spencer
 * 74) Sprite
 * 75) Sydney
 * 76) Tatum
 * 77) Taylor
 * 78) Tegan
 * 79) Terry
 * 80) Tobi
 * 81) Tory
 * 82) Tracy
 * 83) Vivian
 * 84) Wallis
 * 85) Waverly
 * 86) Whitney
 * 87) Wisdom
 * 88) Yeast


text:
LMIs in Control/Matrix and LMI Properties and Tools/Dualization Lemma

Dualization Lemma
Consider $$P_{i}\in \text{S}^{n} $$ and the subspaces $$U, V$$, where $$P$$ is invertible and $$U +V = \text{R}^{n} $$. The following are equivalent.

$$ X^{T}PX<0$$ for all $$X\in \text{U}$$ \ $$\left\{0\right\}$$ and  $$ X^{T}PX \ge 0$$ for all $$X\in \text{V} $$.

$$ X^{T}P^{-1}X>0$$ for all $$X\in \text{U}^{\bot}$$ \ $$\left\{0\right\}$$ and  $$ X^{T}P^{-1}X \le 0$$ for all $$X\in \text{V}^{\bot} $$.

Example
Consider the matrices $$Q\in \text{S}^{n},S\in \text{R}^{n \times m},R\in \text{S}^{m},M\in \text{R}^{m \times n} $$ where $$ R\ge 0 ,$$ which define the quadratic matrix inequality

$$ \begin{align} \qquad \begin{bmatrix} 1 & M\\ \end{bmatrix} \begin{bmatrix} Q & S\\ S^{T} & R\\ \end{bmatrix} \begin{bmatrix} 1\\ M\\ \end{bmatrix} <0. \qquad (1) \end{align} $$

Define $$ \begin{align} P= \begin{bmatrix} Q & S\\ S^{T} & R\\ \end{bmatrix} ,U=R( \begin{bmatrix} 0\\ 1\\ \end{bmatrix} ) \end{align} $$ where $$U+V=R^{n+m}$$. Notice that $$(1)$$ is equivalent to $$X^{T}PX<0$$ for all $$X\in \text{U}$$ \ $$\left\{0\right\}$$.Additionally, $$X^{T}PX<0 \ge$$ for all $$X\in \text{V} $$ is euaivalent to

$$ \begin{align} \qquad \begin{bmatrix} 0 & 1\\ \end{bmatrix} \begin{bmatrix} Q & S\\ S^{T} & R\\ \end{bmatrix} \begin{bmatrix} 0\\ 1\\ \end{bmatrix} =R \ge 0, \end{align} $$

which is satisfied based on the definition of $$R$$. By the dualization lemma, $$(1)$$ is satisfied with $$R \ge 0$$ if and only if

$$ \begin{align} \qquad \begin{bmatrix} -M^{T} & 1\\ \end{bmatrix} \begin{bmatrix} \tilde{Q} & \tilde{S}\\ \tilde{S}^{T} & \tilde{R}\\ \end{bmatrix} \begin{bmatrix} -M^{T}\\ 1\\ \end{bmatrix} >0,\qquad \tilde{Q}\le 0, \end{align} $$

where $$ \begin{align} \qquad \begin{bmatrix} \tilde{Q} & \tilde{S}\\ \tilde{S}^{T} & \tilde{R}\\ \end{bmatrix} = \begin{bmatrix} Q & S\\ S^{T} & R\\ \end{bmatrix}^{-1}, U^{\bot}=N([1\quad M^{T}])=R( \begin{bmatrix} -M^{T}\\ 1\\ \end{bmatrix} )

\end{align} $$, and $$ \begin{align}V^{\bot}=N([0\quad 1])=R( \begin{bmatrix} 1\\ 0\\ \end{bmatrix} )

\end{align} $$.

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


text:
LMIs in Control/Matrix and LMI Properties and Tools/LMIs in Control/pages/Dicrete-time Algebraic Riccati Equation

Algebraic Riccati Equations are particularly significant in Optimal Control, filtering and estimation problems. The need to solve such equations is common in the analysis and linear quadratic Gaussian control along with general Control problems. In one form or the other, Riccati Equations play significant roles in optimal control of multivariable and large-scale systems, scattering theory, estimation, and detection processes. In addition, closed forms solution of Ricatti Equations are intractable for two reasons namely; one, they are nonlinear and two, are in matrix forms.

The System
The system is defined as:

$$y_t = Ay_{t-1} + Bu_{t-1},$$

The Data
A typical specification of the discrete-time linear quadratic control problem is to minimize

 * $$\sum_{t=1}^T (y_t^T Qy_t + u_t^T Ru_t)$$

subject to the state equation

 * $$y_t = Ay_{t-1} + Bu_{t-1},$$

where y is an n × 1 vector of state variables, u is a k × 1 vector of control variables, A is the n × n state transition matrix, B is the n × k matrix of control multipliers, Q (n × n) is a symmetric positive semi-definite state cost matrix, and R (k × k) is a symmetric positive definite control cost matrix.

The LMI: Discrete-Time Algebraic Riccati Inequality
Title and mathematical description of the LMI formulation.

\begin{align} \text{ Consider}\; A_d \in \mathbb{R}^{n \times m}, B_d \in \mathbb{R}^{n \times m}, P, Q \in \mathbb{S}^{n}, \text {where}\;\\ P>0, \\Q \geq 0, \\ R>0.\\ \text{ The Discrete-Time Algebraic Riccati inequality is given by}\;\\ A^T_d PA_d - A^T_d PB_d(R + B^T_d P B_d)^{-1} B^T_d P A_d + Q - P & \geq 0,\\ \\ \text{ can be written using the Schur complement lemma as}\;\\ \begin{bmatrix} A^T_d - P + Q && A^T_d P B_d\\ \star && R+B^T_d P B_d\\ \end{bmatrix}& \geq 0 \\ \end{align}$$

Equivalently, this discrete-time algebraic Riccati inequality is satisfied under any of the following necessary and sufficient conditions.

1. There exist $$P, Q \in \mathbb{S}^{n}, \text {and} \; R \in \mathbb{S}^m,$$ where $$ P>0, Q \geq 0,and R>0, $$ such that

$$ \begin{bmatrix} Q && 0 && A^T_d P && P\\ \star && R && B^T_d P && 0\\ \star && \star && P && 0 \\ \star && \star && \star && P \\ \end{bmatrix}$$ $$\begin{align} \geq 0 \end{align}$$,

2. There exist $$X, Q \in \mathbb{S}^{n}, \text {and} \; R \in \mathbb{S}^m,$$ where $$ X>0, Q \geq 0,and R>0, $$ such that

$$ \begin{bmatrix} Q && 0 && A^T_d && 1\\ \star && R && B^T_d P && 0\\ \star && \star && X && 0 \\ \star && \star && \star && X \\ \end{bmatrix}$$ $$\begin{align} \geq 0 \end{align}$$,

Conclusion:
If the solution exists, LMIs give a unique, stabilizing, symmetric matrix P.

Implementation:
Matlab code for this LMI in the Github repository:


text:
LMIs in Control/pages/H2-Optimal Filter

H2-Optimal Filter

The goal of optimal filtering is to design a filter that acts on the output $$ \bold{z}$$of the generalized plant and optimizes the transfer matrix from $$ \bold{w}$$to the filtered output.

The System
Consider the continuous-time generalized LTI plant, with minimal state-space representation

$$ \bold{\dot x} = \bold{Ax} + \bold{B}_1\bold{w},$$

$$ \bold{z} = \bold{C}_1\bold{x} + \bold{D}_{11}\bold{w},$$

$$ \bold{y} = \bold{C}_2\bold{x} + \bold{D}_{21}\bold{w},$$

where it is assumed that $$ \bold{A}$$is Hurwitz. A continuous-time dynamic LTI filter with state-space representation

$$ \bold{\dot x}_f = \bold{A}_f\bold{x}_f + \bold{B}_f\bold{y},$$

$$ \bold{\hat z} = \bold{C}_f\bold{x}_f + \bold{D}_f\bold{y},$$

is designed to optimize the transfer function from $$ \bold{w}$$to $$ \bold{\tilde z}=\bold{z}-\bold{\hat z}$$, which is given by

$$ \tilde \bold P (s) = \tilde \bold C_1(s\bold{I}-\tilde \bold A)^{-1}\tilde \bold B _1+\tilde \bold D _{11},$$

where

$$ \tilde\bold A = \begin{bmatrix} \bold A & \bold 0 \\ \bold B_f \bold C_2 & \bold A_f \end{bmatrix}, $$

$$ \tilde\bold B_1 = \begin{bmatrix} \bold B_1 \\ \bold B_f \bold D_{21} \end{bmatrix}, $$

$$ \tilde\bold C_1 = \begin{bmatrix} \bold C_1 - \bold D_f\bold C_2 & -\bold C_f \end{bmatrix}, $$

$$ \tilde\bold D_{11} = \bold D_{11}-\bold D_f\bold D_{21}. $$

Optimal Filtering seeks to minimize the given norm of the transfer function $$ \tilde \bold P (s).$$There are two methods of synthesizing the H2-optimal filter.

