text:
LMIs in Control/Controller Synthesis/Discrete Time

 * /State Feedback/
 * /Optimal State Feedback/
 * /Output Feedback/
 * /Static Output Feedback/
 * /Optimal Output Feedback/
 * /Optimal Dynamic Output Feedback/
 * Discrete Time Stabilizability
 * Quadratic Schur Stabilization


text:
LMIs in Control/Observer Synthesis/Continuous Time

 * /Optimal Observer and State Estimation/
 * /Optimal Filter/
 * Detectability LMI
 * LMI for the Observability Grammian
 * H-infinity filtering
 * H2 filtering
 * H2 Optimal Observer
 * HInf Optimal Observer
 * Mixed H2 HInf Optimal Observer
 * H2 Optimal Filter
 * HInf Optimal Filter
 * FDI Filter Design For Systems With Sensor Faults: an LMI
 * H2 Optimal State estimation
 * Hurwitz Detectability
 * /Full-Order State Observer/
 * /Full-Order H-infinity State Observer/
 * /Reduced-Order State Observer/

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


text:
LMIs in Control/Observer Synthesis/Discrete Time

 * H2-Optimal Filter]
 * Hinf-Optimal Filter
 * Optimal Observer; Mixed
 * Optimal Observer; H2
 * Optimal Observer; Hinf
 * Discrete Time Detectability
 * Schur Detectability


text:
LMIs in Control/Stability Analysis/Continuous Time

 * /D-Stability/
 * /Time-Delay Systems/
 * /Parametric, Norm-Bounded Uncertain System Quadratic Stability/
 * /Stability of Structured, Norm-Bounded Uncertainty/
 * /Stability under Arbitrary Switching/
 * /Quadratic Stability Margins/
 * Stability of Linear Delayed Differential Equations
 * H infinity Norm for Affine Parametric Varying Systems
 * Entropy Bond for Affine Parametric Varying Systems
 * Dissipativity of Affine Parametric Varying Systems
 * Hankel Norm of Affine Parameter Varying Systems
 * Positive Orthant Stabilizability
 * LMI For Stabilization Condition for Systems With Unsymmetrical Saturated Control
 * LMI Condition For Exponential Stability of Linear Systems With Interval Time-Varying Delays
 * Conic Sector Lemma
 * Polytopic Quadratic Stability
 * Mu Analysis
 * Optimization Over Affine Family of Linear Systems
 * /Hurwitz Stabilizability/
 * /Quadratic Hurwitz Stabilization for Polytopic Systems/
 * /Output Energy Bound for Autonomous LTI Systems/
 * Transient State Bound for Non-Autonomous LTI Systems
 * Transient Output Bound for Autonomous LTI Systems
 * Transient Output Bound for Non-Autonomous LTI Systems
 * /Output Energy Bound for Non-Autonomous LTI Systems/


text:
LMIs in Control/Stability Analysis/Discrete Time

 * D-Stability
 * Discrete-Time Lyapunov Stability
 * LMI for Schur Stabilization
 * L2-Gain of Systems with Multiplicative noise
 * Discrete-Time Quadratic Stability
 * Output Energy Bound for Discrete-Time Autonomous LTI Systems
 * Transient Bound for Discrete-Time Non-Autonomous LTI Systems
 * Output Energy Bound for Discrete-Time Non-Autonomous LTI Systems
 * Transient Output Bound for Discrete-Time Autonomous LTI Systems


text:
LMIs in Control/Controller Synthesis/Continuous Time/State Feedback

 * /H-infinity/
 * /H-2/
 * /Mixed/
 * Stabilization of Second-Order Systems
 * LQ Regulation via H2 Control
 * Controller to achieve the desired Reachable set; Polytopic uncertainty
 * Controller to achieve the desired Reachable set; Norm bound uncertainty
 * Controller to achieve the desired Reachable set; Diagonal Norm-bound uncertainty


text:
LMIs in Control/Controller Synthesis/Continuous Time/Optimal State Feedback

 * /H-infinity/
 * /H-2/
 * /Mixed/


text:
LMIs in Control/Controller Synthesis/Continuous Time/Output Feedback

 * /H-infinity/
 * /H-2/
 * /Mixed/


text:
LMIs in Control/Controller Synthesis/Continuous Time/Static Output Feedback

 * /H-infinity/
 * /H-2/
 * /Mixed/
 * Continuous-Time Static Output Feedback Stabilizability


text:
LMIs in Control/Controller Synthesis/Continuous Time/Optimal Output Feedback

 * /H-infinity/
 * /H-2/
 * /Mixed/


text:
LMIs in Control/Controller Synthesis/Continuous Time/Optimal Dynamic Output Feedback

 * /H-infinity/
 * /H-2/
 * /Mixed/


text:
LMIs in Control/Controller Synthesis/Discrete Time/State Feedback

 * /H-infinity/
 * /H-2/
 * /Mixed/
 * /Closed-Loop Robust Stability and Controller synthesis of Discrete-Time System with Polytopic Uncertainty/


text:
LMIs in Control/Controller Synthesis/Discrete Time/Optimal State Feedback

 * Discrete Time Hinf Optimal Full State Feedback Control
 * Discrete Time H2 Optimal Full State Feedback Control
 * Discrete Time Mixed H2-Hinf Optimal Full State Feedback Control


text:
LMIs in Control/Controller Synthesis/Discrete Time/Output Feedback

 * /H-infinity/
 * /H-2/
 * /Mixed/


text:
LMIs in Control/Controller Synthesis/Discrete Time/Static Output Feedback

 * /H-infinity/
 * /H-2/
 * /Mixed/
 * Discrete-Time Static Output Feedback Stabilizability


text:
LMIs in Control/Controller Synthesis/Discrete Time/Optimal Output Feedback

 * /H-infinity/
 * /H-2/
 * /Mixed/


text:
LMIs in Control/Controller Synthesis/Discrete Time/Optimal Dynamic Output Feedback

 * Discrete Time Hinf Optimal Dynamic Output Feedback Control
 * Discrete Time H2 Optimal Dynamic Output Feedback Control
 * /Mixed/


text:
LMIs in Control/pages/Notion of Matrix Positivity

Notation of Positivity
A symmetric matrix $$A\in\R^{n\times n}$$ is defined to be:

positive semidefinite, $$(A\ge 0)$$, if $$x^TAx\ge 0 $$ for all $$x\in\R^n, x\neq \mathbf{0} $$.

positive definite, $$(A>0)$$, if $$x^TAx> 0 $$ for all $$x\in\R^n, x\neq \mathbf{0} $$.

negative semidefinite, $$(-A\ge 0)$$.

negative definite, $$(-A>0)$$.

indefinite if $$A$$ is neither positive semidefinite nor negative semidefinite.

Properties of Positive Matricies

 * For any matrix $$M$$, $$M^TM>0$$.
 * Positive definite matricies are invertible and the inverse is also positive definite.
 * A positive definite matrix $$A>0$$ has a square root, $$A^{1/2}>0$$, such that $$A^{1/2}A^{1/2}=A$$.
 * For a positive definite matrix $$A>0$$ and invertible $$M$$, $$M^TAM>0$$.
 * If $$A>0$$ and $$M>0$$, then $$A+M>0$$.
 * If $$A>0$$ then $$\mu A>0$$ for any scalar $$\mu>0$$.


text:
LMIs in Control/pages/Basic Matrix Theory

Basic Matrix Notation
Consider the complex matrix $$A\in \C^{n\times m} $$.
 * $$ A=\begin{bmatrix} a_{11} & \dots & a_{1m} \\

\vdots & \ddots & \vdots \\ a_{n1} & \dots & a_{nm}\end{bmatrix} \in \C^{n\times m} $$

Transpose of a Matrix

The transpose of $$A$$, denoted as $$A^{T}$$ or $$A'$$ is:
 * $$ A^T=\begin{bmatrix} a_{11} & \dots & a_{n1} \\

\vdots & \ddots & \vdots \\ a_{1m} & \dots & a_{nm}\end{bmatrix} \in \C^{m\times n}. $$

Adjoint of a Matrix

The adjoint or hermitian conjugate of $$A$$, denoted as $$A^{*}$$ is:
 * $$ A^*=\begin{bmatrix} a_{11}^* & \dots & a_{n1} ^*\\

\vdots & \ddots & \vdots \\ a_{1m}^* & \dots & a_{nm}^*\end{bmatrix} \in \C^{m\times n}. $$ Where $$a_{nm}^*$$ is the complex conjugate of matrix element $$a_{nm}$$.

Notice that for a real matrix $$A\in\R^{n\times m}$$, $$A^*=A^T$$.

Important Properties of Matricies
Hermitian, Self-Adjoint, and Symmetric Matricies

A square matrix $$A\in \C^{n\times n} $$ is called Hermitian or self-adjoint if $$A=A^*$$.

If $$A\in \R^{n\times n} $$ is Hermitian then it is called symmetric.

Unitary Matricies

A square matrix $$A\in \C^{n\times n} $$ is called unitary if $$A^*=A^{-1}$$ or $$A^*A=I$$.


text:
LMIs in Control/pages/Schur Complement

An important tool for proving many LMI theorems is the Schur Compliment. It is frequently used as a method of LMI linearization.

The Schur Compliment
Consider the matricies $$Q$$, $$M$$, and $$R$$ where $$Q$$ and $$M$$ are self-adjoint. Then the following statements are equivalent:
 * $$Q>0$$ and $$M-RQ^{-1}R^*>0$$ both hold.
 * $$M>0$$ and $$Q-R^*M^{-1}R>0$$ both hold.
 * $$\begin{bmatrix} M & R \\

R^* & Q\end{bmatrix} > 0 $$ is satisfied.

More concisely:
 * $$\begin{bmatrix} M & R \\ R^* & Q\end{bmatrix} > 0 \iff \begin{bmatrix} M & 0 \\ 0 & Q-R^*M^{-1}R\end{bmatrix} > 0 \iff \begin{bmatrix} M-RQ^{-1}R^* & 0 \\ 0 & Q\end{bmatrix} > 0

$$

title: LMIs in Control/pages/Schur Complement


text:
LMIs in Control/KYP Lemma (Bounded Real Lemma)

KYP Lemma (Bounded Real Lemma)

The Kalman–Popov–Yakubovich (KYP) Lemma is a widely used lemma in control theory. It is sometimes also referred to as the Bounded Real Lemma. The KYP lemma can be used to determine the $$H_\infty$$ norm of a system and is also useful for proving many LMI results.

The System

\begin{align} \dot x(t)&=Ax(t)+Bu(t)\\ y(t)&=Cx(t)+Du(t)\\ x(0)&=x_0 \end{align}$$ where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$, $$u(t)\in \R^q$$, at any $$t\in \R$$.

The Data
The matrices $$ A,B,C,D $$ are known.

The Optimization Problem
The following optimization problem must be solved.

 * $$\begin{align}

&\underset{\gamma,\;X}{\operatorname{minimize}}\quad \gamma\\ &\operatorname{subject\;to} & X>0\\ & & \begin{bmatrix} A^TX+XA & XB \\B^TX & -\gamma I\end{bmatrix}+\frac{1}{\gamma}\begin{bmatrix} C^T \\D^T \end{bmatrix}\begin{bmatrix} C & D \end{bmatrix}<0\\ \end{align}$$

The LMI: The KYP or Bounded Real Lemma
Suppose $$\hat G(s)(A,B,C,D)$$ is the system. Then the following are equivalent.
 * $$ 1)\quad \left\| G \right\|_{H_\infty} \leq \gamma $$
 * $$ 2)\quad \text{There exists a}\; X>0 \;\text{such that}$$
 * $$ \begin{bmatrix} A^TX+XA & XB \\B^TX & -\gamma I\end{bmatrix}+\frac{1}{\gamma}\begin{bmatrix} C^T \\D^T \end{bmatrix}\begin{bmatrix} C & D \end{bmatrix}<0

$$

Conclusion:
The KYP Lemma can be used to find the bound $$\gamma$$ on the $$H_\infty$$ norm of a system. Note from the (1,1) block of the LMI we know that $$A$$ is Hurwitz.

