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User:Margav06/sandbox/Click here to continue/Observer synthesis/FDI Filter Design For Systems With Sensor Faults

FDI Filter Design For Systems With Sensor Faults: an LMI

Systems with faulty sensors are a very common type of systems. In many cases, redundancy is added in the form of additional sensors, but in certain cases it could be a costly solution. For general linear system models, the LMI in this section can be utilized to design state estimators which can detect and isolate faulty sensor readings in order to mitigate their effects.

The System

\begin{align} \dot x(t)&=Ax(t)+B_1w(t) + B_2u(t)\\ y(t)&=Cx(t) + v(t) \end{align}$$

where $$ x(t) \in \R^n $$ is the state, $$ u(t) \in \R^m $$ is the control input, $$ w(t) \in \R^r $$ is the process noise, $$ y(t) \in \R^p $$ is the output and $$ v(t) \in \R^q $$ is the measurement noise.

The Data
The state space matrices $$ (A, B_1, B_2, C) $$ are required to be known.

The Optimization LMI
The following LMI is used to design the Fault Detection and Isolation (FDI) filter:

\begin{align} &\text{min}_{\Phi, \Theta, \gamma} \; \gamma, \\ &\text{subj. to: } \Phi > 0, \\ & \quad \quad \begin{bmatrix} A^\top\Phi + \Phi A + C^\top \Theta^\top + \Theta C & \Phi B_1 & \Theta & I \\ *                                       & -\gamma I & 0 & 0 \\ * & * & -\gamma I & 0 \\ * & * & * & -\gamma I \end{bmatrix} <0\\ \end{align}$$

Then the filter is $$ K = \Phi^{-1} \Theta $$.

Conclusion:
The LMI designed in this section is used to design filters that can work on systems that are prone to sensors getting damaged or faulty.

Implementation
To solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:

https://github.com/smhassaan/LMI-Examples/blob/master/FDI_Filter_example.m

Related LMIs
H-infinity Optimal Filter

H-infinity Optimal Observer


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User:Margav06/sandbox/Click here to continue/Observer synthesis/Hurwitz Detectability

Hurwitz Detectability
Hurwitz detectability is a dual concept of Hurwitz stabilizability and is defined as the matrix pair $$ (A, C) $$, is said to be Hurwitz detectable if there exists a real matrix $$ L $$ such that $$ A + LC $$ is Hurwitz stable.

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 system matrices of appropriate dimensions and are known.

The Optimization Problem
There exist a symmetric positive definite matrix $$ P $$ and a matrix $$ W $$ satisfying $$ A^TP + PA + W^TC + C^TW < 0 $$ There exists a symmetric positive definite matrix $$ P $$ satisfying $$ N_c^T ( A^TP + PA) N_c < 0 $$ with $$ N_c $$ being the right orthogonal complement of $$ C $$. There exists a symmetric positive definite matrix $$ P $$ such that $$ A^TP + PA < \gamma C^TC $$ for some scalar $$ \gamma > 0 $$

The LMI:
Matrix pair $$(A,C)$$, is Hurwitz detectable if and only if following LMI holds
 * $$ A^TP + PA + W^TC + C^TW < 0. $$
 * $$ N_c^T ( A^TP + PA) N_c < 0 $$
 * $$ A^TP + PA - \gamma C^TC < 0 $$

Conclusion:
Thus by proving the above conditions we prove that the matrix pair $$(A,C)$$ is Hurwitz Detectable.

Implementation
Find the MATLAB implementation at this link below Hurwitz detectability

Related LMIs
Links to other closely-related LMIs LMI for Hurwitz stability LMI for Schur stability Schur Detectability

Return to Main Page:

 * https://en.wikibooks.org/wiki/LMIs_in_Control

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


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User:Margav06/sandbox/Click here to continue/Observer synthesis/Reduced-Order State Observer

Reduced Order State Observer
The Reduced Order State Observer design paradigm follows naturally from the design of Full Order State Observer.

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 system matrices of appropriate dimensions and are known.