Synthesis 1
Solve for $$ \bold A_n \in \R^{n_x\times n_x}, \bold B_n \in \R^{n_x\times n_y}, \bold C_f \in \R^{n_z\times n_x}, \bold D_f \in \R^{n_z\times n_y}, \bold{X,Y} \in \S^{n_x}, \bold Z \in \S^{n_z},$$and $$ \nu\in\R_{>0}$$ that minimize the objective function $$ J(\nu)=\nu$$, subject to

$$ \bold{X,Y,Z} > 0,$$

$$ \bold Y-\bold X>0, $$

$$ tr(\bold Z)<\nu,$$

$$ \bold D_{11}-\bold D_f\bold D_{21}=\bold 0,$$

$$ \begin{bmatrix} -\bold Z & \bold C_1-\bold D_f\bold C_2 & -\bold C_f \\ \end{bmatrix} <0,$$
 * & -\bold Y & -\bold X \\
 * & * & -\bold X

$$ \begin{bmatrix} \bold{YA}+\bold A^T\bold Y + \bold B_n \bold C_2 +\bold C^T_2\bold B_n^T & \bold A_n + \bold C_2^T\bold B_n^T+\bold A^T\bold X & \bold{YB}_1+\bold B_n\bold D_{21} \\ \end{bmatrix} <0.$$
 * & \bold A_n+\bold A_n^T & \bold{XB}_1+\bold B_n \bold D_{21} \\
 * & * & -\bold{I}

Synthesis 2
Synthesis 2 is identical to Synthesis 1, with the exception of the final two matrix inequality constraints:

$$ \begin{bmatrix} -\bold Z & \bold B_1^T\bold Y^T+\bold D_{21}^T\bold B_n^T & \bold B_1^T\bold X^T+\bold D_{21}^T\bold B_n^T \\ \end{bmatrix} <0,$$
 * & -\bold Y & -\bold X \\
 * & * & -\bold X

$$ \begin{bmatrix} \bold{YA}+\bold A^T\bold Y + \bold B_n \bold C_2 +\bold C^T_2\bold B_n^T & \bold A_n + \bold C_2^T\bold B_n^T+\bold A^T\bold X & \bold C_1^T-\bold C_2^T\bold D_f^T \\ \end{bmatrix} <0.$$
 * & \bold A_n+\bold A_n^T & -\bold C_f^T \\
 * & * & -\bold{I}

Remark
In both cases, if $$ \bold D_{11}=0$$and $$ \bold D_{21}\neq0,$$then it is often simplest to choose $$ \bold D_f=0$$in order to satisfy the equality constraint (above).

Conclusion
In both cases, the optimal H2 filter is recovered by the state-space matrices $$ \bold A_f=\bold X^{-1}\bold A_n, \bold B_f=\bold X^{-1}\bold B_n, \bold C_f, $$and $$ \bold D_f.$$

Remark
The problem of optimal filtering can alternatively be formulated as a special case of synthesizing a dynamic output "feedback" controller for the generalized plant given by

$$ \bold{\dot x} = \bold{Ax} + \bold{B}_1\bold{w},$$

$$ \bold{z} = \bold{C}_1\bold{x} + \bold{D}_{11}\bold{w}-\bold u,$$

$$ \bold{y} = \bold{C}_2\bold{x} + \bold{D}_{21}\bold{w}.$$

The synthesis methods presented in this page take advantage of the fact that the controller in this case is not a true feedback controller, as it only appears as a feedthrough term in the performance channel.

title: LMIs in Control/pages/H2-Optimal Filter


text:
LMIs in Control/pages/Hinf-Optimal Filter

Hinf-Optimal Filter

The goal of optimal filtering is to design a filter that acts on the output $$ \bold{z}$$of the generalized plant and optimizes the transfer matrix from $$ \bold{w}$$to the filtered output.

The System
Consider the continuous-time generalized LTI plant, with minimal state-space representation

$$ \bold{\dot x} = \bold{Ax} + \bold{B}_1\bold{w},$$

$$ \bold{z} = \bold{C}_1\bold{x} + \bold{D}_{11}\bold{w},$$

$$ \bold{y} = \bold{C}_2\bold{x} + \bold{D}_{21}\bold{w},$$

where it is assumed that $$ \bold{A}$$is Hurwitz. A continuous-time dynamic LTI filter with state-space representation

$$ \bold{\dot x}_f = \bold{A}_f\bold{x}_f + \bold{B}_f\bold{y},$$

$$ \bold{\hat z} = \bold{C}_f\bold{x}_f + \bold{D}_f\bold{y},$$

is designed to optimize the transfer function from $$ \bold{w}$$to $$ \bold{\tilde z}=\bold{z}-\bold{\hat z}$$, which is given by

$$ \tilde \bold P (s) = \tilde \bold C_1(s\bold{I}-\tilde \bold A)^{-1}\tilde \bold B _1+\tilde \bold D _{11},$$

where

$$ \tilde\bold A = \begin{bmatrix} \bold A & \bold 0 \\ \bold B_f \bold C_2 & \bold A_f \end{bmatrix}, $$

$$ \tilde\bold B_1 = \begin{bmatrix} \bold B_1 \\ \bold B_f \bold D_{21} \end{bmatrix}, $$

$$ \tilde\bold C_1 = \begin{bmatrix} \bold C_1 - \bold D_f\bold C_2 & -\bold C_f \end{bmatrix}, $$

$$ \tilde\bold D_{11} = \bold D_{11}-\bold D_f\bold D_{21}. $$

Optimal Filtering seeks to minimize the given norm of the transfer function $$ \tilde \bold P (s).$$

Filter Synthesis
Solve for $$ \bold A_n \in \R^{n_x\times n_x}, \bold B_n \in \R^{n_x\times n_y}, \bold C_f \in \R^{n_z\times n_x}, \bold D_f \in \R^{n_z\times n_y}, \bold{X,Y} \in \S^{n_x}, $$and $$ \gamma\in\R_{>0}$$ that minimize the objective function $$ J(\gamma)=\gamma$$, subject to

$$ \bold{X,Y} > 0,$$

$$ \bold Y-\bold X>0, $$

$$ \begin{bmatrix} \bold{YA}+\bold A^T\bold Y + \bold B_n \bold C_2 +\bold C^T_2\bold B_n^T & \bold A_n + \bold C_2^T\bold B_n^T+\bold A^T\bold X & \bold{YB}_1+\bold B_n\bold D_{21} \\ \end{bmatrix} <0.$$
 * & \bold A_n+\bold A_n^T & \bold{XB}_1+\bold B_n \bold D_{21} \\
 * & * & -\bold{I}

Conclusion
The optimal Hinf filter is recovered by the state-space matrices $$ \bold A_f=\bold X^{-1}\bold A_n $$and $$ \bold B_f=\bold X^{-1}\bold B_n. $$

Remark
The problem of optimal filtering can alternatively be formulated as a special case of synthesizing a dynamic output "feedback" controller for the generalized plant given by

$$ \bold{\dot x} = \bold{Ax} + \bold{B}_1\bold{w},$$

$$ \bold{z} = \bold{C}_1\bold{x} + \bold{D}_{11}\bold{w}-\bold u,$$

$$ \bold{y} = \bold{C}_2\bold{x} + \bold{D}_{21}\bold{w}.$$

The synthesis method presented in this page takes advantage of the fact that the controller in this case is not a true feedback controller, as it only appears as a feedthrough term in the performance channel.


text:
LMIs in Control/Stability Analysis/Discrete Time/Output Energy Bound for Autonomous LTI Systems

Output Energy Bound for Autonomous LTI Systems

Autonomous systems are initialized under a given set of initial conditions, and then run without any additional inputs. It is useful to know ahead of time the bounds such a system will operate within. This analysis can be used to determine the upper bound on the output of a given autonomous LTI system.