Implementation
A link to CodeOcean or other online implementation of the LMI (in progress)

Related LMIs
Positive Real Lemma


text:
LMIs in Control/Positive Real Lemma

Positive Real Lemma

The Positive Real Lemma is a variation of the Kalman–Popov–Yakubovich (KYP) Lemma. The Positive Real Lemma can be used to determine if a system is passive (positive real).

The System

\begin{align} \dot x(t)&=Ax(t)+Bu(t)\\ y(t)&=Cx(t)+Du(t)\\ x(0)&=x_0 \end{align}$$ where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$, $$u(t)\in \R^q$$, at any $$t\in \R$$.

The Data
The matrices $$ A,B,C,D $$ are known.

The LMI: The Positive Real Lemma
Suppose $$\hat G(s)(A,B,C,D)$$ is the system. Then the following are equivalent.
 * $$ 1)\quad G \; \text{is passive, i.e.} \; \left\langle u, Gu \right\rangle _{L_2} \ge 0 \; (\hat G(s)+\hat G(s)^*\ge 0) $$
 * $$ 2)\quad \text{There exists a}\; X>0 \;\text{such that}$$
 * $$ \begin{bmatrix} A^TX+XA & XB-C^T \\B^TX-C & -D^T-D\end{bmatrix}\le 0

$$

Conclusion:
The Positive Real Lemma can be used to determine if the system $$G$$ is passive. Note from the (1,1) block of the LMI we know that $$A$$ is Hurwitz.

Implementation
A link to CodeOcean or other online implementation of the LMI (in progress)

Related LMIs
KYP Lemma (Bounded Real Lemma)

title: LMIs in Control/Positive Real Lemma


text:
LMIs in Control/pages/Positive Real Lemma

Positive Real Lemma

The Positive Real Lemma is a variation of the Kalman–Popov–Yakubovich (KYP) Lemma. The Positive Real Lemma can be used to determine if a system is passive (positive real).

The System

\begin{align} \dot x(t)&=Ax(t)+Bu(t)\\ y(t)&=Cx(t)+Du(t)\\ x(0)&=x_0 \end{align}$$ where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$, $$u(t)\in \R^q$$, at any $$t\in \R$$.

The Data
The matrices $$ A,B,C,D $$ are known.

The LMI: The Positive Real Lemma
Suppose $$\hat G(s)(A,B,C,D)$$ is the system. Then the following are equivalent.
 * $$ 1)\quad G \; \text{is passive, i.e.} \; \left\langle u, Gu \right\rangle _{L_2} \ge 0 \; (\hat G(s)+\hat G(s)^*\ge 0) $$
 * $$ 2)\quad \text{There exists a}\; X>0 \;\text{such that}$$
 * $$ \begin{bmatrix} A^TX+XA & XB-C^T \\B^TX-C & -D^T-D\end{bmatrix}\le 0

$$

Conclusion:
The Positive Real Lemma can be used to determine if the system $$G$$ is passive. Note from the (1,1) block of the LMI we know that $$A$$ is Hurwitz.

Implementation
This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/Positive_Real_Lemma.m

Related LMIs
KYP Lemma (Bounded Real Lemma)


text:
LMIs in Control/pages/KYP Lemma (Bounded Real Lemma)

KYP Lemma (Bounded Real Lemma)

The Kalman–Popov–Yakubovich (KYP) Lemma is a widely used lemma in control theory. It is sometimes also referred to as the Bounded Real Lemma. The KYP lemma can be used to determine the $$H_\infty$$ norm of a system and is also useful for proving many LMI results.

The System

\begin{align} \dot x(t)&=Ax(t)+Bu(t)\\ y(t)&=Cx(t)+Du(t)\\ x(0)&=x_0 \end{align}$$ where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$, $$u(t)\in \R^q$$, at any $$t\in \R$$.

The Data
The matrices $$ A,B,C,D $$ are known.

The Optimization Problem
The following optimization problem must be solved.

 * $$\begin{align}

&\underset{\gamma,\;X}{\operatorname{minimize}}\quad \gamma\\ &\operatorname{subject\;to} & X>0\\ & & \begin{bmatrix} A^TX+XA & XB \\B^TX & -\gamma I\end{bmatrix}+\frac{1}{\gamma}\begin{bmatrix} C^T \\D^T \end{bmatrix}\begin{bmatrix} C & D \end{bmatrix}<0\\ \end{align}$$

The LMI: The KYP or Bounded Real Lemma
Suppose $$\hat G(s)(A,B,C,D)$$ is the system. Then the following are equivalent.
 * $$ 1)\quad \left\| G \right\|_{H_\infty} \leq \gamma $$
 * $$ 2)\quad \text{There exists a}\; X>0 \;\text{such that}$$
 * $$ \begin{bmatrix} A^TX+XA & XB \\B^TX & -\gamma I\end{bmatrix}+\frac{1}{\gamma}\begin{bmatrix} C^T \\D^T \end{bmatrix}\begin{bmatrix} C & D \end{bmatrix}<0

$$

Conclusion:
The KYP Lemma can be used to find the bound $$\gamma$$ on the $$H_\infty$$ norm of a system. Note from the (1,1) block of the LMI we know that $$A$$ is Hurwitz.

Implementation
Since the KYP lemma shown above is nonlinear in gamma, in order to implement it in MATLAB we must first linearize it by using the Schur Complement to arrive at the dual formulation below:
 * $$ \begin{bmatrix} A^TX+XA & XB & C^T \\B^TX & -\gamma I & D^T \\ C & D & -\gamma I \end{bmatrix} <0

$$. This dual KYP LMI is now linear in both $$X$$ and $$\gamma$$.

This implementation requires the use of Yalmip and Sedumi.

https://github.com/eoskowro/LMI/blob/master/KYP_Lemma_LMI.m

Related LMIs
Positive Real Lemma


text:
User talk:Jorge.abad1

edit suggestion: creating one part for anthropology and social sciences to show how that approach opposes scientific one? Also economic and political might not be the most crucial and might make it too long as word count is not very large

title: User talk:Jorge.abad1


text:
LMIs in Control/pages/Stabilizability LMI

Stabilizability LMI

A system is stabilizable if all unstable modes of the system are controllable. This implies that if the system is controllable, it will also be stabilizable. Thus, stabilizability is a essentially a weaker version of the controllability condition. The LMI condition for stabilizability of pair $$(A,B)$$ is shown below.

The System

\begin{align} \dot x(t)&=Ax(t)+Bu(t),\\ x(0)&=x_0, \end{align}$$ where $$x(t)\in \R^n$$, $$u(t)\in \R^m$$, at any $$t\in \R$$.

The Data
The matrices necessary for this LMI are $$A$$ and $$B$$. There is no restriction on the stability of A.

The LMI: Stabilizability LMI
$$(A,B)$$ is stabilizable if and only if there exists $$X>0$$ such that
 * $$AX+XA^T-BB^T<0$$,

where the stabilizing controller is given by
 * $$u(t)=-\frac{1}{2} B^TX^{-1}x(t)$$.

Conclusion:
If we are able to find $$X>0$$ such that the above LMI holds it means the matrix pair $$(A,B)$$ is stabilizable. In words, a system pair $$(A,B)$$ is stabilizable if for any initial state $$x(0)=x_0$$ an appropriate input $$u(t)$$ can be found so that the state $$x(t)$$ asymptotically approaches the origin. Stabilizability is a weaker condition than controllability in that we only need to approach $$x(t)=0$$ as $$t\rightarrow \infty$$ whereas controllability requires that the state must reach the origin in a finite time.

Implementation
This implementation requires Yalmip and Sedumi.

https://github.com/eoskowro/LMI/blob/master/Stabilizability_LMI.m

Related LMIs
Hurwitz Stability LMI

Detectability LMI

Controllability Grammian LMI

Observability Grammian LMI


text:
LMIs in Control/Matrix and LMI Properties and Tools/KYP Lemmas/Continuous Time

 * KYP Lemma (Bounded Real Lemma)
 * Positive Real Lemma
 * KYP Lemma for QSR Dissipative Systems
 * KYP Lemma witout Feedthrough
 * KYP Lemma for Descriptor Systems
 * Generealized KYP (GKYP) Lemma for conic Sectors


text:
LMIs in Control/Matrix and LMI Properties and Tools/Continuous Time

 * /Schur Complement/
 * LMI for Eigenvalue Minimization
 * LMI for Matrix Norm Minimization
 * LMI for Generalized Eigenvalue Problem
 * LMI for Linear Programming
 * LMI for Feasibility Problem
 * /Structured Singular Value/
 * /Eigenvalue Problem/
 * LMI for Minimizing Condition Number of Positive Definite Matrix
 * Continuous Quadratic Stability
 * Exterior Conic Sector Lemma
 * Modified Exterior Conic Sector Lemma
 * DC Gain of a Transfer Matrix


text:
LMIs in Control/pages/Peak to Peak norm

Peak-to-peak norm performance of a system

The System
Considering the following system:

\begin{align} \dot{x} = A x + B u\\ z     = C x + D u\\ \end{align}$$

Where $$ x(t) \in \mathbb{R}^{n} $$ is the state signal, $$ u(t) \in \mathbb{R}^{m} $$ is the input signal, and $$ z(t) \in \mathbb{R}^{p} $$ is the output. When given an initial condition $$ x(0) = 0 $$, the system can be defined to map the output and input signals for the peak-to-peak performance.

\begin{align} ||T||_{\infty,\infty} := \sup\limits_{0<||u||_{\infty}<\infty} \frac{||z||_{\infty}}{||u||_{\infty}}\\ \end{align}$$

The Data
The matrices $$ A $$, $$ B $$, $$ C $$, and $$ D $$ are the only data sets required for this optimization problem.

The Optimization Problem
Consider a continuous-time LTI system, $$ G: L_{2e} \to L_{2e} $$, given that: $$ A $$, $$ B $$, $$ C $$, and $$ D $$ $$ A \in \mathbb{R}^{n \times n} $$, $$ B \in \mathbb{R}^{n \times m} $$, $$ C \in \mathbb{R}^{p \times n} $$, and $$ D \in \mathbb{R}^{p \times m} $$. Given that the matrix $$ A $$ is Hurwitz,The peak-to-peak norm of $$ G $$ is given as:

\begin{align} ||T||_{\infty,\infty} := \sup\limits_{0<||w||_{\infty}<\infty} \frac{||z||_{\infty}}{||u||_{\infty}}\\ \end{align}$$

The LMI: Peak-to-Peak norm
There exists a matrix $$ P \in \mathbb{S}^{n} $$ and $$ \gamma $$, $$ \epsilon $$, $$ \mu \in \mathbb{R}_{>0} $$, where the following constraints are used: $$ \min \mu $$

\begin{align} P < 0 \\ \begin{bmatrix} A^T P + P A + \gamma P & P B \\ B' P & \epsilon I\end{bmatrix}&<0\\ \begin{bmatrix} \gamma P & 0 & C^T \\ 0 & (\mu - \epsilon) I & D^T \\ C & D & \mu I\end{bmatrix}&>0\\ \end{align}$$

Since this optimization has $$ \gamma P $$ in the constraints, this does make this optimization bi-linear. attempting to solve this LMI is not feasible unless some type of substitute is implemented to the variables $$ \gamma P $$.