The Problem Formulation
Given a State-space representation of a system given as above. First an arbitrary matrix $$ R \in \R^{(n-m) x n} $$ is chosen such that the vertical augmented matrix given as

\begin{align} T=\begin{bmatrix} C \\ R \\ \end{bmatrix} \end{align} $$

is nonsingular, then

\begin{align} CT^{-1}=\begin{bmatrix} I_{m} & 0 \end{bmatrix} \end{align} $$

Furthermore, let

\begin{align} TAT^{-1}=\begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \\

\end{bmatrix} , A_{11} \in \R^{m x m} \end{align} $$

then the matrix pair $$ (A_{22},A_{12}) $$ is detectable if and only if $$ (A,C) $$ is detectable, then let

\begin{align} Tx=\begin{bmatrix} x_{1} \\ x_{2} \\ \end{bmatrix} , TB=\begin{bmatrix} B_{1} \\ B_{2} \\ \end{bmatrix} \end{align} $$ then a new system of the form given below can be obtained

\begin{align} \begin{bmatrix} \dot x_{1} \\ \dot x_{2} \\ \end{bmatrix} =\begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{bmatrix} \begin{bmatrix} \dot x_{1} \\ \dot x_{2} \\ \end{bmatrix} + \begin{bmatrix} B_{1} \\ B_{2} \\ \end{bmatrix}u , y=x_{1} \end{align} $$

once an estimate of $$ x_{2} $$ is obtained the the full state estimate can be given as

\begin{align} \hat x= T^{-1}\begin{bmatrix} y \\ \hat x_{2} \\ \end{bmatrix} \end{align} $$ the the reduced order observer can be obtained in the form.

\begin{align} \dot z&=Fz+Gy+Hu,\\ \hat x_{2}&=Mz+Ny\\ \end{align} $$ Such that for arbitrary control and arbitrary initial system values, There holds

\begin{align} lim_{t \to \infty} (x_{2}-\hat x_{2}) = 0 \end{align} $$

The value for $$ F,G,H,M,N $$ can be obtain by solving the following LMI.

The LMI:
The reduced-order observer exists if and only if one of the two conditions holds.

1) There exist a symmetric positive definite Matrix $$ P $$ and a matrix $$ W $$ that satisfy
 * $$ A_{22}^TP + PA_{22} + W^A_{12} + A_{12}^TW < 0. $$

Then $$ L = P^{-1}W $$ 2) There exist a symmetric positive definite Matrix $$ P $$ that satisfies the below Matrix inequality Then $$ L = -\frac{1}{2}P^{-1}A_{12}^{T} $$.
 * $$ A_{22}^TP + PA_{22} - A_{12}^TA_{12} < 0 $$

By using this value of $$ L $$ we can reconstruct the observer state matrices as

\begin{align} F=A_{22}+LA{12}, G=(A_{21}+LA_{11}) - (A_{22}+LA_{12})L, H= B_{2}+LB_{1} , M=I , N=-L, \end{align} $$

Conclusion:
Hence, we are able to form a reduced-order observer using which we can back of full state information as per the equation given at the end of the problem formulation given above.


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User:Margav06/sandbox/Click here to continue/Observer synthesis/Discrete Time Detectability

Discrete-Time Detectability

A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.

Discrete-Time LTI systems can be made detectable using observer gain L, which can be found using LMI optimization, such that the close loop system is detectable.

The System
Discrete-Time LTI System with state space realization $$(A_d,B_d,C_d,D_d)$$ $$ \begin{align} &A_d \in \bf{R^{n*n}}, &B_d \in \bf{R^{n*m}}, &C_d \in \bf{R^{p*n}}, &D_d \in \bf{R^{p*m}} \;\\ \end{align}$$

The Data
The matrices: System $$ (A_d,B_d,C_d,D_d), P, W $$.

The Optimization Problem
The following feasibility problem should be optimized:

Maximize P while obeying the LMI constraints. Then L is found.

The LMI:
Discrete-Time Detectability

The LMI formulation

$$ \begin{align} P \in {S^{n}}; W \in {R^{m*n}}\;\\ &P>0\\ \begin{bmatrix}P & A_d^TP+C_d^TW\\
 * & P\end{bmatrix}&>0,\\

L = P^{-1}W

\end{align}$$

Conclusion:
The system is detectabe iff there exits a $$P$$, such that $$P>0$$. The matrix $$A_d+LC_d$$ is Schur with $$L = P^{-1}W$$

Implementation
A link to CodeOcean or other online implementation of the LMIMATLAB Code

Related LMIs
- Continuous time Detectability


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User:Margav06/sandbox/Click here to continue/Observer synthesis/Schur Detectability

Schur Detectability

Schur detectability is a dual concept of Schur stabilizability and is defined as follows, the matrix pair $$ (A, C), $$ is said to be Schur detectable if there exists a real matrix $$ L $$ such that $$ A + LC $$ is Schur stable.