The System
Consider the continuous-time, LTI autonomous system with state space representation

$$ \bold{\dot x} = \bold{Ax}$$,

$$ \bold{y}=\bold{Cx}$$,

where $$ \bold{A} \in \R^{n\times n}, \bold{C} \in \R^{p\times n}$$ and $$ \bold{x}(0)=\bold{x}_0.$$

Determining an Upper Bound
The output of this system will satisfy

$$ \sqrt{\int_{0}^{T} \bold{y^Ty}dt}=\left\vert\left\vert \bold{y} \right\vert\right\vert_{2T} \leq \gamma \left\vert\left\vert \bold{x}_0 \right\vert\right\vert_2 ,\forall T \in \R_{\geq0}$$

if there exists some matrix $$ \bold{P} \in \S^p$$and scalar $$ \gamma \in \R_{>0}$$ such that

$$ \bold{P} > 0$$,

$$ \bold{P} - \gamma\bold{I} \leq0$$,

$$ \begin{bmatrix} \bold{PA}+\bold{A^TP} & \bold{C^T} \\ \end{bmatrix} \leq0$$.
 * & -\gamma\bold{I}

Conclusion
Given an autonomous system with an initial operating condition, the parameter $$ \gamma$$ can be used to determine the largest feasible bound on the output of that system.


text:
LMIs in Control/Stability Analysis/Continuous Time/Output Energy Bound for Discrete-Time Autonomous LTI Systems

Output Energy Bound for Discrete-Time Autonomous LTI Systems

Autonomous systems are initialized under a given set of initial conditions, and then run without any additional inputs. It is useful to know ahead of time the bounds such a system will operate within. This analysis can be used to determine the upper bound on the output of a given autonomous LTI system operating in discrete time.

The System
Consider the discrete-time, LTI autonomous system with state space representation

$$ \bold{x}_{k+1} = \bold{A}_d\bold{x}_k$$,

$$ \bold{y}_k=\bold{C}_d\bold{x}_k$$,

where $$ \bold{A}_d \in \R^{n\times n}$$ and $$ \bold{C}_d \in \R^{p\times n}.$$

Determining an Upper Bound
The output of this system will satisfy

$$ \left\vert\left\vert \bold{y} \right\vert\right\vert_{2k} \leq \gamma \left\vert\left\vert \bold{x}_0 \right\vert\right\vert_2 ,\forall k \in \Z_{\geq0}$$

if there exists some matrix $$ \bold{P} \in \S^n$$and scalar $$ \gamma \in \R_{>0}$$ such that

$$ \bold{P} > 0$$,

$$ \bold{P} - \gamma\bold{I} \leq0$$,

$$ \begin{bmatrix} \bold{A_d^T P A_d}-\bold{P} & \bold{C_d^T} \\ \end{bmatrix} \leq0$$.
 * & -\gamma\bold{I}

Conclusion
Given an autonomous system operating in discrete-time conditions, the parameter $$ \gamma$$ can be used to determine the largest feasible bound on the output of that system.


text:
Lentis/Atlantic Coast Pipeline

Introduction
The Atlantic Coast Pipeline was a natural gas pipeline proposed by Atlantic in 2014. The line would deliver natural gas to customers in West Virginia, Virginia, and North Carolina. Atlantic faced opposition from activist groups because of environmental and health concerns. The project was cancelled in 2020 due to financial and legal risk.

The Atlantic Coast Pipeline case alludes to the power dynamic between corporate America and the constituents of corporate America.

Natural Gas
The Atlantic Coast Pipeline was a proposed natural gas pipeline. Hydrocarbons in natural gas include methane, ethane, propane, and butane. The thermal energy from natural gas combustion is used to power electricity generators. Residents and businesses can use the natural gas from the Atlantic Coast Pipeline to heat buildings and homes, power appliances, and cook food. Atlantic seeks to increase the accessibility of natural gas.

The Atlantic Coast Pipeline LLC
Duke Energy, Dominion Energy, Piedmont Gas, and Southern Company Gas formed the Atlantic Coast Pipeline LLC (Atlantic) to handle the Atlantic Coast Pipeline.

This proposal originated from Duke Energy and Piedmont Gas, with Dominion Energy brought in later to build the pipeline. Duke energy is a Fortune 150 company dedicated to modernizing the energy grid, promoting clean energy, and increasing natural gas use. Dominion Energy produces, transmits, stores and distributes electricity and natural gas. The pipeline would serve both their interests by expanding their servicing capabilities. It is considered the Dominion-Duke Energy joint venture since Dominion and Duke Energy hold the majority of shares in the project.

The Proposal
Atlantic cited growing energy demands as the motivation for the pipeline. The pipeline was to run 600 miles starting in Harrison County, WV and ending in Robeson County, NC. The lateral pipeline was to branch off in southern Virginia and end in Chesapeake, VA. The pipeline was projected to throughput 1.5 billion cubic feet of natural gas daily.

Land surveying would begin in May 2014. Construction would begin in 2017 and end in 2019. Atlantic planned for full operation in late 2019.

Financial
Atlantic states in their proposal that constructing the 600 mile pipeline creates 17,240 jobs and 2.7 billion dollars of financial activity. Many of the jobs created are short lived because labor requirements decrease once the pipeline is built. Construction expenses would create 4.2 million dollars per year of tax revenue for local governments. Atlantic pays 30 million annually in property taxes everywhere the pipe passes.

Electricity
Natural gas prices may stabilize from the pipeline. Atlantic claims VA and NC residents could save 377 million dollars a year on electric bills. Atlantic's estimates for electricity savings do not account for Jevon's Paradox which implies that consumption will increase with the efficiency of the system. Cheaper electricity incentivizes greater consumption.

Pipeline Detriments
Living near a pipeline is dangerous. Properties near the pipeline can lose value due to the risk of gas leaks and explosions. Landowners may lose their property in acquisitions for building rights. In the US from 2010 to 2019, there were 6,298 pipeline incidents, 140 fatalities, and 5.3 billion dollars in damages. In the US from 2010 to 2019, there were 922 fires and 329 explosions caused by fossil fuel lines. The Atlantic Coast Pipeline endangers the local ecosystems it passes.

Government Involvement
The pipeline's frequent legal issues and its construction across several states led to government involvement in its development. The Federal Energy Regulatory Commission (FERC) monitors environmental impact in determining authorization of pipelines. The Trump Administration declared the pipeline to be one of its top infrastructure deals because of potential economic impact and the creation of new jobs, and signed executive orders to expedite construction. The Virginia State Congress sent a bipartisan letter of support to the FERC advocating for pipeline approval.

Consumer Energy Alliance (CEA)
The CEA is a lobbying organization founded by oil and gas industry insiders, positioned as a pro-consumer group. The CEA sent an open letter to the FERC in support of the pipeline, emphasizing its importance in bringing energy justice to the non-drilling state of North Carolina. The CEA also created two pipeline-related reports, claiming that burning natural gas saves money, and that the development of the pipeline would be good for the environment.

The Waterkeeper Alliance
The Waterkeeper Alliance is a trade association focused on protecting waterways across the country. They opposed the environmental hazards of the pipeline, such as deforestation increasing erosion and the risk of broken pipes leaking contaminants into the soil. There are several individual waterkeepers of note within this organization, who educated and mobilized citizens, developed pipeline and construction monitoring programs, organized grassroots opposition, filed lawsuits against the FERC, and more.

Alliance for the Shenandoah Valley
The Alliance for the Shenandoah Valley includes the Augusta County Alliance and the Shenandoah Valley Network. They opposed the damage and devaluation of historical, residential, and conservation sites, as well as the safety hazards introduced. They distributed anti-pipeline fact sheets, citing damages and risks proven by existing pipelines.

Union Hill, Virginia
The citizens of Union Hill lived in the path of the pipeline, and they opposed the risk of property damage and air pollution to their historically black town built upon reclaimed plantation land from the Civil War era. They argued against the pipeline in the 4th Circuit Court of Appeals in Richmond and won the case to vacate the pipeline's construction permits with the court writing "Environmental Justice is not merely a box to be checked". In January 2020, they were recognized as an environmental justice community.

United States Forest Service v. Cowpasture River Preservation Association
Cowspasture River Preservation Association filed a petition with the Forest Service to review the 4th Circuit Court of Appeals' decision for the special use permit issued by the agency. The 4th Circuit Court determined the Forest Service did not have jurisdiction to grant the right-of-way according to the Mineral Leasing Act because the Appalachian Trail is a part of the National Park System. The Supreme Court case overruled that decision and determined the Forest Service did have the right to grant building permissions.

Cessation of the Atlantic Coast Pipeline
North Carolina, Virginia, and West Virginia united to protest the Atlantic Coast Pipeline construction. Even though Atlantic won the Supreme court hearing, “it was not enough to turn the tide on their flailing project”. Primarily due to public disapproval and legal uncertainty, Atlantic announced the cessation of the Atlantic Coast Pipeline on July 5, 2020.

Pipeline Future
On December 18, 2020, Atlantic proposed a Disposition and Restoration Plan to the FERC, giving detailed plans for closing the project over the next 24 months. Starting with the statement, "Atlantic will comply with all the terms and conditions set forth in its existing and planned state and federal permits". Atlantic is seeking approval from FERC to retire any pipes currently installed underground and approval to de-commit facilities across construction sites, to begin and finish the restoration in areas of the right-of-way that might have been disturbed. Atlantic confirmed that customers will not be held responsible for costs associated with the clean, "No, the companies have no plans to seek cost recovery from customers for this project" as stated in their FAQ. Atlantic will continue efforts to make natural gas more accessible across the country since they believe it plays a considerable role in the path to net-zero emissions.