Conclusion:
The results from this LMI will give the peak to peak norm of the system:

\begin{align} \sup\limits_{0<||u||_{\infty}<\infty} \frac{||z||_{\infty}}{||u||_{\infty}} < \mu \end{align}$$

Implementation

 * Peak-to-Peak Norm Example

title: LMIs in Control/pages/Peak to Peak norm


text:
LMIs in Control/pages/S Procedure

S-Procedure

The Optimization Problem
In general procedures, considering following quadratic function $$ F_0(x) : \mathbb{R}^{n} \to \mathbb{R} $$, $$ F_i(x) : \mathbb{R}^{n} \to \mathbb{R} $$ where $$ i = 1,...,m $$. The inequality $$ F_0(x) \leq 0 $$ is satisfied when all $$ F_i(x) \geq 0 $$.

\begin{align} F_0(x)+\sum^{m}_{i=1}\tau_i F_i(x) \leq 0 \end{align}$$

Where $$ x \in \mathbb{R}^{n} $$, and $$ \tau_i \in \mathbb{R}_{\geq 0} $$

This type of procedure is used to help solve problems that were originally NP-hard problems. An example of this is the following inequality: $$ x^T F x \geq 0: \forall x \geq 0 $$. By using the defined problem above, an LMI can be constructed using the S-Procedure:

\begin{align} F - \tau G \succeq 0 \end{align}$$

Where the scalar $$ \tau \geq 0 $$.

The Data
The data is dependent on the type of problem being solved, and is used more as a tool to solve complex problems that were difficult to solve before.

The LMI: S-Procedure
There exists a scalar $$ \tau \geq 0 $$ where:

\begin{align} F - \tau G \succeq 0 \end{align}$$

Conclusion:
The results from this LMI will help construct quadratic stability as quadratic stability requires matrix positivity on a subset. Examples of this implementation include creating a controller based on parametric, norm-bounded uncertainties for robust problems.

Implementation

 * S-Procedure Example

title: LMIs in Control/pages/S Procedure


text:
Z80 Assembly/Z80 Architecture

The Z80 architecture is based on that of the Intel 8080.

Registers
The Z80 has eight main registers, six of which are paired to form three 16-bit general-purpose registers.
 * A (accumulator) is the main 8-bit general-purpose register.
 * F (flags) stores flags used to control the flow of a program that are set based on the last arithmetic instruction:
 * Bit 7 (S; sign bit) indicates whether the result was negative.
 * Bit 6 (Z; zero bit) indicates whether the result was zero.
 * Bit 4 (H; half-carry bit) indicates whether the result required a carry from a lower to an upper nybble.
 * Bit 2 (P; parity bit) indicates the parity of the result.
 * Bit 1 (N; subtraction bit) indicates whether the operation was a subtraction.
 * Bit 0 (C; carry bit) indicates whether a carry occurred.
 * BC is a 16-bit register, made up of B (most significant byte) and C (least significant byte).
 * DE is a 16-bit register, made up of D (most significant byte) and E (least significant byte).
 * HL is a 16-bit register, made up of H (most significant byte) and L (least significant byte). HL is generally used for addressing locations in memory, as there are a number of instructions that use HL for this purpose that don't have counterparts for BC and DE.

Each of these main registers has a shadow register counterpart. These registers are accessed by swapping with the main registers using the EX and EXX instructions. EX can be used to swap A and F with their shadow counterparts A' and F', while EXX swaps the three pairs BC, DE, HL with their counterparts BC', DE', HL'.

In addition to the 8-bit registers there are a number of special 16-bit registers:
 * IX and IY are the index registers, designed to be used as base offsets for addressing memory.
 * PC is the program counter, indicating the location of the currently executing instruction.
 * SP is the stack pointer, indicating the location of the top of the processor stack.

title: Z80 Assembly/Z80 Architecture


text:
Z80 Assembly/Introduction

Why Learn z80?
While most computers today are 64 bit with Intel x86 or AMD processors, there are still many devices that use Z80 processors. Most notable of such devices are GameBoy, Sega Genesis and Handheld Calculators such as Casio and the TI-series. Several other consumer electronics devices such as MIDI synthesizers also have a Z80 processor.

But why 8-bits?
The Z80 microprocessor was designed in the 70s. Its competitor was Intel's 8008. Most computers in the 70s were 8-bit.

title: Z80 Assembly/Introduction


text:
Issues in Interdisciplinarity 2019-20/History of the Nuclear Family in Britain

This chapter will tackle the debate around the emergence of the nuclear family in Britain, within and between disciplines. The nuclear family is the basic type of family, composed of a conjugal pair and their children. To understand the current debates surrounding the changing nature of the family and the reasons for the apparent decline of the nuclear family, studying its emergence is crucial.

Historical Context
The History of the Family only formed after 1958. Initial research assigned the emergence of the nuclear family to the "structural modernisation of western societies since the 19th century". The pre-nuclear family was seen as more complex in structure, changing due to nuclearization, individualism, and emotionalism. From the 1970s, research became more defined, accounting the shift in family structures to a point earlier in History.

Stone's 1977 research locates the late Medieval family, already concentrated around the nuclear family, within a complex system of kinship relations. The family structure began forming more defined boundaries between 1500–1700, resulting in the isolated nuclear family between 1600–1700. The transformation was due to the diminishing importance of kinship and clientage, the increasing state power, and the missionary success of Protestantism.

Macfarlarne (1987) provides a contradicting view, arguing against a change in the structure of the English family in the early modern period, and that the family's key features were not caused by urbanisation, but actually "made such a development possible".

While it is important to recognise that both change and continuity were present in the history of family, this debate within history adds further complexity to taking an interdisciplinary approach.

Anthropological Approach
The term "nuclear family" first emerged within Anthropology during the 1940s. Anthropology, however, distinguishes that the common focus on the nuclear family as a modern institution is not justified. In the late Roman Empire, the nuclear family already existed as the smallest unit of society. There have even been discoveries of the graves of nuclear families dating back to 2600 BCE, such as the one by Haak et al found in 2006 in a late Neolithic community in Germany.

Malinowski and other functionalists viewed the nuclear family as a "universal human institution" and ascribed nurturing as its prime function. Later Anthropologists disagreed, identifying 'The Family' as a construct linked to the state's ideology and arguing that it is the mother-child relation that serves the function of child nurturing. However, a conjugal pair has always been the centre of kin relations and served as "the basis of a nuclear family".

Goody points out how the language around kinship highlights the crucial role of the nuclear family in England: While Anglo-Saxon words for nuclear family members and kin members were closely related, this changed after the conquest of England in 1066. The Norman-French terms for kin members were adapted, while Germanic roots of the closest kind remained. This merge of languages isolated the nuclear family within society.

Anthropologists consider an earlier time period than other disciplines, often ignoring the possibility of later origins.

Sociology approach
Research on the nuclear family developed significantly from the early 1900s, with major shifts during the 20th century. Sociologists have always fixated on a linear progression from extended families pre-industrialisation to the nuclear family post-industrialisation. Thus, focusing on Contemporary History in their analysis of the nuclear family has meant that they failed to consider factors pre-industrialisation.

Earlier sociologists, such as Parson championed this theory. They believed industrialisation of Britain caused a complete transformation in family structure; a reduction in size, and prioritising of conjugal bonds rather than extended family. Furthermore, it was thought that urbanisation, loosened kinship ties. The nuclear family was perceived as more suited to the higher demands of an industrial society due to their increased social and geographical mobility. The popular theories of functionalism and modernism during the 1950s, formed a hierarchy of societies based on the progression towards nuclear families, which were seen as superior

Revisionist Theory (1960s-1970s)
Revisionists challenged importance of industrialisation in the formation of the nuclear family. Anderson explains how huge changes in production were only possible after or simultaneous to changing family order. The use of more historical sources and a new emphasis on class in Sociology, helped create a more complex theory of the origins of the nuclear family. Gordon highlighted how working class families extended to deal with the hardships of industrialisation, showing that earlier theories were oversimplified. Ultimately both revisionists and traditional sociologists were limited by their fixation on the industrialisation and didn't appreciate the influence of other events before and after it.

Pre-Industrial Revolution Beliefs
Within Economics, there is little research into the origination of the nuclear family before the Industrial Revolution. Economists such as Alfred Marshall and Adam Smith, however, outlined the presence of neolocal residences, which form the foundations of a nuclear family.

19th Century Origins
As outlined in the Sociological Approach, Parsons' theory draws upon the economic impact of the Industrial Revolution on families. Parsons' lack of empirical evidence, however, makes this theory weak from an economic perspective. Anderson's research also opposes this as he found 23% of households contained extended family in an 1851 census, at the peak of industrialisation.

21st Century Research
Recent research into the History of the Nuclear Family focuses more on the shift in family structure, especially the transition into the 'typical' Nuclear Family seen in the 21st Century of a couple with two children. This shift is most apparent between 1881 and 1928 when the mean number of children within a family fell from 5.27 to 2.08.

The main catalysts of the shift to a smaller family structure originated during World War I. With 40-60% of recruits failing health checks during the Boer War, it became clear that social reform was necessary to have a strong labour force. The Liberal Government placed the pressure of financing these reforms on the wealthy through tax policies. Many families saw a break down of their wealth and a deterioration of the man's position within the home, paired with the increase in a women's independence during and after WWI, the typical roles within a family changed. Many landowners sold parts of their estates to compensate their loss of income, and these properties were largely partitioned into smaller units to help defeat the housing crisis. The combination of these factors led to a fall in the size of the family as a focus on children and wellbeing also began to dominate families' thinking.

Overall Economic research mostly investigates the shifts in family structures and their significance within society as opposed to its origins, thus it is less significant to the debate.

Conclusion
Anthropology differs in its temporal approach to the other disciplines. Economy, History, and especially Sociology tend to use historical dichotomies of before and after, regarding the transformation from extended to nuclear families, disregarding periods of history that could be evaluated into analysis.

These disciplines have all evolved their arguments about the emergence of the nuclear family, especially over the last century. Whilst there are conflicts within disciplines, and more importantly between the disciplines about the emergence, an interdisciplinary approach allows a broader analysis of the topic, and a multi-faceted understanding of the implications of family structure in the ever-changing world around us.


text:
Issues in Interdisciplinarity 2019-20/Power in Guilt

Introduction
Guilt can be defined as doing something which breaks legal or moral laws. However, the feeling of guilt does not always overlap with factual guilt. The study of guilt is difficult, insofar as there are power conflicts between the social construction of guilt and its biological basis. This WikiBook will focus on the perception of guilt through the perspective of (the sociology of) religion (Catholic and Eastern Orthodox Christianity), law and biology.

Biology of Guilt
The neurological root of guilt as a withdrawal emotion remains somewhat ambiguous. However, studies comparing the brain circuitry of individuals with psychiatric disorders against those without such conditions furthered research on the Ventromedial Prefrontal Cortex (vmPFC) and its interactions with the Orbitofrontal Cortex (OFC) as a determinant in guilt-specific emotion. A correlation could be drawn between the increased frequency of antisocial behaviour, seemingly void of such mediating emotions as guilt, and the dysfunction of the OFC and vmPFC regions of the brain. With such dysfunctions arising from both injury and developmental abnormalities, it could be argued that the role of physical anatomy in guilt perception is conclusive. Even so, any external, social influence must be mediated via such structures and so is inevitably susceptible to their influence.