The System
We consider the following system:

$$ \begin{align} x(k+1) = Ax(k) + Bu(k)\\ y(k) = Cx(k) + Du(k)\\ \end{align}$$

where the matrices $$ A \in \mathbb{R}^{n\times n} $$, $$B \in \mathbb{R}^{n \times r} $$, $$ C \in \mathbb{R}^{m\times n} $$,$$ D \in \mathbb{R}^{m\times r} $$$$ x \in \mathbb{R}^{n} $$,$$ y \in \mathbb{R}^{m} $$, and $$ u \in \mathbb{R}^{r} $$ are the state matrix, input matrix, state vector, and the input vector, respectively.

Moreover, $$k$$ represents time in the discrete-time system and $$k+1$$ is the next time step.

The state feedback control law is defined as follows:

$$ \begin{align} u(k) = Kx(k) \end{align}$$

where $$K \in \mathbb{R}^{n\times r} $$ is the controller gain. Thus, the closed-loop system is given by:

$$ \begin{align} x(k+1) = (A + BK)x(k) \end{align}$$

The Data

 * The matrices $$ A,B,C,D $$ are system matrices of appropriate dimensions and are known.

The Optimization Problem
There exist a symmetric matrix $$ P $$ and a matrix W satisfying $$\begin{align} \begin{bmatrix}  -P &  A^TP + C^TW^T    \\ PA + WC  & P \end{bmatrix} < 0\\ \end{align}$$ There exists a symmetric matrix $$ P $$ satisfying $$\begin{align} \begin{bmatrix}  -N_c^TPN_c &  N_c^TA^TP    \\ PAN_c  & -P \end{bmatrix} < 0\\ \end{align}$$ with $$ N_c $$ being the right orthogonal complement of $$ C $$. There exists a symmetric matrix P such that $$\begin{align} \begin{bmatrix}  -P &  PA    \\ A^TP  & -P-\gamma C^TC \end{bmatrix} < 0\\ \end{align}$$ $$ \gamma > 1 $$

The LMI:
The LMI for Schur detecability can be written as minimization of the scalar, $$\gamma$$, in the following constraints:

$$ \begin{align} & \text{min} \quad \gamma\\ & \text{s.t.} \end{align}$$ $$\begin{align} \begin{bmatrix}  -P &  A^TP + C^TW^T    \\ PA + WC  & P \end{bmatrix} < 0\\ \end{align}$$ $$\begin{align} \begin{bmatrix}  -N_c^TPN_c &  N_c^TA^TP    \\ PAN_c  & -P \end{bmatrix} < 0\\ \end{align}$$ $$\begin{align} \begin{bmatrix}  -P &  PA    \\ A^TP  & -P-\gamma C^TC \end{bmatrix} < 0\\ \end{align}$$

Conclusion:
Thus by proving the above conditions we prove that the matrix pair $$(A,C)$$ is Schur Detectable.

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

Related LMIs
LMI for Hurwitz stability LMI for Schur stability Hurwitz Detectability

Return to Main Page
LMIs in Control/Tools


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User:Margav06/sandbox/Click here to continue/Observer synthesis/Full-order state Observer

Full-Order State Observer
The problem of constructing a simple full-order state observer directly follows from the result of Hurwitz detectability LMI's, Which essentially is the dual of Hurwitz stabilizability. If a feasible solution to the first LMI for Hurwitz detectability exist then using the results we can back out a full state observer $$ L $$ such that $$ A + LC $$ is Hurwitz stable.

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 system matrices of appropriate dimensions and are known.

The Optimization Problem
The full-order state observer problem essential is finding a positive definite $$ P $$ such that the following LMI conclusions hold.

The LMI:
1) The full-order state observer problem has a solution if and only if there exist a symmetric positive definite Matrix $$ P $$ and a matrix $$ W $$ that satisfy Then the observer can be obtained as $$ L = P^{-1}W $$ 2) The full-state state observer can be found if and only if there is a symmetric positive definite Matrix $$ P $$ that satisfies the below Matrix inequality In this case the observer can be reconstructed as $$ L = -\frac{1}{2}P^{-1}C^{T} $$. It can be seen that the second relation can be directly obtained by substituting $$ W= -\frac{1}{2}C^{T} $$ in the first condition.
 * $$ A^TP + PA + W^TC + C^TW < 0. $$
 * $$ A^TP + PA - C^TC < 0 $$

Conclusion:
Hence, both the above LMI's result in a full-order observer $$L$$ such that $$ A + LC $$ is Hurwitz stable.


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User:Margav06/sandbox/Click here to continue/Observer synthesis/Full-order Hinf state Observer

In this section, we design full order H-$$ \infty $$ state observer.