Energy Alternatives
Compared to other fossil fuels Atlantic's proposed natural gas pipeline would have reduced energy emissions so its cancellation leaves a need for alternative solutions to combat climate change. Although the solution isn't known to man, it will take a combination of high-tech and low-tech solutions. Solar, wind, nuclear, and hydroelectric power are the most promising renewable energy sources sought after today. Solar is an abundant and inexhaustible energy, and comes in various forms: solar heat, solar photovoltaics, solar thermal electricity, and solar fuels. Wind power is the most cost effective energy source. Nuclear energy has a low cost of operation and high energy density. Hydroelectric is highly reliable and efficient, provides flood control, and clean drinking water.

Conclusion
Natural gas serves many valuable purposes as the most eco-friendly fossil fuel. Expanding its usage and providing more natural gas to power the grid through the Atlantic Coast Pipeline seems good on paper. However, the environmental and human cost to building the pipeline outweigh the energy benefits provided by natural gas. Forgetting to take into account the incidental effects of building the pipeline motivated participants to protest, and led to the litigation and financial pressure that ultimately doomed the pipeline. Further research into the topic includes minimizing the environmental issues with constructing the pipeline and investigating the barriers preventing the pursuit of alternative energy sources by giant corporations.

title: Lentis/Atlantic Coast Pipeline


text:
Radiation Oncology/Ecological Footprint

Ecological Footprint of Health Care

General Carbon Footprint

 * University of Bergen, Norway; 2021 PMID 34711584 -- "Priority setting and net zero healthcare: how much health can a tonne of carbon buy?" (Bhopal A, BMJ. 2021 Oct 28;375:e067199.); Link
 * Proposal to use carbon as part of health care priority setting
 * Clinical decision-making; health system perspective; and a global perspective

 * UK NHS; 2021 PMID 33581070 -- "Health care's response to climate change: a carbon footprint assessment of the NHS in England" (Tennison I, Lancet Planet Health. 2021 Feb;5(2):e84-e92.). Overview Link. PDF Link
 * Model using Scope 1 (direct emissions from owned or controlled sources), Scope 2 (indirect emissions from purchased electricity, heat, cooling), Scope 3 (other indirect costs associated with organization's operations, including medical equipment and food/catering, as well as staff commute), and Non-Protocol (patient and visitor travel)
 * Outcome: In 2019, NHS emissions were 25 megatones of CO2 equivalent (reduction of 26% since 1990). Footprint included supply chain (62%), direct delivery of care (24%), staff and patient/visitor travel (10%), and commissioned private health care services (4%)

 * US National Expenditure Accounts
 * 2020 PMID 33284703 -- "Health Care Pollution And Public Health Damage In The United States: An Update" (Eckelman MJ, Health Aff (Millwood). 2020 Dec;39(12):2071-2079.)
 * US health care emissions 1,693 kg CO2 per person in 2018, slowly risking since 2012
 * Contribution to emissions: 1) Direct emissions from health care operations (7%), 2) Indirect emissions from purchased energy (11%), 3) Indirect emissions from supply chain (82%). Largest supply chain contribution from pharmaceuticals/chemicals, food, energy, plastics
 * Variation in state-level emissions, which were not highly correlated to quality
 * 2016 PMID 27280706 -- "Environmental Impacts of the U.S. Health Care System and Effects on Public Health" (Eckelman MJ, PLoS One. 2016 Jun 9;11(6):e0157014.). Article Link
 * Estimate of direct and indirect emissions attributable to health care sector using economic input-output life cycle assessment modeling using National Health Expenditures from 2003-2013
 * Greenhouse gas emissions 655 million metric tons CO2 equivalent (9.8% national total). Health care (36%), physician and clinical services (12%) and prescription drugs (10%) were largest categories.
 * Direct emissions from on-site operation 2.5%; majority of the emission from supplies of energy, goods, and services. Power generation (36%)
 * Additional impact in producing acid rain (12%), GHG (10%), smog formation (10%), air pollutants (9%) causing 470,000 DALYs lost. The are commensurate with 44k-98k of patients who die in hospitals from preventable medical errors
 * 2009 PMID 19903917 -- "Estimate of the carbon footprint of the US health care sector" (Chung J, JAMA. 2009 Nov 11;302(18):1970-2.)
 * US health sector contributed 8% of greenhouse gas emission nationally, and 7% of CO2 emission.

 * NHS, 2016 PMID 31098080 -- "Reducing the carbon footprint of hospital-based care" (Tomson C, Future Hosp J. 2015 Feb;2(1):57-62.). Article Link

Radiation Oncology

 * Princess Margaret, 2023 (2019-2021) PMID 36309074 -- "Evaluating the Short-term Environmental and Clinical Effects of a Radiation Oncology Department's Response to the COVID-19 Pandemic" (Cheung R, Int J Radiat Oncol Biol Phys. 2023 Jan 1;115(1):39-47. doi: 10.1016/j.ijrobp.2022.04.054. Epub 2022 Oct 26.)
 * Retrospective. 10,175 patients resulting in 176,423 fractions.
 * Outcome: Hypofractionation 17% to 27%. Carbon footprint reduced by 39%. Reduction in visits to outpatient clinic (39% vs 25%, SS) or ED department (6% vs 2%, SS)
 * Conclusion: Environmental benefit of hypofractionation and virtual care, with no added acute adverse events

 * TARGIT-A Trial
 * Randomized trial. Traditional EBRT vs TARGIT single fraction
 * UK data; 2016 PMID 27160842 -- "Environmental and social benefits of the targeted intraoperative radiotherapy for breast cancer: data from UK TARGIT-A trial centres and two UK NHS hospitals offering TARGIT IORT" (Coombs NJ, BMJ Open 2016 May 9;6(5):e010703.) Paper link
 * Retrospective analysis. 505 UK patients with geographic data. Google Maps. Fuel efficiency of 40 mpg, CO2e ~285 g/mile.
 * Outcome: Distance travelled TARGIT mean 87 miles vs mean 392 miles (SS). CO2 emissions 24.7 kg vs 111 kg (SS). Time travelling 3 hr vs 14 hr (SS). Patients in 2 semi-rural hospitals larger journeys 753 miles, 30 hrs, and 215 kg CO2 per patient
 * TARGIT intraoperative radiotherapy significant reduces journeys for treatment and has environmental benefits
 * Global impact; 2022 PMID 36033486 -- "Global adoption of single-shot targeted intraoperative radiotherapy (TARGIT-IORT) for breast cancer-better for patients, better for healthcare systems" (Vaidya JS, Front Oncol. 2022 Aug 11;12:786515.)
 * Retrospective analysis. Data from 242 of 260 centers in 35 countries. Approximately 44,752 patients treated with TARGIT between 1998 and 2020. Travel data from the UK paper were used. Assumed 66% patients were urban (travel distance saved 305 miles) vs rural (750 miles saved)
 * Outcome: Estimate >20 million miles travel saved, representing 5.6 million kg of CO2 equivalent
 * Conclusion: TARGIT IORT estimates of ecological benefit

Cataract Surgery

 * Cochin University, France; 2021 PMID 34823888 -- "The carbon footprint of cataract surgery in a French University Hospital" (Ferrero A, J Fr Ophtalmol. 2021 Nov 22;S0181-5512(21)00538-6. doi: 10.1016/j.jfo.2021.08.004.)
 * One day of surgery. Greenhouse gas emission related to patient and staff transportation, building energy consumption, and life cycle assessment (LCA) of equipment and sterilization processes
 * Outcome: LCA of disposable items was 59.5 kg CO2 per procedure. Single procedure generated 2.8 kg of CO2. Average patient transportation per procedure was 7.3 kg of CO2. Sterilization was 2.1 kg. Energy consumption of the building was 0.8 kg and staff transport 0.08 kg.
 * Conclusion: Total carbon footprint of one cataract procedure was 81.1 kg CO2 equivalent.