From an evolutionary perspective, guilt can be seen as an indicator of altruism and so, in turn, a promoter of social cohesion. This perspective, however, ignores the individualist advantage that is key to allow guilt to act as an evolutionarily stable strategy in natural selection. The most frequent conclusion is that guilt serves to mediate the behaviour of an individual against transgressive actions, to establish a stable role within their social structures and so optimise the likelihood of survival to reproduce.

Perhaps unlike social factors, that vary among different communities, the influence of biology on emotional decision making holds power in its universality and ability to provide empirical evidence. This is, however, a potential issue in terms of research. Whilst the discipline is universal, it is not individually comprehensive. To study the topic of guilt from a solely biological perspective ignores the vast social structures that drive perceptions of guilt across the world.

Socio-Theological Perspective
Christianity remains a prominent religion today, with its 2 billion followers projected to rise to nearly 3 billion by 2050. Thus, Christian notions such as guilt and penance remain powerful for many people. One would assume that, with religion being dogmatic, their definitions would be constant, but differences in interpretation of Scripture and outside influences mean that this is not the case. This complicates their study.

Case-studies

The condemnation of (acts of ) homosexuality has led, for example, to a lot of internal conflict for religious homosexuals and complications for the legalisation of gay marriage. It has, however, been suggested that there is no reason to interpret Scripture as “anti-homosexual”. It may then be the case that it is not necessarily the dogma itself that perpetuates the homophobia in religion, but rather an interpretation assigned by intolerant individuals which has remained dominant today.

The church has historically had to tailor its doctrines to economic and political factors to retain its power; this can be seen in the case of "indulgences" as penance (giving money to the Catholic Church to receive partial pardon of sins), which was encouraged by Pope Innocent III in order to finance his Crusade projects. Indulgences (especially on behalf of the wealthy), however, do not seem to quite fit the usual definition of penance: "...undergoing some penalty as an expression of sorrow for sin or wrongdoing". A similar argument may be made for the construction of the church Our Lady of Kazan, whereby an entrepreneur was convinced by a priest to contribute heavily through being told this would constitute penance.

Overview

Religion is a powerful factor that influences people's perceptions of guilt, but its own definitions are arguably heavily influenced by factors like economics, politics, and the prejudices of its interpreters. This makes it harder to arrive at a definite conclusion as to what said definitions are and thus even harder to study them and solve pressing real-world problems.

Legal Perspective
Legally speaking, there is a distinction between legal guilt and factual guilt. Factual guilt is concerned with the plain fact whether the suspect committed a crime, i.e. broken the law, or not. However, in order to convict someone of a crime, it needs to be proven. Therefore, someone can be factually guilty, but without proper evidence, they would be considered legally innocent.

The Case of the Norfolk Four
On July 7, 1997, in Norfolk, Virginia, eighteen-year-old Michelle Bosko was raped and murdered in her apartment. Her body was found by her husband the next day. The first suspect, Danial Williams, confessed after eight hours of interrogation and threats of the death penalty. However, his DNA did not match the crime scene’s and police turned to Joseph Dick, Jr. He confessed under similar circumstances, but his DNA was not a match either. The pattern continued until, seven men were accused of the crime, four of which confessed. Even after the actual killer, Omar Ballard, confessed to having committed the crime alone, the four suspects were charged and the last ones only exonerated in 2016.

Surprisingly, ¼ of wrongful convictions involve false confessions. This is mainly due to the psychological tactics police officers use to get confessions via the power imbalance between suspect and interrogator. Often, police officers are over-confident in their ability to tell apart those who are lying or not. Together with long interrogations, threats of harsher punishments, or even promises of leniency, coupled with the provision of false evidence, they are able to get confessions out of innocent victims, either because they see no way out, or they might have become convinced of their own guilt. This conviction can go as far as apologising to the victim's family for a crime one did not commit, as was the case with Joseph Dick.

The suspect's changed perspective on their own guilt makes it even more difficult to study. Confessions are not reliable, in or outside the court of law, if the suspects themselves are not sure of the facts anymore. The determination of legal guilt, therefore, needs to rely more on biological evidence than the malleable memory of the suspects.

Conclusion
Whilst biology may have apparent power in its ability to rationalise an ambiguous concept and produce data that is universal to all humans, no matter of location or religion, it would be arrogant to extrude it as the sole influence in guilt and so ignore the power of huge structures (i.e. religion and the legal system) that drive social perceptions of guilt.

Ultimately, no-one specific discipline can produce complete research into the perception of guilt. Such research requires the evaluation of power dynamics that twist both how we perceive guilt in our social communities and then how such perceptions are processed through our biology. Therefore the recognition and critique of sources of power in such disciplines is essential to allow comprehensive and representative research.


text:
Talk:Issues in Interdisciplinarity 2019-20/Evidence in Addiction

Group contribution
Introduction: written together by Anna and Rita

Neuroscience: written by Adèle

Economics: written together by Charlotte and Rita

Psychology: written by Anna

Combining Evidence in Government Intervention: written together by Adèle and Charlotte

Overall editing: done throughout all members of the group

drafts
Anna and Rita

Addiction is a complex topic which has created a lot of controversy over its meaning, effects and treatment methods. The topic of addiction has yielded a lot of research from several different disciplines: Neuroscience, Psychology and Economics. Due to the different methodologies, sources, intent and data used by the disciplines, the issue of evidence has created tensions between the disciplines. However, when brought together, they can facilitate a more holistic understanding of the cause, nature and effect of addiction and therefore make useful contributions to government decisions on interventions to decrease the harm caused by it. Just as addiction has various effects on individuals and their surroundings, addiction is studied across several disciplines. Neuroscience, Psychology and Economics are three examples of disciplines with particular interest in the study of addiction. Through their different models and methodologies, they produce different evidence describing different aspects of addiction and its effects. Brought together, it can facilitate a more holistic understanding of the cause, nature and effect of addiction and therefore make useful contributions to government decisions on interventions to decrease the harm caused by it.

possible paragraph: looking at the difference/ tensions between real world (historical and stat) evidence vs hypothetical/ social experiment/ model evidence. hypothetical psychological models which show pattern of behaviour can sometimes not work in real life. e.g. number of deaths due to overdose could be in those with addictions but also many first time users who have limited education and take drugs of high potency/ impurities etc. genetics - contribute to higher dopamine levels naturally which doesn't directly cause addiction however it can cause more high risk behaviour leading to the possibility of drug use ( psychological argument but the wider context has to be considered.)

Conflict between qualitative (ex psychology) and quantitative (ex neuroscience) evidence. - language: what is the definition of addiction

physical effect on brain e.g. release of dopamine can equal addiction however from a psychological perspective can look at the meaning of addiction e.g effect on relationships, spending, quality of life. These view points can be created using either quantitative evidence or qualitative evidence. methodologies.

essential to combine evidence to have different uses: clinical use (drugs against addiction) but also psycho therapies - is the evidence representative and does it have meaning (e.g. for clinical use). Combining evidence from different disciplines enables a more efficient treatment against addiction.

Quantitative evidence can sometimes be more easily obtained for alcohol and opiate addictions because, as depressant drugs, they are the only ones which will exhibit physical withdrawal symptoms. How can we measure e.g. a gambling addiction. addiction has a relationship with environment. All physical symptoms of addiction can disappear after withdrawal however what is causing the addiction will still be there - must look from a wider context (a psychological approach).

behavioural economics - war on drugs - shipments of drugs obviously can contribute to addiction however due to the economic situation, complete control of drugs is near impossible. estimated worth of the UK drug market is £5 billion however in reality it is much more (the illegality means the economic evidence can never be truly valid and therefore cannot always reflect the relationship between economic trends and addiction.

Psychological methods for obtaining evidence: negative and positive reinforcement models suggest that people become addicted to substances to gain the positive reward and to avoid the negative withdrawal effects. this model has limitations as drugs such as psycho-stimulants do not produce any withdrawal symptoms. this shows how in psychology models are constantly edited (process of elimination).

Limitations of quantitative methods used in neuroscience (brain imaging): can’t measure emotions, perception (interpretativist) - can you extrapolate results from primate testing to apply to humans, with more complex society and interactions?

Possible Structure:

Introduction: Each of the disciplines uses specific methodologies and different types of evidence to support theories (with some overlaps) - focus more on positivist/interpretivist approaches using qualitative/quantitative evidence (usually a combination of both) Understanding real world complex problems - need a combination of evidence and methodology

1. Neuroscience evidence >already have a good start by Adele

2. Economic evidence >Lotti and Rita

3. Psychology (as a possible overlap with both) >Adele and Anna

4. Government decision making in the real world - what types of evidence should be used to inform decisions > come together to draw together research

Conclusion: To respond to rising levels of addiction, the disciplinary methods and evidence need to be combined to give governments the best understanding ...

Difference between drug addiction and dependence is defined by DrugRehab.com as “compulsively seeking the medication despite obvious harms” But this is a source with a specific aim (curing drug addiction)... worth considering if the definition of ‘addiction’ varies depending on the organisation/research aims.

Economics

Evidence for addiction in Economics usually focuses on addiction in the case of drugs, including illegal substances as well as alcohol and tobacco - does psychology and neuroscience take a more varied base of evidence for addiction, not just to substances but to social media, excercise, sex etc...??

Economic article which has a section on government policy-making - can use it later: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2896982/pdf/nihms188513.pdf

Economic models and research methodologies can bring about:

-	Quantitative evidence of quantities consumed of addictive substances (simple demand/supply models)

-	Quantitative evidence of consumers’ behavior in response to price change of the addictive substances (price elasticity of demand)

-	Quantitative evidence of the effectiveness of government legislation on quantity demanded (and therefore consumed) over time (simple supply/demand) ex: cigarette pricing in Australia >> https://www.tobaccoinaustralia.org.au/chapter-13-taxation/13-5-impact-of-price-increases-on-tobacco-consumpt

-  Quantitative evidence of the cost of consumption of addictive substances (to economy?); direct costs include medical costs but also costs of social services and informal care while indirect costs include "premature mortality, short-term absence, early retirement, crime-related costs" >> https://link-springer-com.libproxy.ucl.ac.uk/content/pdf/10.1007%2Fs10198-005-0282-5.pdf

However, purely quantitative economic evidence has weaknesses:

-	Fails to distinguish between misuse/dependence/addiction of consumers to the studied product, as it doesn’t consider individuals’ life circumstances

-	Lack of data for illegal substances makes it difficult to make precise estimates

Possibly helpful paper: https://books.google.co.uk/books?hl=de&lr=&id=5hZY4e0L_OcC&oi=fnd&pg=PR9&dq=economics+addiction&ots=m16zNpDA3T&sig=CTBjt6Xda0__D_nm3ruz3Pt4ztk&redir_esc=y#v=onepage&q=economics%20addiction&f=false

Several economic models and methodologies are used in producing evidence contributing to studies of addiction. Cost-of-illness methodologies give rise to quantitative evidence of the costs to a nation generated by consumption of addictive substances. These encompass both direct costs, such as those of health-care and social services, as well as indirect costs such as premature mortality, early retirement and costs relating to crime (CITE). Limitations in the cost-of-illness methodology include difficulties in making precise estimations, especially for the indirect costs (CITE). Additionally, these indirect costs do not contain all negative effects of addiction, for instance on family members of the one affected. Simple demand/supply models can show hypothetical information but also real world quantitative evidence of quantities of addictive substances consumed and consumer’s behavior to price changes of these substances. Price elasticity models can then more specifically compare the ratio of change in quantity consumed to change in price, giving a more precise idea of the behavior of consumers of addictive substances (CITE).

Ideas for Part on Government Intervention:

Economics study a whole market rather than individuals, and economic evidence is therefore very valuable to consider when making policies that will affect a whole society (Caulkin). >> Should compliment not dominate study of addiction. Combined with psychological evidence focusing on individuals gives more complete idea of the problem.