The System
Given a state-space representation of a linear system

\begin{align} \ \dot x(t) = Ax(t) + B_{1}u(t) + B_{2}w(t), x(0) = x_{0} \\ \ y(t) = C_{1}x(t) + D_{1}u(t) + D_{2}w(t) \\ \ z(t) = C_{2}x(t)\\ \end{align}$$

 * $$ x \in \mathbb{R}^{n}, y \in \mathbb{R}^{l}, z in \mathbb{R}^{m} $$ are the state vector, measured output vector and output vectors of interest.
 * $$ w \in \mathbb{R}^{p}, u \in \mathbb{R}^{r}, $$ are the disturbance vector and control vector respectively.

The Data
$$ A,B_{1},B_{2},C_{1},C_{2},D_{1},D_{2}$$ are system matrices

Definition
For the system, a full order state observer of the form of equation (1) is introduced and the estimate of interested output is given by.

The estimate of interested output is

Given the system and a positive scalar $$ \gamma $$, L is found such that

LMI Condition
The $$ H_{\infty}$$ state observers problem has a solution if and only if there exists a symmetric positive definite matrix $$ P $$ and a matrix $$ W $$ satisfying the below LMI

When such a pair of matrics is found, the solution is

Implementation
This implementation requires Yalmip and Mosek.
 * https://github.com/ShenoyVaradaraya/LMI--master/blob/main/hinf_obs.m

Conclusion
Thus, an $$H_{\infty} $$ state observer is designed such that the output vectors of interest are accurately estimated.


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User:Margav06/sandbox/Click here to continue/Time-Delay Systems

Time-Delay Systems

 * 1) Continuous Time
 * 2) /LMI for Time-Delay system on delay Independent Condition/
 * 3) /LMI for Time-Delay system on delay Dependent Condition/
 * 4) /LMI for Stability of Retarded Differential Equation with Slowly-Varying Delay/
 * 5) /LMI for Robust Stability of Retarded Differential Equation with Norm-Bounded Uncertainty/
 * 6) /Bounded Real Lemma under Slowly-Varying Delay/
 * 7) /LMI for L2-Optimal State-Feedback Control under Time-Varying Input Delay/
 * 8) Discrete Time
 * 9) /Stability Condition for Discrete-Time TDS/


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User:Margav06/sandbox/Click here to continue/Time-Delay Systems/LMI for Time-Delay system on delay Independent Condition

The System
The problem is to check the stability of the following linear time-delay system

\begin{align} \begin{cases} \dot x(t)&=Ax(t)+A_dx(t-d)\\ x(t)&=\phi(t), t\in[-d,0], 0 < d \leq \bar{d},\\ \end{cases} \end{align}$$

where

\begin{align} {A, A_d}\in\mathbb{R}^{n\times n}\text{, }{A}\in\mathbb{R}^{n\times r}\text{ are the system coefficient matrices,}\\ \end{align}$$

$$ \phi (t) $$ is the initial condition $$d$$ represents the time-delay $$\bar d$$ is a known upper-bound of $$d$$

The Data
The matrices $$A, A_d$$ are known

The LMI: The Time-Delay systems (Delay Independent Condition)
From the given pieces of information, it is clear that the optimization problem only has a solution if there exists two symmetric matrices $$P,S\in\mathbb{S}^{n}$$ such that

$$P>0$$

\begin{bmatrix} A^TP+PA+S & PA_d \\ A_d^TP & -S \end{bmatrix}$$$$\begin{align}< 0\end{align}$$

This LMI has been derived from the Lyapunov function for the system. By Schur Complement we can see that the above matrix inequality is equivalent to the Riccati inequality $$ A^TP+PA+PA_dS^{-1}A_d^TP+S<0$$

Conclusion:
We can now implement these LMIs to do stability analysis for a Time delay system on the delay independent condition

Implementation
The implementation of the above LMI can be seen here

https://github.com/yashgvd/LMI_wikibooks

Related LMIs
Time Delay systems (Delay Dependent Condition)


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User:Margav06/sandbox/Click here to continue/Time-Delay Systems/LMI for Time-Delay system on delay Dependent Condition

The System
The problem is to check the stability of the following linear time-delay system on a delay dependent condition

\begin{align} \begin{cases} \dot x(t)&=Ax(t)+A_dx(t-d)\\ x(t)&=\phi(t), t\in[-d,0], 0 < d \leq \bar{d},\\ \end{cases} \end{align}$$

where

\begin{align} {A, A_d}\in\mathbb{R}^{n\times n}, A\in\mathbb{R}^{n\times r}\text{ are the system coefficient matrices,}\\ \end{align}$$

$$ \phi (t) $$ is the initial condition $$d$$ represents the time-delay $$\bar d$$ is a known upper-bound of $$d$$