 * University of Otago Welligton, NZ; 2021 PMID 34531593 -- "The carbon footprint of cataract surgery in Wellington" (Latta M, N Z Med J. 2021 Sep 3;134(1541):13-21.)
 * Use of Eyefficiency tool. Carbon footprint at 4 hospitals. 142 cataract surgeries.
 * Outcome: a cataract surgery produced on average 152 kg of CO2 equivalent. Greatest contribution from disposable materials and from patient travel

 * Multi-institutional; 2021 PMID 34104796 -- "Improving productivity, costs and environmental impact in International Eye Health Services: using the 'Eyefficiency' cataract surgical services auditing tool to assess the value of cataract surgical services" (Goel H, BMJ Open Ophthalmol. 2021 May 20;6(1):e000642.)
 * Use of Eyefficiency tool. Carbon footprint at 9 cataract surgical facilities worldwide. 1 week or 30 consecutive cataract surgeries continuously.
 * Outcome: 475 cataract extractions (mix phacoemulsification and manual incision). Greenhouse gases ranged from 41 kg of CO2 equivalent to 130 kg CO2 equivalent
 * Conclusion: Global diversity in metrics

 * University Hospital of Wales; 2013 PMID 23429413 -- "The carbon footprint of cataract surgery" (Morris DS, Eye (Lond). 2013 Apr;27(4):495-501.)
 * Component analysis for one patient, including building/energy use, travel, and procurement
 * Outcome: Carbon footprint 182 kg CO2eq. Given total 2230 patient treated in 2011, overall footprint 405.5 tonnes CO2eq. Building/energy use 36%, travel 10%, procurement 54% (medical equipment 33%).

Head&Neck Surgery

 * Dalhousie University, Canada; 2021 PMID 33875009 -- "Carbon footprint reduction associated with a surgical outreach clinic" (Forner D, J Otolaryngol Head Neck Surg. 2021 Apr 19;50(1):26.)
 * Self-administered survey of 113 patient travel patterns to outreach clinic vs regional centre. 12/2019 - 02/2020.
 * Outcome: Own car (86%). Median distance to outreach clinic 29 kg vs regional centre 327 km. Mean CO2 emission 117 kg to 144 kg per person; equivalent to individual's 2.5% of annual carbon footprint.
 * Conclusion: Surgical outreach clinics decrease carbon footprint associated with patient travel

Hysterectomy

 * University of Pittsburgh; 2015 PMID 25517602 -- "Environmental impacts of surgical procedures: life cycle assessment of hysterectomy in the United States" (Thiel C, Environ Sci Technol. 2015 Feb 3;49(3):1779-86.) Link
 * LCA analysis
 * Outcome: Different mix of hysterectomies, impact ~400 kg CO2 per case

Operating Rooms

 * Multi-Institutional, Canada; 2017 PMID 29851650 -- "The impact of surgery on global climate: a carbon footprinting study of operating theatres in three health systems" (MacNeill AJ, Lancet Planet Health. 2017 Dec;1(9):e381-e388.)
 * Three academic hospitals in Canada (Vancouver General Hospital), USA (University of Minnesota) and UK (John Radcliffe), 1 year period in 2011
 * Outcome: Total annual footprint 5,188 tons kg at JRH, 4,182 tons kg at UMMC, 3,219 tons kg at VGH. Per unit area in m2 JRH 1,702 kg/m2, UMMC 2,284 kg/m2, VGH 1,951 kg/m2. Per operation JRH 173 kg, UMMC 232 kg, VGH 146 kg. Largest sources of consumption anaesthetic gases and energy consumption. Surgical theatres 3x-6x more energy-intense than hospitals as a whole due to HVAC requirements.

General Surgery Practice

 * University of Leeds; 2013 PMID 23998839 -- "Carbon footprint of patient journeys through primary care: a mixed methods approach" (Andrews E, Br J Gen Pract. 2013 Sep;63(614):e595-603.) Link
 * 650 daily journeys. 306 patient filled travel survey in March 2012.
 * Outcome: 55% by car (62% <2 miles), 30% walked (64% <1 mile), 10% bus. Total carbon footprint 63 tonnes CO2 per year
 * Comment: PMID 24152453 -- "Carbon footprint of patient journeys" (Clayton M, Br J Gen Pract. 2013 Oct;63(615):517. Paper link
 * Patient choice regarding seeing physicians far away

Minimally Invasive Surgery

 * University of Western Ontario, Canada; 2012 PMID 22845049 -- "Environmental impact of minimally invasive surgery in the United States: an estimate of the carbon dioxide footprint" (Power NE, J Endourol. 2012 Dec;26(12):1639-44.)
 * Retrospective. Incremental use of CO2 in minimally invasive surgery (MIS). Scope 1 emissions was CO2 used during surgery. 2.5 million procedures in the US in 2009 identified.
 * Outcome: Scope 1 emissions 303 tonnes CO2. Scope 2 and 3 emissions 355,621 tonnes including gas transport and biomedical waste
 * Conclusion: Direct CO2 use minimal, but indirect CO2 use in minimally invasive surgeries large environmental impact

The aim of this study was to estimate the effect that the expansion of laparoscopic surgery has had on global
 * Wansbeck Hospital, Newcastle, UK; 2008 PMID 18095020 -- "The carbon footprint of laparoscopic surgery: should we offset?" (Gilliam AD, Surg Endosc. 2008 Feb;22(2):573.) Paper link
 * Retrospective. Laparoscopic procedures performed. CO2 cylinders identified.
 * Outcome: Surgeries 1,629. OR time 1.01 hours. CO2 Cylinders 415. Operative time per cylinder 4 hours. Each cylinder contributes 0.0009 tones CO2. Total 450 liters = 0.0009 tonnes
 * Conclusion: Direct CO2 impact minimal

Renal Service

 * MetroHealth System, Cleveland, Ohio; 2022 PMID 35654600 -- "Sources of Variation in the Carbon Footprint of Hemodialysis Treatment" (Sehgal AR, J Am Soc Nephrol. 2022 Jun 2
 * Retrospective. 15 Ohio hemodialysis facilities, in 2020.
 * Outcome: Annual emissions ~769 tonnes CO2e, and 59 kg CO2e per treatment. Largest contributors patient/staff transportation 28%, electricity 27%, natural gas 15%. Equivalent to 93 home energy use
 * Variability: 3x across facilities; largest contributor transportation
 * Conclusion: Similar treatments in similar geographic region show substantial variation in carbon footprint

 * Hassan II University, Cacablanca, Morocco; 2021 PMID 33159433 -- "Carbon footprint of a hemodialysis unit in Morocco" (Mtioui N, Ther Apher Dial. 2021 Oct;25(5):613-620.
 * Retrospective, 2019 data. 3x4 hours, 424 m2, 24 generators, 80 patients
 * Outcome: 409 tonnes CO2e per year; or 5.11 tonnes CO2e per patient per year. Largest contributions electricity (28%), equipment purchase/service (27%), staff and patient travel (22%)

 * Dorset County Hospital, UK; 2010 PMID 20719900 -- "The carbon footprint of a renal service in the United Kingdom" (Connor A, QJM. 2010 Dec;103(12):965-75.)
 * Activity data for building energy use, travel, and procurement
 * Outcome: Carbon footprint 3,006 tonnes CO2 eq per year; building energy 13%, travel 15%, procurement 72% (pharmaceuticals 35%, medical equipment 25%). Emissions 161 kg CO2 eq per bed per day inpatient and 22 kg outpatient.

 * Geelong Hospital, Australia; 2013 PMID 23731962 -- "The carbon footprint of an Australian satellite haemodialysis unit" (Lim AEK, Aust Health Rev. 2013 Jun;37(3):369-74.)
 * Suburban satellite hemodialysis unit in 2011.
 * Outcome: Annual per-patient 10.2 tonnes CO2 equivalent. Contributions pharmaceuticals 36%, medical equipment 23%. Contribution of water (4% to 12%) and electricity (5% to 19%) varies throughout Australia

Meta-Analysis

 * Dalhousie University, Canada; 2021 PMID 34251875 -- "Operating in a Climate Crisis: A State-of-the-Science Review of Life Cycle Assessment within Surgical and Anesthetic Care" (Drew J, Environ Health Perspect. 2021 Jul;129(7):76001.)
 * Overview of life cycle assessment for surgery and anesthesia. 44 studies eligible
 * Outcome: Annual impact between 3,200,000 kg and 5,200,000 kg of CO2. Per case, range 6 kg to 1,007 kg CO2 equivalent. Main emission sources anesthetic gases, single use equipment, HVAC

 * Wichita State; 2020 no PMID -- "Quantifying hospital services by carbon footprint: A systematic literature review of patient care alternatives" Alshqaqeeq F, Resources, Conservation and Recycling, 2020 March. Link
 * Peer reviewed articles Jan 2000 - Dec 2016. 48 studies identified. Environmental scopes: 1) Travel-related energy, 2) Direct energy, 3) Procedural goods and equipment energy

 * NHS Systematic Review; 2020 PMID 32516230 -- "The Carbon Footprint of Surgical Operations: A Systematic Review" (Rizan C, Ann Surg. 2020 Dec;272(6):986-995.) Link
 * 8 eligible publications.
 * Outcome: Carbon footprint of single operation range 6 kg to 814 kg of CO2 equivalent. Major sources were electricity use and consumables