Moreover, economic models can not only consider real-world case studies but also the hypothetical, which should be useful in foreseeing the possible impact of certain policies (Caulkin). It is difficult, however, to make precise estimations, especially for the consumption of illegal substances.


text:
Talk:Issues in Interdisciplinarity 2019-20/Truth in the Formation of Social Groups

(testing) yeah I think this bit works fine - we should all be able to edit it Melisa.Ozturk.00 (discuss • contribs) 14:23, 2 December 2019 (UTC)

Outline:

-Sociology : explaining social bonds and the passage for example from mechanic to organic solidarity

-Biology : what biological / genetic phenomena causes humans to naturally regroup ( survival, instinct ... )

-Psychology : the need for humans to belong and feel as though they are part of a greater whole

-Geography : people form groups depending on their geographic proximity

Introduction structure:

-social bonds are everywhere -social groups at the base of society -evolutionary process, from couples, to clans to larger communities -but how do you explain their formation? -Different approaches to the study of formation of social groups, which discipline truly explains it?

Intro (Maia): Social groups are considered to have followed a gradual evolution, originating from couples and developing into clans and larger communities. Although their existence and progression is undeniable, how exactly do you explain their formation? Various approaches can be adopted to study this topic. Disciplines such as geography, biology, psychology and sociology have come up with different explanations for this phenomenon. While some of these are sources of contradiction, others on the contrary, compliment one another and help us understand where truth resides in the formation of social groups.

Intro (Melisa): Human interaction and development are governed by our social groupings - from smaller familial groups to whole nations. How and why these groups form are contested among different disciplines, are our groupings predisposed by our ethnicities and where we have grown up, staying fixed over time, or are they more volatile and prone to change, being affected by varying factors like favourite sports clubs and reading preferences?

Intro (Julia) : It has been long believed that social groups are the base of a prosperous society, human interaction evolving from individuals to larger communities. Commonly, a social group is defined as two or more people who interact with one another, share similar characteristics, and collectively have a sense of unity. Although there are many studies in modern society related to social groups, the truth in the formation of a social group has has not been yet discussed. Being a gradual process, it can be analysed from an interdisciplinary perspective, the truths of various disciplines fusing together to form the actual definition of a social group. What are the disciplines that contribute in this specific case? Do their truths work together in a dynamic way?

Introduction combined: Human interaction and development are governed by our social groupings. Commonly, a social group is defined as two or more people who interact with one another, share similar characteristics, and collectively have a sense of unity. Social groups are considered to have followed a gradual evolution, originating from couples and developing into clans and larger communities. Although their existence and progression is undeniable, how exactly do you explain their formation? Various approaches can be adopted to study this topic. Disciplines such as geography, biology, psychology and sociology have come up with different explanations for this phenomenon. While some of these are sources of contradiction, others on the contrary, compliment one another and help us understand where truth resides in the formation of social groups.

Discipline order:

-geography (the most obvious : people regroup because of proximity)

-biology (molecular analysis of the genes that affect your social behaviour)

-psychology (the need to belong is "natural")

-sociology (social interactions)

Discussion:

Hey Melissa, it's Maïa writing :) This is the little part I wrote in sociology, it's very simplified but I think you can transfer the idea to geography:

"Often times social groups formed based on age, sex, way of life … Today with the rise of globalisation and social media creation of “social groups” formed of people from all over the world, different cultures, different languages, different ages, but similar interests… " - I feel this would be a better fit for the concluding paragraph? Melisa.Ozturk.00 (discuss • contribs) 14:45, 8 December 2019 (UTC)

I think we should change the headings to proximity, genes, belonging and solidarity (or whatever you see fit for your section) and talk about why the chosen discipline's truth is relevant to that idea Melisa.Ozturk.00 (discuss • contribs) 14:45, 8 December 2019 (UTC)

Oh true, it could work in the conclusion as well. And I agree I think we need to change the headings...

2A02:C7F:C030:F800:DD3:CB78:D240:CD83 (discuss) 17:15, 8 December 2019 (UTC) Maïa

Conclusion:

Julia conclusion: It can be easily observe that the truth in formation of social groups can be interpreted from an interdisciplinary perspective, with humanities point of view being more closely related to each other than to the natural sciences. Although biological explanation is highly concerned on the molecular analyses of the genes involved in brain behaviour, other disciplines, such as psychology and sociology complete harmoniously the positivist truth of the science, by illustrating the need of the humans to belong and feel as part of a great whole, respectively clarifying the origins of social bonds.

something I want to add in conc: proximity doesn't always mean you come to like those people, more about evoking strong feelings, could also be people you dislike the most

Emma conclusion idea:

Through looking at the formation of social groups in the perspective of multiple disciplines, a dizzying array of often contradictory conclusions can be found. For example, geography pins down proximity as a key incentive, and psychology backs this up through an evolutionary perspective. However, sociology also concludes that difference and independence from others are important for a social group to function. These disagreements arise from the individual “truths” of each discipline and their varying positivist and interpretive views.

However, as contradictory as it may seem, these differences in “truths” allow us to see the phenomena of social groups from a more holistic view. Disciplines such as geography, psychology, and sociology take on a positivist attitude when regarding their "truths"; however, they tend to draw conclusions from general trends from empirical data, thus ignoring phenomena that deviate from the general conclusion. For example, in geography, proximity may be a cause of social groups forming. On the other hand, it also may be a cause for social division and ostracism. These holes in "truths", however, can be filled by looking at the issue from another discipline such as sociology and the human tendency to form social groups with people who share a commonality. Additionally, when asked why social groups form at all, biology can answer; because it is coded in our genes to do so. Psychology can back this up through an explanation of the evolutionary factors that favoured our ancestors who were more inclined to form social groups. When looking at each disciplines' conclusions at a glance, they may seem unconnected or even contradictory, but upon deeper inspection, they fit together to create a much bigger whole. Fundamentally, social behaviour cannot be explained through one set of truths; it’s a complex interplay of systems that may often appear contradictory.

Although proximity does in fact play a large part in the formation of groups it would be more accurate to say that it affects interpersonal relations both positively and negatively. While those in closest proximity are likely to be the ones we voluntarily associate with more regularly, they could equally be the people who we dislike the most.

In more modern times urbanisation, globalisation and the development of the internet have resulted in growing diversity and thus the creation of cross-cultural social groups. ^ contradictions to the truths Melisa.Ozturk.00 (discuss • contribs) 22:20, 8 December 2019 (UTC)


text:
Issues in Interdisciplinarity 2019-20/Evidence in Addiction

Introduction
We define addiction as a persistent compulsive need for a substance or an activity that, in the long term, is detrimental to an individual's quality of life. Addiction is a complex topic which has created much controversy over its meaning, effects and treatment methods. This has yielded various research from different disciplines: Neuroscience, Psychology and Economics. Due to different methodologies, sources, intent and data, evidence has created tensions between the disciplines. However, when combined, they facilitate a more holistic understanding of the cause, nature and effect of addiction and therefore make useful contributions to governments' decision making to decrease the harm caused.

Evidence in Neuroscience
Brain imaging technologies provide most of the evidence used in neuroscience to study addiction. Imaging shows which parts of the brain are activated during craving, satisfaction, and withdrawal. Imaging studies show that during satisfaction or craving, the prefrontal regions of the brain are activated in a pattern that involves sections of the brain associated with reward, motivation, memory, and cognitive control. Imaging also shows that dopamine levels in the brain increase during craving and satisfaction and that this increase deregulates the prefrontal regions of the brain which causes a loss of rational behavior and compulsive drug intake.

The main imaging technologies are positron emission tomography (PET) and functional magnetic resonance imaging (fMRI). PET measures brain activity by detecting the radioactivity emitted by injected positron-emitting isotopes —regions with high radioactive activity are associated with brain activity. fMRI measures the brain activity by detecting the changes in blood oxidation —when a region of the brain is more active, it consumes more oxygen. Other imaging technologies include Computed tomography (CT), Electroencephalography (EEG), Magnetoencephalography (MEG) and Near infrared spectroscopy (NIRS). However, there are limitations in the quantitative methods used in brain imaging. They do not seem to measure subjective emotions and perception.

Another source of evidence for the study addiction in neuroscience are animal models of substance use disorder. For example, a study in nonhuman primates showed how environmental factors affect neuronal circuits, in turn influencing addictive behavior. The study showed that social status, dominant or subordinate, affects dopamine receptors' expression in the brain, which then influences an individual's drug intake. However is it rigorous to extrapolate results from primate testing to apply to humans, with more complex society and interactions?

Evidence in Economics
Economics traditionally assumes humans displaying fully informed autonomous rational behaviour, thus understanding addiction is problematic when building supply-demand models of consumption. Evidence for addiction is usually in the context of drugs, both legal and illegal, taken from real-world case studies where independent variables are already in place for experimental analysis.

There is still a lack of control variables, but these ‘natural experiments’ are valuable because they cannot easily be synthetically recreated, for example, economists cannot simply change the drinking age for selected groups of teenagers simply to study the result on behaviour. An example of identifying causality from a real-world case study is where researchers Norstrom and Skog took advantage of a phase-in of retail alcohol sales in Sewden, and studied the link of this policy with crime rates.

Data studying crime rates is also used in cost-of-illness methodologies, which provide quantitative evidence of the financial costs generated by consumption of addictive substances. These encompass both direct costs, such as those of health-care and social services, as well as indirect costs such as premature mortality, early retirement and costs relating to crime. Limitations in the cost-of-illness methodology include difficulties in making precise estimations, especially for the indirect costs. Additionally, indirect costs do not consider qualititative data such as burden on family members.

Price elasticity models show the ratio of change in quantity consumed to change in price, giving a more precise idea of the behavior of consumers of addictive substances. For example, Caulkin discusses studies focusing on specific groups of consumers, demonstrating that marijuana consumption of high school students in the U.S. was less responsive to price changes than that of Australian adults. Nevertheless, looking at elasticity evidence as an average concerning figures for drugs, addicts are grouped in with recreational drug users, so the individual personal elasticities are skewed by non-addicted users.

Evidence in Psychology
Psychological methods for obtaining evidence include creating hypothetical yet functional models, for example, positive and negative reinforcement models, based on controlled qualitative data provided by social experiments to identify patterns in human behavior for addiction. These experiments can be specific, measuring differing effects on individuals obtained through patient-practitioner interactions. Broader case studies are also used as evidence to support models. One study looked at treatment programs, finding that patients who had supportive and constructed relationships were more likely to benefit from the treatment, it showed a 41% variance and helps to support functional models.

However, various data needs to be collected to continuously modify models. For example, the positive and negative reinforcement model which is highly based on quantitative data from studies on alcoholism shows that addicts stay addicted to maintain the positive effects of the addiction and to avoid the negative side effects of withdrawal. Though this model is supported by studies done on alcohol addiction, it cannot explain why psychoactive drugs which do not have withdrawal effects can be highly addictive.

Combining Evidence in Government Intervention
An interdisciplinary approach is essential in government policy-making. Governments should have a holistic view of addiction, considering evidence used by different disciplines to reduce cost to society.

Government intervention to reduce addiction has included tax increase for addictive substances, marketing bans, raising awareness through education, and legislation such as imposing a minimum age to buy a substance.

Decisions tend to be based on evidence of past world events, much of which is provided by economic data. To create innovative ideas of combatting abuse of addictive substances, using prediction models such as those used in psychology is beneficial. Rather than designing policies which fix the results of drug overuse, governments can consider ways to combat addiction at the source, by considering both qualitative and quantitative evidence, as well as ‘natural’ and ‘hypothetical’ experimental data used by the different disciplines.