 * James Cook University, Australia; 2012 PMID 23078475 -- "The energy burden and environmental impact of health services" (Brown LH, Am J Public Health. 2012 Dec;102(12):e76-82.). Link
 * 38 publications
 * Review of National Level Health Service Emissions (16 studies); Hospital / Hospital Facilities (8 studies); Surgery (4 studies); Water consumption (4 studies)

Telemedicine

 * UK NHS Lothian; 2022 PMID 35134839 -- "Calculating the carbon footprint of a Geriatric Medicine clinic before and after COVID-19" (Bartlett S, Age Ageing. 2022 Feb 2;51(2). Paper link
 * Retrospective. 48 patient prior to social distancing and 39 patients after social distancing (7 seen via telemed). Estimates for virtual (0.99 kg CO2e) and face-to-face (4.82 kg CO2e) consults, which included use of PPE. Scope 2 and Scope 3. UK petrol car 0.3 kgCO2e per mile. UK grid 0.233 kgCO2e per kWh. PPE 0.15 kgCO2e per outfit. Video call 2.9 kgCO2 per hour
 * Outcome: Mean travel distance 4.7 miles. Hybrid (face-to-face + telemed) patient travel 24 kg CO2e vs face-to-face 42 kg CO2e. Clinic overhead ~30 kg CO2e.
 * Conclusion: Environmental benefit of hybrid model with telemedicine consultations

 * St James Hospital, Dublin, Ireland; 2021 PMID 34464571 -- "Time, cost and carbon-efficiency: a silver lining of COVID era virtual urology clinics?" (Croghan SM, Ann R Coll Surg Engl. 2021 Sep;103(8):599-603.)
 * Retrospective. 1,016 consultations, 72% virtual.
 * Outcome: Travel saving 93.8 miles round trip. Cost saving 25.98 pounds. CO2e reduction 6.1 tonnes
 * Conclusion: Virtual clinics provide efficiency across number of domains

 * University of Southampton, UK; 2021 PMID 35372776 -- "The environmental impacts of telemedicine in place of face-to-face patient care: a systematic review" (Ravindrane R, Future Healthc J. 2022 Mar;9(1):28-33.) Paper link
 * Systematic review. 14 studies.
 * Outcome: All studies found benefit

 * University of Cambridge, UK; 2021 PMID 33791483 -- "Does telemedicine reduce the carbon footprint of healthcare? A systematic review" (Purohit A, Future Healthc J. 2021 Mar;8(1):e85-e91. Paper link
 * Systematic review. 14 studies.
 * Outcome: savings ranged from 0.7 to 372 kg CO2e, but were highly context specific. The highest was airplane travel saved from Scotland to California H&N surgeon, who watched nasopharyngoscopy virtually

 * Sorbonne University; 2021 PMID 34454847 -- "Positive environmental impact of remote teleconsultation in urology during the COVID-19 pandemic in a highly populated area" (Filfilan A, Prog Urol. 2021 Dec;31(16):1133-1138). Paper link
 * Prospective, 2 weeks, patients scheduled for remote teleconsultation. 80 patients. Conversion 1kWh = 0.1 kg CO2e. 1 km car = 0.175 kg CO2e, while 1 km metro = 0.003 kg CO2e.
 * Outcome: Face-to-face would have resulted in 84 km per patient, total 6699 km. Cars usual mode of transport (97%). CO2 avoided was 1141 kg CO2e (99% reduction).
 * Secondary outcomes: Patients saved 974 euro on transport and 112 hours travel time (1.4 hours per patient). Good experience with teleconsultation in 86%
 * Conclusion: Teleconsultation reduces environmental impact

 * University of Cambridge; 2021 PMID 33791483 -- "Does telemedicine reduce the carbon footprint of healthcare? A systematic review" (Purhoit A, Future Healthc J. 2021 Mar;8(1):e85-e91.)
 * Systematic review. CO2 savings per consultation 0.7 - 372 kg, highly context specific. Telemedicine unit energy use very low

 * University Health Network, Toronto; 2010 PMID 20958198 -- "The impact of telemedicine on greenhouse gas emissions at an academic health science center in Canada" (Masino C, Telemed J E Health. 2010 Nov;16(9):973-6.)
 * Travel distances for each appointment evaluated by Google Maps
 * Outcome: Avoided travel 757,234 km; equivalent to 185 metric tons of CO2e. Videoconference unit consumption 42 kg CO2. Air pollutants 360 kg avoided.

 * Umea University, Sweden, 2014 PMID 25188322 -- "Carbon footprint of telemedicine solutions--unexplored opportunity for reducing carbon emissions in the health sector" (Holmner A, PLoS One. 2014 Sep 4;9(9):e105040.) Paper link
 * Lifecycle inventory of 2 rehab units in 2012. 238 telemedicine visits. One hour telemed appointment between 1.9 and 8.4 kgCO2e, depending choice of telemed hardware, bandwidth, meeting duration
 * Outcome: Avoided travel distance 82,310 km. Avoided carbon cost 602 kgCO2e (~98% reduction)
 * Conclusion: Telemedicine is a potent carbon reduction strategy

Environmental Impact

 * The Lancet Countdown
 * 2021 PMID 34687662 -- "The 2021 report of the Lancet Countdown on health and climate change: code red for a healthy future" (Romanello M, Lancet. 2021 Oct 30;398(10311):1619-1662.) Link
 * 2020 PMID 33278353 -- "The 2020 report of The Lancet Countdown on health and climate change: responding to converging crises" (Watts N, Lancet. 2021 Jan 9;397(10269):129-170.)
 * 2019 PMID 31733928 -- "The 2019 report of The Lancet Countdown on health and climate change: ensuring that the health of a child born today is not defined by a changing climate" (Watts N, Lancet. 2019 Nov 16;394(10211):1836-1878.)
 * 2018 PMID 29096948 -- "The Lancet Countdown on health and climate change: from 25 years of inaction to a global transformation for public health" (Watts N, Lancet. 2018 Feb 10;391(10120):581-630.) Link
 * 2017 PMID 27856085 -- "The Lancet Countdown: tracking progress on health and climate change" (Watts N, Lancet. 2017 Mar 18;389(10074):1151-1164.)
 * 2015 PMID 26111439 -- "Health and climate change: policy responses to protect public health" (Watts N, Lancet. 2015 Nov 7;386(10006):1861-914.) Link
 * Formation of The Lancet Commission on Health and Climate Change

 * Northeastern University/Yale University; 2018 PMID 29072942 -- "Estimated Global Disease Burden From US Health Care Sector Greenhouse Gas Emissions" (Eckelman M, Am J Public Health. 2018 Apr;108(S2):S120-S122.)
 * Estimate of disease burden from US health care sector LCA emissions of 614 million tons of CO2 equivalent in 2013
 * Outcome: 123,000 to 381,000 disability adjusted life-years of future health damage; malnutrition largest damage

title: Radiation Oncology/Ecological Footprint


text:
LMIs in Control/Matrix and LMI Properties and Tools/Discrete Time/Discrete Time Negative Imaginary Lemma

The System
Given a square, discrete-time LTI system G: L2e --> L2e with state-space realization (Ad, Bd, Cd, Dd) where

$$A_d \in \mathbb{R}^{n x n}$$, $$B_d \in \mathbb{R}^{n x m}$$, $$C_d \in \mathbb{R}^{m x n}$$, and $$D_d \in \mathbb{R}^{m x m}$$.

In this system, $$C_d(z1 - A_d)^{-1} B_d + D_d = B_d^T (z1 - A_d^T)^{-1} C_d^T + D_d^T$$ and $$det(1+A) \neq 0$$  and $$det(1-A) \neq 0$$.

The Data
$$A_d \in \mathbb{R}^{n x n}$$, $$B_d \in \mathbb{R}^{n x m}$$, $$C_d \in \mathbb{R}^{m x n}$$, and $$D_d \in \mathbb{R}^{m x m}$$

The LMI:
The system G posed above is considered to be negative imaginary under either of the sufficient and necessary conditions:

 * 1) There exists $$P \in \mathbb{S}^{n}$$, where P > 0 such that

$$A_d^T P A_d - P \leq 0,$$

$$C_d + B_d^T (A_d^T - 1)^{-1} P (A_d +1) = 0$$

2. There exists $$Q \in \mathbb{S}^{n}$$, where Q > 0 such that

$$A_d Q A_d^T - Q \leq 0,$$

$$B_d + (A_d -1)^{-1} Q (A_d^T + 1)C_d^T = 0$$

Conclusion
By using the LMI described above, a discrete LTI system can be evaluated for the negative imaginary condition.

Implementation
This LMI can be implemented in any LMI solver such as YALMIP, using an algorithmic solver like MOSEK.

Related LMIs

 * Negative Imaginary Lemma


text:
LMIs in Control/Matrix and LMI Properties and Tools/Discrete Time/Discrete Time System Zeros With Feedthrough

The System
Given a square, discrete-time LTI system G: L2e --> L2e with minimal state-space realization (Ad, Bd, Cd, Dd) where

$$A_d \in \mathbb{R}^{n x n}$$, $$B_d \in \mathbb{R}^{n x m}$$, $$C_d \in \mathbb{R}^{p x n}$$, and $$D_d \in \mathbb{R}^{p x m}$$ with m $$\leq$$ p. Dd is full rank.

The transmission zeros of $$G(z) = C_d(z1 - A_d)^{-1} B_d + D_d $$ are the eigenvalues of: $$A_d - B_d (D_d^T D_d)^{-1} D_d^T C_d$$.

The Data
$$A_d \in \mathbb{R}^{n x n}$$, $$B_d \in \mathbb{R}^{n x m}$$, $$C_d \in \mathbb{R}^{p x n}$$, and $$D_d \in \mathbb{R}^{p x m}$$ with m $$\leq$$ p. Dd is full rank.

The LMI:
With the system defined above, it can be seen that G(z) is minimum phase if and only if there exists $$P \in \mathbb{S}^{n}$$, where P > 0, such that:

$$\begin{bmatrix} P & (A_d - B_d(D_d^T D_d)^{-1} D_d^T C_d)P \\ * & P \end{bmatrix} > 0$$.

If the system G is square (m = p), then full rank Dd implies Dd-1 exists and the above LMI simplifies to:

$$\begin{bmatrix} P & (A_d - B_d D_d^{-1} C_d)P \\ * & P \end{bmatrix} > 0$$.

Conclusion
With the LMI constructed above, the system zeros for a discrete-time LTI system with feedthrough can be found and verified.

Implementation
The LMI can be implemented using a platform like YALMIP along with an LMI solver such as MOSEK to compute the result.

Related LMIs

 * System Zeros without feedthrough
 * System zeros with feedthrough


text:
LMIs in Control/Stability Analysis/Continuous Time/Transient State Bound for Non-Autonomous LTI Systems

The System
For a continuous-time LTI system with a state-space representation of:

$$\dot{x} = Ax + Bu$$

where $$A \in \mathbb{R}^{n x n}$$, $$B \in \mathbb{R}^{n x m}                               $$ and x(0) = x0,

the transient bound can be evaluated with the following LMI.

The Data
$$A \in \mathbb{R}^{n x n}$$, $$B \in \mathbb{R}^{n x m}                               $$ and x(0) = x0.

The LMI:
The Euclidean norm of the state satisfies:

$$\lVert x(T) \rVert _2^2 \leq \gamma^2 (\lVert x_0 \rVert _2^2 + \lVert u \rVert _{2T}^2), \forall T \in \mathbb{R}_{\geq 0}$$

if there exists $$P \in \mathbb{S}^{n} $$ and $$\gamma \in \mathbb{R}_{> 0}$$, where P > 0, such that:

 * $$P - \gamma 1 \leq 0, $$
 * $$\begin{bmatrix} P & 1 \\ * & \gamma 1 \end{bmatrix} \geq 0, $$
 * $$\begin{bmatrix} PA + A^T P & PB \\ * & -\gamma 1 \end{bmatrix} \leq 0 . $$

if x0 = 0 and u is a unit-energy input ($$\lVert u \rVert _{2T} \leq 1, \forall T \in \mathbb{R} _{\geq 0} $$), then the above LMIs ensure that $$\lVert x(T) \rVert _{2} \leq  \gamma , \forall T \in \mathbb{R} _{\geq 0} $$

Conclusion
By using this LMI the transient state bound can be analyzed for a given non-autonomous LTI system.

Implementation
The LMI given above can be implemented and solved using a tool such as YALMIP, along with an LMI solver such as MOSEK.

Related LMIs

 * Output Energy Bound for Autonomous LTI Systems


text:
LMIs in Control/Stability Analysis/Discrete Time/Transient Bound for Discrete-Time Non-Autonomous LTI Systems

The System
For a discrete-time LTI system with a state-space representation of:

$$x_{k+1} = A_d x_k + B_d u_k, $$

where $$A_d \in \mathbb{R}^{n x n}$$, and $$B_d \in \mathbb{R}^{n x m}$$,

the transient bound can be analyzed with the LMI below.

The Data
$$A_d \in \mathbb{R}^{n x n}$$, and $$B_d \in \mathbb{R}^{n x m}$$

The LMI:
The Euclidean norm of the state satisfies:

$$\lVert x_k \rVert _2^2 \leq \gamma^2 (\lVert x_0 \rVert _2^2 + \lVert u \rVert _{2k}^2), \forall k \in \mathbb{Z}_{\geq 0}$$

if there exists some $$P \in \mathbb{S}^{n}$$ and $$\gamma \in \mathbb{R}_{> 0}$$, where P > 0 such that:

 * $$P - \gamma 1 \leq 0, $$
 * $$\begin{bmatrix} P & 1 \\ * & \gamma 1 \end{bmatrix} \geq 0, $$
 * $$\begin{bmatrix} A_d^T P A_d - P & A_d^T P B_d \\ * & B_d^T B_d- \gamma 1 \end{bmatrix} \leq 0. $$

if x0 = 0 and u is a unit-energy input ($$\lVert u \rVert _{2k} \leq 1, \forall k \in \mathbb{Z} _{\geq 0} $$), then the above LMIs ensure that $$\lVert x_k \rVert _{2} \leq  \gamma , \forall k \in \mathbb{Z} _{\geq 0} $$

Conclusion
By using this LMI the transient state bound can be analyzed for a given non-autonomous LTI system.

Implementation
The LMI given above can be implemented and solved using a tool such as YALMIP, along with an LMI solver such as MOSEK.

Related LMIs

 * Output Energy Bound for Discrete-Time Autonomous LTI Systems


text:
LMIs in Control/Matrix and LMI Properties and Tools/Lyapunov Stability of a System With Polynomial Dynamics

[UNDER CONSTRUCTION] - CME

Find the local Lyapunov function for a system with known polynomial dynamics.

Required Data
The system dynamics to be tested in the form $$ \dot x = f(x) $$ where $$ f(x) $$ is a polynomial function of the states $$ x $$.

A function or functions representing the local domain over which stability is to be determined, $$ X := \{x : p_i(x) \geq 0, i = 1,...,k\} $$. Commonly this is a ball of specified radius such that $$p(x) = R^2 - x^Tx $$. [Is this assuming P-Compactness of the set?]

Find the Lyapunov Function
Suppose there exists Lyapunov function $$V(x)$$, a scalar $$\epsilon > 0$$, and $$s_0, s_i, t_0, t_i \in \Sigma_s$$ (where $$ \Sigma_s $$ is the set of Sum-of-Squares polynomials, all of which are positive) such that:

$$ V(x) = s_0(x) + \sum \limits_i s_i(x)p_i(x) + \epsilon x^T x $$

$$ -\dot V(x) = -\nabla V(x)^T f(x) = t_0(x) + \sum \limits_i t_i(x)p_i(x) + \epsilon x^T x $$

The Lyaponov function, if it exists, will be positive and always decreasing over the desired set $$ X $$

Example
Code example in SOStools probably worthwhile!

Reference
SOS tools manual is a good one

Tie in the reference from SOS tools? The Khalil?

Related LMIs
Add things here about SOS methods

Global Lyapunov Search


text:
LMIs in Control/Global Minimum of Polynomial via SOS Method

[UNDER CONSTRUCTION] - CME

The global minimum of a certain polynomial functions $$f(x)$$ can be found using Sum-of-Squares (SOS) methods. This is a useful starting point to the more-useful but less straightforward issue of local minimums.

Required Data
A polynomial function $$f(x)$$ whose minimum is desired.

The Problem
To find the minimum value of $$f(x)$$, we suppose that there exists a scalar $$ p $$ such that

$$ f(x) - p \geq 0 $$

This is equivalent to determining if $$(f(x)-p)$$ has a SOS representation, since a SOS polynomial is never negative. This becomes an optimization problem by attempting to find the largest possible $$ p $$ such that a representation exists. In other words, the optimization problem becomes

$$ \min_p -p $$ subject to $$ f(x)-p \geq 0 $$

Example
Code example in SOStools probably worthwhile!

Reference
SOS tools manual is a good one

Related LMIs
SOS Basics

Local Minimum is a solid follow-up


text:
LMIs in Control/Local Minimum of Polynomial via SOS Method

[UNDER CONSTRUCTION] - CME

The local minimum of a certain polynomial functions f(x) can be found using Sum-of-Squares (SOS) methods. Here we assume the local set has a P-Compact representation, and that this representation is used.

Required Data
A polynomial function f(x) whose minimum is desired.