To incorporate neuroscientific evidence, governments could consider theories developed from studying brain scans, explaining causes of addiction. It can also help understanding addicts' decision making. In addition, when looking at aspects of psychology such as cognition, the evidence can be more ambiguous than is presented by neuroscience. This could support by keeping ideas concrete and potentially creating effective treatments that would work physically, but also find the root cause of addiction.

Furthermore, psychology can build on neuroscience by understanding how individuals’ particular environments are more vulnerable to addiction, or to relapse after treatment. This can consequently allow economists to understand the psychology of addiction when it considers real-world case study evidence to build policies. For example, Bernheim and Rangel discuss a policy of subsidising rehabilitation centers and treatment programs, with evidence from psychology illustrating that addicts are in fact rational enough to understand their condition, yet environmental factors play a large role in fostering that addiction.

This short study of evidence in addiction shows how an interdisciplinary approach is crucial when using academic research to solve real world complex issues.


text:
TeX/penalty

Synopsis
\penalty \binoppenalty= \brokenpenalty= \clubpenalty= \displaywidowpenalty= \exhyphenpenalty= \floatingpenalty= \hyphenpenalty= \interlinepenalty= \linepenalty= \postdisplaypenalty= \predisplaypenalty= \relpenalty= \widowpenalty=

Description
`\penalty` sets the penalty for a line or page break at that point.

Some penalties are built in to the TeX system and inserted automatically:

 * `\binoppenalty` for a line break in math mode after a binary operator.
 * `\brokenpenalty` for a page break, where the last line of the previous page contains a hyphenation.
 * `\clubpenalty` for a broken page, with a single line of a paragraph remaining on the bottom of the preceding page.
 * `\displaywidowpenalty` for a break before last line of a paragraph.
 * `\exhyphenpenalty` for hyphenating a word which already contains a hyphen.
 * `\floatingpenalty` for splitting an insertion.
 * `\hyphenpenalty` for line breaking at an automatically inserted hyphen.
 * `\interlinepenalty` for the penalty added after each line of a paragraph.
 * `\linepenalty` the badness of each line within a paragraph.
 * `\postdisplaypenalty` for a break after a display.
 * `\predisplaypenalty` for a break before a display.
 * `\relpenalty` for a line break at a relation.
 * `\widowpenalty` for a broken page, with a single line of a paragraph (called "widow") remaining on the top of the succeeding page.

Default
LaTeX for example sets these default values for built in penalties:

\binoppenalty=700 \brokenpenalty=100 \clubpenalty=150 \displaywidowpenalty=50 \exhyphenpenalty=50 \floatingpenalty=20000 \hyphenpenalty=50 \interlinepenalty=0 \linepenalty=10 \postdisplaypenalty=0 \predisplaypenalty=10000 \relpenalty=500 \widowpenalty=150

title: TeX/penalty


text:
LMIs in Control/pages/Robust Stabilization of Second-Order Systems

Stabilization is a vastly important concept in controls, and is no less important for second order systems with perturbations. Such a second order system can be conceptualized most simply by the model of a mass-spring-damper, with added perturbations. Velocity and position are of course chosen as the states for this system, and the state space model can be written as it is below. The goal of stabilization in this context is to design a control law that is made up of two controller gain matrices $$ K_{p}$$, and $$ K_{d}$$. These allow the construction of a stabilized closed loop controller.

The System
In this LMI, we have an uncertain second-order linear system, that can be modeled in state space as:

\begin{align} (A_{2}+\Delta A_{2}) \ddot x + (A_{1} + \Delta A_{1})\dot x + (A_{0} + \Delta A_{0})x &= Bu )\\ y_{d} &= C_{d} \dot x\\ y_{p} &= C_{p} x \end{align}$$

where $$ x \in R^{n}$$ and $$ u \in R^{r}$$ are the state vector and the control vector, respectively, $$ y_{d} \in R^{m_{p}}$$ and $$ y_{d} \in R^{m_{p}}$$ are the derivative output vector and the proportional output vector, respectively, and $$ A_{2}, A_{1}, A_{0}, B, C_{d}, C_{p}$$ are the system coefficient matrices of appropriate dimensions.

$$ \Delta A_{2}, \Delta A_{1},$$ and $$ \Delta A_{0}$$ are the perturbations of matrices $$ A_{2}, A_{1},$$ and $$A_{0}$$, respectively, are bounded, and satisfy

\begin{align} \end{align}$$
 * \Delta A_{2}|_{2} \leq \epsilon_{2}, |\Delta A_{1}|_{2} \leq \epsilon_{1}, |\Delta A_{0}|_{2} \leq \epsilon_{0},

or

\begin{align} max\{\|\Delta a_{2ij}\|\} \leq \eta_{2}, max\{\|\Delta a_{1ij}\|\} \leq \eta_{1}, max\{\|\Delta a_{0ij}\|\} \leq \eta_{0}, \end{align}$$

where $$\epsilon_{2}, \epsilon_{1}, \epsilon_{0}$$ and $$\eta_{2}, \eta_{1}, \eta_{0}$$ are two sets of given positive scalars, $$\Delta a_{2ij}, \Delta a_{1ij},$$ and $$\Delta a_{0ij}$$ are the i-th row and j-th collumn elements of matrices $$\Delta A_{2}, \Delta A_{1}, $$ and $$\Delta A_{0}, $$, respectively. Also, the perturbation notations also satisfy the assumption that $$ \Delta A_{2}, \Delta A_{0} \in S^{n} $$ and $$ A_{2} + \Delta A_{2} > 0$$.

To further define: $$x$$ is$$\in R^{n}$$ and is the state vector, $$A_{0} $$ is $$\in R^{n*n}$$ and is the state matrix on $$ x $$, $$A_{1} $$ is $$\in R^{n*n}$$ and is the state matrix on $$ \dot x $$ , $$A_{2} $$ is $$\in R^{n*n}$$ and is the state matrix on $$ \ddot x $$, $$B$$ is $$\in R^{n*r}$$ and is the input matrix, $$u$$ is $$\in R^{r}$$ and is the input, $$C_{d} $$ and $$C_{p} $$ are $$\in R^{m*n}$$ and are the output matrices, $$y_{d}$$ is $$\in R^{m}$$ and is the output from $$C_{d} $$, and $$y_{p}$$ is $$\in R^{m}$$ and is the output from $$C_{p} $$.

The Data
The matrices $$ A_{2}, A_{1}, A_{0}, B, C_d, C_p $$ and perturbations $$ \Delta A_{2}, \Delta A_{1}, \Delta A_{0}, $$ describing the second order system with perturbations.

The Optimization Problem
For the system defined as shown above, we need to design a feedback control law such that the following system is Hurwitz stable. In other words, we need to find the matrices $$ K_{p}$$ and $$ K_{d}$$ in the below system.

\begin{align} (A_{2}+\Delta A_{2}) \ddot x + (A_{1} - BK_{p}C_{p} + \Delta A_{1})\dot x + (A_{0} - BK_{d}C_{d} + \Delta A_{0})x &= 0 \\ \end{align}$$

However, to do proceed with the solution, first we need to present a Lemma. This Lemma comes from Appendix A.6 in "LMI's in Control systems" by Guang-Ren Duan and Hai-Hua Yu. This Lemma states the following: if $$A_{2} > 0, A_{1} + A_{1}^{T} > 0, A_{0} > 0$$, then the following is also true for the system described above:

The system is hurwitz stable if

\begin{align} \lambda_{min}(A_{2}) > |\Delta A_{2}|_{2}, \lambda_{min}(A_{1} + A_{1}^{T}) > |\Delta A_{1}|_{2}, \lambda_{min}(A_{0}) > |\Delta A_{0}|_{2}\end{align}$$,

or

the system is hurwitz stable if

\begin{align}\lambda_{min}(A_{2}) > \sqrt{l_{2}}max\{\|\Delta a_{2ij}\|\}, \lambda_{min}(A_{1} + A_{1}^{T}) > \sqrt{l_{1}}max\{\|\Delta a_{1ij}\|\}, \lambda_{min}(A_{0}) > \sqrt{l_{0}}max\{\|\Delta a_{0ij}\|\}\end{align}$$

, where $$l_{2}, l_{1}, l_{0}$$ are the numbers of nonzero elements in matrices $$ \Delta A_{2}, \Delta A_{1}, \Delta A_{0},$$ respectively.

The LMI: Robust Stabilization of Second Order Systems
This problem is solved by finding matrices $$K_{p} \in R^{r*m_{p}}$$ and $$K_{d} \in R^{r*m_{d}}$$ that satisfy either of the following sets of LMIs.

\begin{align} A_{0} - BK_{d}C_{d} &> \epsilon_{0}I,\\ (A_{1} - BK_{p}C_{p}) + (A_{1} - BK_{p}C_{p})^{T} &> \epsilon_{1}I. \end{align}$$

or

\begin{align} A_{0} - BK_{d}C_{d} &> \eta_{0}\sqrt{l_{0}}I,\\ (A_{1} - BK_{p}C_{p}) + (A_{1} - BK_{p}C_{p})^{T} &> \eta_{1}\sqrt{l_{1}}I. \end{align}$$

Conclusion:
Having solved the above problem, the matrices $$K_{p}$$ and $$K_{d}$$ can be substituted into the input as $$ u = K_{p}C_{p}\dot x + K_{d}C_{d}x $$ to robustly stabilize the second order uncertain system. The new system is stable in closed loop.

Implementation
This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/ROBstab2ndorder.m

Related LMIs
Stabilization of Second-Order Systems


text:
User:SibyllePoli/sandbox

Wikibooks : Disney’s power over the cultural appropriation (on which the company builts its empire)

Concerning the isssues the Walt Diney Company meets about the new platform Disney +, measures must be taken. Does the company need to censure some classics Disney ? Is the « warning » at the beginning of movies efficient ?

« Disney Bashing »

The polemics mainly concern classic movies, such as Dumbo, Peter Pan, the Lady and the Tramp, the Jungle Book, the Lion King.. For those movies which are severly criticzed, the company must take some measures. Indeed, with the concept of the ‘politically correct’, a lot of scenes which were accepted at the time are not anymore, because they comport racism, stereotypes or cultural appropriation, and the new interpretations are inspired by the post-colonial studies, and this indictment is called « Disney bashing ». Thus, the company tries to avoid this phenomen of critics by putting a warning at the beginning of the movies. This warning is also displayed on the platform Disney +, and it prove that Disney took in consideration the fact that public opinion has evolved, and so has sensibility. Putting a warning shows that the company realizes the racist, sexist or stereotyped statements in the movie which may schok the telespecators. However, this warning is not considered good enough ; indeed, many specialist tend to think that the message is inadequate and does not touche important subjects such as sexual preferences or gender. Moreover some of the warning can be backgrounded. A question appears : what is the politic of the firm ? Why making all of the controversial movies exposed to critics and polemics ? What are the solutions ? Leonard Maltin (movie’s commentateur) explains the company’s politics : « Some people would sweep film like this under the rug, pretend they never existed. We certainly would not want children to get the wrong idea by seeing some of those stereotypes but disney fans and enthusiasts should be able to enjoy them intact and concerned parents might use this opportunity to talk about the way things were many years ago. » Concerning the censur (which can be seen as a solution), Disney applies the « all or nothing » principle. Indeed, either a movie is completely censured such as « the South Melody » because the movie is judged « guilty » of racism between blacks and whites after the secession war,( disney let the movie diffused in europe until 2001 and annonced that it would not be on Disney+ ) either a movie is broadcast without interruption. And even with the new platform, this principle hasn’t changed. But today, the walt disney company seems engaged in the fight for diversity and against discriminations. The company took some measures. Indeed, by removing the south melody, or by producing movies which concerns stereotypes discriminations and racism such as Zootopia, the company tries to destroy the critics about old movies.

title: User:SibyllePoli/sandbox


text:
Issues in Interdisciplinarity 2019-20/Evidence in the Causes of Homosexuality

This article examines evidence within the disciplines of developmental psychology, epigenetics and endocrinology when studying the causes of homosexuality, providing a wider understanding of the topic, due to the issues presented by evidence in these disciplines.

Developmental Psychology
A theory in Developmental Psychology, labelled ‘Exotic becomes Erotic’ (EBE) states that a child’s nature affects their interest for gender-conforming activities and peers. Applied to both homosexual and heterosexual individuals, children who do not conform to gender norms supposedly feel unfamiliarity with same-sex peers, producing increased autonomic arousal (i.e. erotic or romantic attraction) for members of the same sex.

Data taken from a study in San Francisco concludes conformity to gender norms in childhood as being the most significant childhood indicator of sexual orientation for adult men and women. The study supporting this theory uses an observational research methodology - commonly found within developmental psychology.

The data from the San Francisco study presents evidence in the form of quantitative data, with no clarification on the classification of ‘gay’, ‘lesbian’ or ‘heterosexual’ individuals in the study. Individuals who were bisexual or did not fit into these categories may have been included in the table. Moreover, the study defines ‘sex-typical activities’ as ‘baseball and football’ for boys and ‘hopscotch, playing house or jacks’ for girls. With no proof supporting these assumptions, we see a common weakness in observational studies within developmental psychology - where evidence is susceptible to researcher bias, and accurate quantitative data cannot be produced.

The study in San Francisco suggests a clear causal link between gender non-conformity in childhood and a homosexual orientation in adulthood, yet observational studies cannot permit cause and effect conclusions. Developmental psychology often uses the case study research method - which do not allow conclusions to be generalised to a larger population. Therefore, there are clear flaws existing in the evidence presented within this discipline.

Epigenetics
According to epigenetics, sexual orientation is determined by the epigenome. The evidence that support this hypothesis lies in a series of empirical studies conducted on identical twins and the analysis of their DNA. Tung Ngun looked at the DNA of 47 set of twins, where 37 of them were discordant --meaning that one was gay and the other heterosexual-- and 10 where both twins identified as gay. By collecting and analyzing their DNA, he found that discordant twins had indeed the same code, but the way it functioned differed.

Epigenetic marks (epi-marks) settle themselves on top of the genome and produce a chemical reaction which activates or deactivates parts of the gene. This process is called DNA methylation and acts according to the needs of a situation, affecting the overall expression of the gene. In this case, epi-marks are responsible for switching on and off sex-specific traits: whether they be sexual identity, sexual preference or the development of genitals. Indeed, they are closely linked to the amount of testosterone one is exposed to. The more testosterone one is exposed to, the higher the likeliness to be sexually inclined towards women (and vice versa).

Ngun and his team later looked at “at the associations between specific epi-marks and sexual orientation in one group, then tested how well those results could predict sexual orientation in the second group”. They reached a 70% of accuracy, which compared to previous studies is a leap forward, but still demonstrates a gap in the evidence manipulated.

As epigenetics remains a relatively new discipline, the studies may lack core information about epigenomes, which might hinder the validity of results when studying external subjects such as the etiology of sexual orientation. Furthermore, the small sample size render the results inapplicable to wider populations. It has also not looked at other degrees of sexual orientation on the Kinsey scale, such as bisexuality. Finally, although empirical evidence employed in genetics is presumably objective, the experts manipulating it are inevitably biased, and might alter end results.

Endocrinology
According to the field of endocrinology, and more specifically neuroendocrinology, sexual orientation is determined prenatally by the action of hormones in the human brain, precisely involving the sexually dimorphic nucleus (SDN) located in the hypothalamus. The evidence supporting this thought lies in various experiments conducted on ferrets, among other animals. This SDN which is usually around 2.2 times larger in male than in female brains, according to neurobiology, defines sexual orientation: A larger SDN as seen in men will lead to an attraction to females and a smaller SDN as seen in women will lead to an attraction to males.

Endocrinological studies and experiments have shown that the size of the SDN is conditioned by natural testosterone levels during the prenatal and perinatal period of a human’s life, thus explaining the usual preference of men for women and vice versa. However, it is important to note that the field of endocrinology considers not only testosterone important to the development of the SDN but also estrogens formed from testosterone by aromatase. These claims are demonstrated through the evidence provided by multiple studies on test subject animals: rats or rams which possess sexually specific average SDN size differences similar to humans.

Homosexuality, then, would be caused by abnormalities regarding hormonal development affecting in turn the growth of the SDN, once again supported by the use of experimental evidence conducted on test subject animals, more specifically lab rats treated perinatally with hormones such as testosterone and estrogen or ATD, an aromatase inhibitor, to reverse their sexual partner preference.

As a response to the apparent fluctuation in human sexuality that is observable in our societies, endocrinologists affirm that sexual orientation and sexual behavior are separate: the prior being a definite set in stone biological preference determined by endocrinology which cannot be changed after development, and the latter being a non-reflective behavior adopted by humans in specific contexts. Thus, they deny the possible use of sexual behavior as evidence on the cause and nature of homosexuality and choose to rely solely on empirical endocrinological experimentation.

Finally, Endocrinology falls short of fully explaining animal sexual orientation; humans have defined not only heterosexuality and homosexuality, but attraction to both sexes: bisexuality. Endocrinology cannot in the present day provide an explanation of bisexuality, as it lacks the required evidence to emit or test any hypotheses. This situation begs that question of whether it is therefore wise to make use of only empirical and endocrinological evidence to explore the causes of homosexuality, and if the attachment to this form of evidence only does not hold research back.

Evaluating evidence
Evidence can be presented in different manners through a range of disciplines, with there being no absolute method to collect data. Positivistic ideologies produce representative results which generalise and summarise in certain disciplines, whilst those with interpretivistic viewpoints produce results giving meaning and experience. Through all disciplines however, evidence can be manipulated to verify the truth or falsity of one's claim or the ideologies of their own discipline. The evidence disciplines choose to use differs from one another, creating differences in their views on the roots of homosexuality. By gathering the evidence of the studies made by developmental psychology, epigenetics and endocrinology, and looking at them together, one can achieve a more holistic vision of the issue.


text:
LMIs in Control/pages/H-infinity filtering

For systems that have disturbances, filtering can be used to reduce the effects of these disturbances. Described on this page is a method of attaining a filter that will reduce the effects of the disturbances as completely as possible. To do this, we look to find a set of new coefficient matrices that describe the filtered system. The process to achieve such a new system is described below. The H-infinity-filter tries to minimize the maximum magnitude of error.

The System
For the application of this LMI, we will look at linear systems that can be represented in state space as

\begin{align} \dot x &= Ax + Bw, x(0) = x_{0}\\ y &= Cx + Dw\\ z &= Lx \end{align}$$

where $$x \in R^{n}, y \in R^{l}, z \in R^{m}$$ represent the state vector, the measured output vector, and the output vector of interest, respectively, $$w \in R^{p}$$ is the disturbance vector, and $$A, B, C, D$$ and $$L$$ are the system matrices of appropriate dimension.

To further define: $$x$$ is $$\in R^{n}$$ and is the state vector, $$A $$ is $$\in R^{n*n}$$ and is the state matrix, $$B$$ is $$\in R^{n*r}$$ and is the input matrix, $$w$$ is $$\in R^{r}$$ and is the exogenous input, $$C $$ is $$\in R^{m*n}$$ and is the output matrix, $$D$$ and $$L$$ are $$\in R^{m*r}$$ and are feedthrough matrices, and $$y$$ and $$z$$ are $$\in R^{m}$$ and are the output and the output of interest, respectively.

The Data
The data are $$w$$ (the disturbance vector), and $$A, B, C, D$$ and $$L$$ (the system matrices). Furthermore, the $$A$$ matrix is assumed to be stable

The Optimization Problem
We need to design a filter that will eliminate the effects of the disturbances as best we can. For this, we take a filter of the following form:

\begin{align} \dot \sigma &= A_{f}\sigma + B_{f}\sigma, \sigma(0) = \sigma_{0}\\ \hat z &= C_{f}\sigma + D_{f}y, \end{align}$$

where $$\sigma \in R^{n}$$ is the state vector, $$\hat z \in R^{m}$$ is the estimation vector of z, and $$A_{f}, B_{f}, C_{f}, and D_{f}$$ are the coefficient matrices of appropriate dimensions.

Note that the combined complete system can be represented as

\begin{align} \dot x_{e} &= \tilde{A}x_{e} + \tilde{B}w, x_{e}(0) = x_{e0}\\ \tilde{z} &= \tilde{C}x_{e} + \tilde{D}w, \end{align}$$

where $$ \tilde{z} = z - \hat z \in R^{m}$$ is the estimation error,

\begin{align} x_{e} = \begin{bmatrix} x \\ \sigma\end{bmatrix}\\ \end{align}$$ is the state vector of the system, and $$\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D} $$ are the coefficient matrices, defined as:

\begin{align} \tilde{A} = \begin{bmatrix} A & 0 \\ B_{f}C & A_{f} \end{bmatrix}, \tilde{B} = \begin{bmatrix} B \\ B_{f}D\end{bmatrix},\\ \tilde{C} = \begin{bmatrix} L-D_{f}C & -C_{f} \end{bmatrix}, \tilde{D} = -D_{f}D \end{align}$$

In other words, for the system defined above we need to find $$A_{f}, B_{f}, C_{f}, and D_{f}$$ such that

\begin{align} \end{align}$$ where $$ \gamma $$ is a positive constant, and
 * G_{\tilde{z}w}(s)|_{\inf} < \gamma,

\begin{align} G_{\tilde{z}w}(s) = \tilde{C}(sI-\tilde{A})^{-1}\tilde{B} + \tilde{D}. \end{align}$$

The LMI: H-inf Filtering
The solution can be obtained by finding matrices $$ R, X, M, N, Z, D_{f}$$ that obey the following LMIs:

\begin{align} X&>0\\ R - X &>0\\ \begin{bmatrix} RA + A^{T}R + ZC + C^{T}Z^{T} & *&*&*\\ M^{T}+ZC+XA & M^{T}+M &*&*\\ B^{T}R+D^{T}Z^{T} & B^{T}X+D^{T}Z^{T} & -\gamma I &*\\ L-D_{f}C & -N & -D_{f}D & -\gamma I\end{bmatrix}&<0\\ \end{align}$$

Conclusion:
To find the corresponding filter, use the optimized matrices from the solution to find:

A_{f} = X^{-1}M, B_{f}=X^{-1}Z, C_{f} = N, D_{f} = D_{f} $$

These matrices can then be used to produce $$\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D}$$ to construct the filter described above, that will best eliminate the disturbances of the system.

Implementation
This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/hinf_filtering.m

Related LMIs
H-2_filtering


text:
LMIs in Control/pages/Insensitive Strip Region Design

Insensitive Strip Region Design

Suppose if one were interested in robust stabilization where closed-loop eigenvalues are placed in particular regions of the complex plane where the said regions has an inner boundary that is insensitive to perturbations of the system parameter matrices. This would be accomplished with the help of 2 design problems: the insensitive strip region design and insensitive disk region design (see link below for the latter).