A P-Compact representation of the set.

First Steps
Touch up on the Putinar Positivstellensatz and what it means for us.

Why does it work
Based in S-Procedure, with the Positivstellensatz all about the right representation.

Example
Code example in SOStools probably worthwhile!

Reference
SOS tools manual is a good one

Related LMIs
SOS Basics

Global Minimum


text:
Wikibooks:Reading room/Administrative Assistance/Archives/2021/October

Justanothergenericusernamenothingtoseehere
Please block this vandalism-only account. SHB2000 (discuss • contribs) 02:08, 12 October 2021 (UTC)

 * ✅ —Atcovi (Talk - Contribs) 18:51, 14 October 2021 (UTC)
 * Your favorite emperor also seems to be the same person as well. SHB2000 (discuss • contribs) 00:18, 15 October 2021 (UTC)
 * Dealt with accordingly, thanks. —Atcovi (Talk - Contribs) 14:31, 15 October 2021 (UTC)


text:
History of wireless telegraphy and broadcasting in Australia/Topical/Publications/Radio in ANZ

BEWARE, JUST A CUT AND PASTE OF AUSTRALASIAN RADIO WORLD AT THIS STAGE, FOR USE AS A TEMPLATE

Australasian Radio World was an early radio periodical which circulated primarily in Australia but also to a lesser extent in New Zealand. It commenced publication in April 1936 and its content was initially focused on the amateur and broadcast listener community with a variety of technical articles as well of reports of amateur radio activities and details of recent loggings of amateur and broadcast (medium wave and shortwave) stations.

Publishers, Owners and Editors
Australasian Radio World was a relatively small publication and throughout its print run the roles of ownership, publisher and editor were mostly commonly held and driven by a single passionate individual, often with little additional support. At its commencement in May 1936, it was published by Alfred Earl "Earl" Read, an expatriate New Zealander and former editor of the New Zealand Radio Times. From the December 1936 issue, the publisher formally became Trade Publications Proprietary Limited, but that firm had been behind the publication from its outset. The company was formed in April 1936 as printers and publishers, with initial subscribers Edward Marriott, Alfred Earl Read, Frederick S. Wales, Archie N. A. Wales, John C. L. Driver, Doris Wood and Clive B. Mellor. First directors were Frederick S. Wales and Archie N. A. Wales. Frederick S. Wales, Archie N. A. Wales and John C. L. Driver also had major interests in the magazine's printers Bridge Printery Pty. Ltd. From the April 1939 issue, with the magazine in financial difficulties, the publisher transitioned to Read Publications, a partnership solely between Alfred Earl Read and Frederick Stephen Wales. The rights to the publication were acquired by Allan Galbraith "Braith" Hull, with the first issue under his proprietorship being March 1940.

Publishers
The magazine was initially published by Trade Publications of Sydney, with A. Earl Read as Editor (he was an expatriate New Zealander). Allen Galbraith Hull (brother of the more widely known Ross Hull) purchased the magazine in 1940 and resigned his editorship of Radio and Hobbies. Hull became editor of Australasian Radio World and continued in that capacity until he sold the publication in November 1950. The new owner was Radio and Electronics (N.Z.) Ltd.

Notes prepared to develop this portion of the article (including transcriptions of publication notes from each magazine issue) can be found here:

Radio in Australia & New Zealand - Publication Notes - all available issues complete

Printers
Printing of the magazine was undertaken by Bridge Printery from the initial issue in May 1936 until February 1949 and there was a close relationship between this printer and the two publishers of the period. At commencement the printer was located at 214 George St, Sydney, but in January 1940 the firm relocated to 117 Reservoir St., Sydney. Bridge Printery's period of responsibility for production was almost 13 years, including all but the final two years of the magazine's print run. In the magazine's tenth anniversary issue on May 1946, Braith Hull paid tribute to Bridge Printery's service in glowing terms, noting that production was personally supervised by the principal Fred Wales and his son Noel. As well as the actual printing, the firm took responsibility for magazine wrapping and bulk postage. Publisher Hull had relocated to Melbourne in the mid 1940s to assist with the war effort and struggled with liaison with a Sydney printer. Hull's plan was to establish his own printshop at his Mornington mansion. As an interim measure, Crusader Press of Auburn, Victoria printed two issues in early 1949. The following ten issues till March 1950 were printed by Hull's local newspaper, the Mornington Post. As a result of post World War 2 problems with paper supply and printshop capacity, there were no issues produced for April and May of 1950. Hull again reallocated the placement of printing to The Clyde Press of Thornbury and that firm produced the remaining magazine issues for 1950. However, the magazine was sold late 1950 with transfer of ownership commencing 1 December 1950. The new owners Cranch / Radio & Electronics (NZ), returned printing to Sydney, commencing January 1951. Printing was undertaken by R. V. Byers (an ARW advertiser) of North Lidcombe for the first half of 1951. When Cranch's own company Radio & Electronics (Aust.) assumed control, printing was undertaken by Acacia Press of Annandale, New South Wales. However Cranch's financial difficulties resulted in cessation of production, with the last issue being October 1951.

Notes prepared to develop this portion of the article (including transcriptions of statements as to printer from each magazine issue) can be found here:

Radio in Australia & New Zealand - Printers - all available issues transcribed

Editorials
There were three editors over the lifetime of the magazine: Read, Hull and Cranch. Read only chose to include editorials sporadically, while both Hull and Cranch rarely missed an editorial each month.

For ease of reference, a complete set of the text-corrected editorials is being progressively developed here:

Radio in Australia & New Zealand - Editorials (all available issues complete)

Issues
An almost complete set of magazines (primarily links to the AmericanRadioHistory webpages) can be found here, a small number have text cut and pasted from the OCR'd PDF, and an even smaller number have been fully text corrected:

Radio in Australia & New Zealand - Issues

Contents
The contents pages of each issue have been amalgamated here as an index and finding aid to specific articles and authors:

Radio in Australia & New Zealand - Contents - A brief start

Issue List
For ease of reference, to supplement the transcriptions of each monthly issue, particular types of magazine content have been brought together in single pages (eg. publication notes, printer statements, editorials etc.). To assist the creation of this type of content a list of issues has been prepared for use as an initial proforma:

Radio in Australia & New Zealand - List of Issues ARW

Guide to Editing
Several sections of ARW changed little in format and even content from month to month. To assist in achieving greater consistency of presentation in this Wikibook, a proforma approach has been developed:

Radio in Australia & New Zealand - Guide to Editing ARW


text:
History of wireless telegraphy and broadcasting in Australia/Topical/Publications/Radio in ANZ/Issues

The following issues have been transcribed (at least in part) to date:

1923

 * Radio in Australia & New Zealand - Vol. I No. 01 - 4 April 1923 - A brief start
 * Radio in Australia & New Zealand - Vol. I No. 02 - 18 April 1923 - A brief start
 * Radio in Australia & New Zealand - Vol. I No. 03 - 2 May 1923 - A brief start
 * Radio in Australia & New Zealand - Vol. I No. 04 - 16 May 1923 - A brief start
 * Radio in Australia & New Zealand - Vol. I No. 05 - 30 May 1923 - A brief start
 * Radio in Australia & New Zealand - Vol. I No. 06 - 13 June 1923 - A brief start
 * Radio in Australia & New Zealand - Vol. I No. 07 - 27 June 1923 - A brief start
 * Radio in Australia & New Zealand - Vol. I No. 08 - 11 July 1923 - Coming real soon now!
 * Radio in Australia & New Zealand - Vol. I No. 09 - 25 July 1923 - Coming real soon now!
 * Radio in Australia & New Zealand - Vol. I No. 10 - 8 August 1923 - Coming real soon now!
 * Radio in Australia & New Zealand - Vol. I No. 11 - 22 August 1923 - Coming real soon now!
 * Radio in Australia & New Zealand - Vol. I No. 12 - 5 September 1923 - Coming real soon now!
 * Radio in Australia & New Zealand - Vol. I No. 13 - 19 September 1923 - Coming real soon now!
 * Radio in Australia & New Zealand - Vol. I No. 14 - 3 October 1923 - Coming real soon now!
 * Radio in Australia & New Zealand - Vol. I No. 15 - 17 October 1923 - Coming real soon now!
 * Radio in Australia & New Zealand - Vol. I No. 16 - 31 October 1923 - Coming real soon now!
 * Radio in Australia & New Zealand - Vol. I No. 17 - 14 November 1923 - Coming real soon now!
 * Radio in Australia & New Zealand - Vol. I No. 18 - 28 November 1923 - Coming real soon now!
 * Radio in Australia & New Zealand - Vol. I No. 19 - 12 December 1923 - Coming real soon now!
 * Radio in Australia & New Zealand - Vol. I No. 20 - 26 December 1923 - Coming real soon now!