The System
Suppose we consider the following continuous-time linear system that needs to be controlled:

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

where $$x\in\mathbb{R}^{n}$$, $$y\in\mathbb{R}^{m}$$, and $$u\in\mathbb{R}^{r}$$ are the state, output and input vectors respectively. Then the steps to obtain the LMI for insensitive strip region design would be obtained as follows.

The Data
Prior to obtaining the LMI, we need the following matrices: $$A$$, $$B$$, and $$C$$.

The Optimization Problem
Consider the above linear system as well as 2 scalars $$\gamma_1$$ and $$\gamma_2$$. Then the output feedback control law $$u=Ky$$ would be such that $${\gamma_1} < {\lambda_i}(A^s_c) < {\gamma_1}$$, where:

\begin{align} {A^s_c}&\triangleq\frac{1}{2}{\langle{A_c}\rangle_s}=\frac{{(A+BKC)^T}+(A+BKC)}{2}\\ \end{align}$$

Letting $$K$$ being the solution to the above problem, then

\begin{align} {\gamma_1}&<{\alpha_1}\leq{Re({\lambda_i}(A+BKC))\leq{\alpha_2}<{\gamma_2}}, &i=1,2,...,n\\ \end{align}$$ where

\begin{align} \begin{cases} {\alpha_1}&={\lambda_{min}}({A^s_c})\\ {\alpha_2}&={\lambda_{max}}({A^s_c})\\ \end{cases} \end{align}$$

The LMI: Insensitive Strip Region Design
Using the above info, we can simplify the problem by setting $$\gamma_1$$ to $$-\infty$$ for all practical applications. This then simplifies our problem and results in the following LMI:

\begin{align} \begin{cases} \text{min }\gamma\\ \text{s.t. }{(A+BKC)^T}+(A+BKC)<{\gamma}I\\ \end{cases} \end{align}$$

Conclusion:
If the resulting solution from the LMI above produces a negative $$\gamma$$, then the output feedback controller $$K$$ is Hurwitz-stable. Hoewever, if $$\gamma$$ is a really small positive number, then $${\alpha_2}=\lambda_{max}(A^s_c)$$ must be negative for the controller to be Hurwitz-stable.

Implementation

 * Example Code - A GitHub link that contains code (titled "InsensitiveStripRegion.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIs

 * Insensitive Disk Region Design - Equivalent LMI for Insensitive Disk Region Design
 * $H_2$ Strip Region Design - LMI for $$H_2$$ Strip Region Design


text:
LMIs in Control/pages/H-2 filtering

For systems that have disturbances, filtering can be used to reduce the effects of these disturbances. Described on this page is a method of attaining a filter that will reduce the effects of the disturbances as completely as possible. To do this, we look to find a set of new coefficient matrices that describe the filtered system. The process to achieve such a new system is described below. The H2-filter tries to minimize the average magnitude of error.

The System
For the application of this LMI, we will look at linear systems that can be represented in state space as

\begin{align} \dot x &= Ax + Bw, x(0) = x_{0}\\ y &= Cx + Dw\\ z &= Lx \end{align}$$

where $$x \in R^{n}, y \in R^{l}, z \in R^{m}$$ represent the state vector, the measured output vector, and the output vector of interest, respectively, $$w \in R^{p}$$ is the disturbance vector, and $$A, B, C, D$$ and $$L$$ are the system matrices of appropriate dimension. To further define: $$x$$ is $$\in R^{n}$$ and is the state vector, $$A $$ is $$\in R^{n*n}$$ and is the state matrix, $$B$$ is $$\in R^{n*r}$$ and is the input matrix, $$w$$ is $$\in R^{r}$$ and is the exogenous input, $$C $$ is $$\in R^{m*n}$$ and is the output matrix, $$D$$ and $$L$$ are $$\in R^{m*r}$$ and are feedthrough matrices, and $$y$$ and $$z$$ are $$\in R^{m}$$ and are the output and the output of interest, respectively.

The Data
The data are $$w$$ (the disturbance vector), and $$A, B, C, D$$ and $$L$$ (the system matrices). Furthermore, the $$A$$ matrix is assumed to be stable

The Optimization Problem
We need to design a filter that will eliminate the effects of the disturbances as best we can. For this, we take a filter of the following form:

\begin{align} \dot \sigma &= A_{f}\sigma + B_{f}y, \sigma(0) = \sigma_{0}\\ \hat z &= C_{f}\sigma, \end{align}$$

where $$\sigma \in R^{n}$$ is the state vector, $$\hat z \in R^{m}$$ is the estimation vector, and $$A_{f}, B_{f}, C_{f}$$ are the coefficient matrices of appropriate dimensions.

Note that the combined complete system can be represented as

\begin{align} \dot x_{e} &= \tilde{A}x_{e} + \tilde{B}w, x_{e}(0) = x_{e0}\\ \tilde{z} &= \tilde{C}x_{e}, \end{align}$$

where $$ \tilde{z} = z - \hat z \in R^{m}$$ is the estimation error,

\begin{align} x_{e} = \begin{bmatrix} x \\ \sigma\end{bmatrix}\\ \end{align}$$ is the state vector of the system, and $$\tilde{A}, \tilde{B}, \tilde{C}$$ are the coefficient matrices, defined as:

\begin{align} \tilde{A} = \begin{bmatrix} A & 0 \\ B_{f}C & A_{f} \end{bmatrix}, \tilde{B} = \begin{bmatrix} B \\ B_{f}D\end{bmatrix},\\ \tilde{C} = \begin{bmatrix} L & -C_{f} \end{bmatrix} \end{align}$$

In other words, for the system defined above we need to find $$A_{f}, B_{f}, C_{f}$$ such that

\begin{align} \end{align}$$ where $$ \gamma $$ is a positive constant, and
 * G_{\tilde{z}w}(s)||_{2} < \gamma,

\begin{align} G_{\tilde{z}w}(s) = \tilde{C}(sI-\tilde{A})^{-1}\tilde{B} \end{align}$$

The LMI: H-2 Filtering
For this LMI, the solution exists if one of the following sets of LMIs hold:

Matrices $$ R, X, M, N, Z, Q$$ exist that obey the following LMIs:

\begin{align} R - X &>0,\\ trace(Q) &< \gamma^{2},\\ \begin{bmatrix} -Q& *&*\\ L^{T} &-R & *\\ -N^{T} & -X & -X\end{bmatrix}&<0,\\ \begin{bmatrix} RA + A^{T}R + ZC + C^{T}Z^{T} & *&*\\ M^{T}+ZC+XA & M^{T}+M &*\\ B^{T}R+D^{T}Z^{T} & B^{T}X+D^{T}Z^{T} & -I\end{bmatrix}&<0.\\ \end{align}$$

or

Matrices $$ \bar{R}, \bar{X}, \bar{M}, \bar{N}, \bar{Z}, \bar{Q}$$ exist that obey the following LMIs:

\begin{align} \bar{R} - \bar{X} &>0,\\ trace(\bar{Q}) &< \gamma^{2},\\ \begin{bmatrix} -\bar{Q}& *&*\\ \bar{R}B+\bar{Z}D &-\bar{R} & *\\ \bar{X}B + \bar{Z}D & -\bar{X} & -I\end{bmatrix}&<0,\\ \begin{bmatrix} \bar{R}A + A^{T}\bar{R} + \bar{Z}C + C^{T}\bar{Z}^{T} & *&*\\ \bar{M}^{T}+\bar{Z}C+\bar{X}A & \bar{M}^{T}+\bar{M} &*\\ L & -\bar{N} & -I\end{bmatrix}&<0.\\ \end{align}$$

Conclusion:
To find the corresponding filter, use the optimized matrices from the first solution to find:

A_{f} = X^{-1}M, B_{f}=X^{-1}Z, C_{f} = N $$

Or the second solution to find:

A_{f} = \bar{X}^{-1}\bar{M}, B_{f}=\bar{X}^{-1}\bar{Z}, C_{f} = \bar{N} $$

These matrices can then be used to produce $$\tilde{A}, \tilde{B}, \tilde{C} $$ to construct the final filter below, that will best eliminate the disturbances of the system.

\begin{align} \dot x_{e} &= \tilde{A}x_{e} + \tilde{B}w, x_{e}(0) = x_{e0}\\ \tilde{z} &= \tilde{C}x_{e}, \end{align}$$

Implementation
This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/H2_Filtering.m

Related LMIs
H-infinity filtering

title: LMIs in Control/pages/H-2 filtering


text:
LMIs in Control/pages/Insensitive Disk Region Design

Insensitive Disk Region Design

Similar to the insensitive strip region design problem, insensitive disk region design is another way with which robust stabilization can be achieved where closed-loop eigenvalues are placed in particular regions of the complex plane where the said regions has an inner boundary that is insensitive to perturbations of the system parameter matrices.

The System
Suppose we consider the following linear system that needs to be controlled:

\begin{align} \begin{cases} \rho x&=Ax+Bu,\\ y&=Cx\\ \end{cases} \end{align}$$

where $$x\in\mathbb{R}^{n}$$, $$y\in\mathbb{R}^{m}$$, and $$u\in\mathbb{R}^{r}$$ are the state, output and input vectors respectively, and $$\rho$$ represents the differential operator (in the continuous-time case) or one-step shift forward operator (i.e., $$\rho x(k)=x(k+1)$$) (in the discrete-time case). Then the steps to obtain the LMI for insensitive strip region design would be obtained as follows.

The Data
Prior to obtaining the LMI, we need the following matrices: $$A$$, $$B$$, and $$C$$.

The Optimization Problem
Consider the above linear system as well as 2 positive scalars $$\gamma$$ and $$q$$. Then the output feedback control law $$u=Ky$$ would be designed such that:

\begin{align} {\eta}&=||A+BKC+qI||<{\gamma}\\ \end{align}$$

Recalling the definition, we have:

\begin{align} {\mathbb{D}_{q,\eta}}&=\{ {s|s\in\mathbb{C},|s+q|<{\eta}}\}\\ &=\{ x+jy|x,y\in\mathbb{R},{(x+q)^2}+{y^2}<{\eta^2}\}\\ \end{align}$$

and

\begin{align} {\mathbb{D}_{q,r}}&=\{ {s|s\in\mathbb{C},|s+q|<{\gamma}}\}\\ &=\{ x+jy|x,y\in\mathbb{R},{(x+q)^2}+{y^2}<{\gamma^2}\}\\ \end{align}$$

Letting $$K$$ being the solution to the above problem, then

\begin{align} {\lambda_i}(A+BKC)&\in{\mathbb{D}_{q,\eta}}\subset{\mathbb{D}_{q,r}}, &i=1,2,...,n\\ \end{align}$$

The LMI: Insensitive Strip Region Design
Using the above info, we can convert the given problem into an LMI, which - after using Schur compliment Lemma - results in the following:

\begin{align} \begin{cases} \text{min }\gamma\\ \text{s.t. }\begin{bmatrix}-{\gamma}I&&(A+BKC+qI)\\{(A+BKC+qI)^T}&&-{\gamma}I\end{bmatrix}&<0\\ \end{cases} \end{align}$$

Conclusion:
For Schur stabilization, we can choose to solve the problem with $$q=0$$. Schur stability is achieved when $${\gamma}{\leq}1$$. Alternately, if $${\gamma}$$ is greater than (but very close to) 1, then Schur stability is also achieved when $${\eta}=||A+BKC+qI||_2{\leq}1$$.

Implementation

 * Example Code - A GitHub link that contains code (titled "InsensitiveDiskRegion.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIs

 * $H_2$ Disk Region Design - LMI for Disk Region Design with minimal $$H_2$$ gain
 * Insensitive Strip Region Design - Equivalent LMI for Insensitive Strip Region Design