For the purpose of the delay dependent system we rewrite the system as

\begin{cases} \dot x(t) &= Ax(t)+A_dx(t-d)\\ \dot x(t) &= (A+A_d)x(t)-A_d(x(t)-x(t-d) \end{cases} $$

The Data
The matrices $$A, A_d$$ are known

The LMI: The Time-Delay systems (Delay Dependent Condition)
From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a symmetric positive definite matrix $$X$$ and a scalar $$0< \beta <1$$ such that

\begin{bmatrix} \Phi(X) & \bar dXA^T & \bar dXA_d^T\\ \bar dAX & -d \beta I & 0\\ \bar dA_dX & 0 & -\bar d(1- \beta )I \end{bmatrix}$$$$\begin{align}< 0\end{align}$$

Here $$ \Phi(X) =X)A+A_d)^T + (A+A_d)X+ \bar d A_dA_d^T<0 $$

This LMI has been derived from the Lyapunov function for the system. It follows that the system is asymptotically stable if

$$ P(A+A_d) + (A+A_d)^TP + \bar dPA_dA_d^TP + \frac {\bar d}{\beta}A^TA +\frac {\bar d}{1- \beta} A_d^TA_d<0$$ This is obtained by replacing $$X$$ with $$P^{-1}$$

Conclusion:
We can now implement these LMIs to do stability analysis for a Time delay system on the delay dependent condition

Implementation
The implementation of the above LMI can be seen here

https://github.com/yashgvd/LMI_wikibooks

Related LMIs
Time Delay systems (Delay Independent Condition)


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User:Margav06/sandbox/Click here to continue/Time-Delay Systems/LMI for Stability of Retarded Differential Equation with Slowly-Varying Delay

This page describes an LMI for stability analysis of a continuous-time system with a time-varying delay. In particular, a delay-independent condition is provided to test uniform asymptotic stability of a retarded differential equation through feasibility of an LMI. The system under consideration pertains a single discrete delay, with the extent of the delay at any time bounded by some known value. Moreover, the delay is assumed to vary only slowly in time, with a temporal derivative bounded by a value less than one. Solving the LMI for a particular value of this bound, uniform asymptotic stability can be shown for any time-delay satisfying this bound.

The System
The system under consideration is one of the form:

\begin{align}

\dot{x}(t) &= Ax(t) + A_1x(t-\tau(t)) & t&\geq t_0, & 0&\leq\tau(t) \leq h, & \dot{\tau}(t)&\leq d<1

\end{align} $$

In this description, $$ A $$ and $$ A_1 $$ are matrices in $$ \mathbb{R}^{n\times n} $$. The variable $$ \tau(t) $$ denotes a delay in the state at time $$ t\geq t_0 $$, assuming a value no greater than some $$ h\in\mathbb{R}_+ $$. Moreover, we assume that the function $$ \tau(t) $$ is differentiable at any time, with the derivative bounded by some value $$ d<1 $$, assuring the delay to be slowly-varying in time.

The Data
To determine stability of the system, the following parameters must be known:

$$ \begin{align} A&\in\mathbb{R}^{n\times n} \\ A_1&\in\mathbb{R}^{n\times n} \\ d&\in [0,1) \end{align} $$

The Optimization Problem
Based on the provided data, uniform asymptotic stability can be determined by testing feasibility of the following LMI:

The LMI: Delay-Independent Uniform Asymptotic Stability for Continuous-Time TDS

\begin{align} &\text{Find}:\\ &\qquad P, Q \in\mathbb{R}^{n\times n}\\ &\text{such that:}\\ &\qquad P>0,\quad Q>0  \\ &\qquad \begin{bmatrix} A^T P+PA+Q & PA_1 \\ A_1^TP & -(1-d)Q \end{bmatrix}<0 \\ \end{align}

$$

Conclusion:
If the presented LMI is feasible, the system will be uniformly asymptotically stable for any delay function $$ \tau(t) $$ satisfying $$ \dot{\tau}(t)\leq d<1 $$. That is, independent of the values of the delays $$ \tau(t) $$ and the starting time $$ t_0\in\mathbb{R} $$:

 * For any real number $$ \epsilon>0 $$, there exists a real number $$ \delta>0 $$ such that:

 * $$ \|x_{t_0}\|_{\mathcal{C}}<\delta \quad \Rightarrow \quad \|x(t)\|<\epsilon \qquad \forall t\geq t_0 $$

 * There exists a real number $$ \delta_a>0 $$ such that for any real number $$ \eta>0 $$, there exists a time $$ T(\delta_a,\eta) $$ such that:

 * $$ \|x_{t_0}\|_{\mathcal{C}}<\delta_a \quad \Rightarrow \quad \|x(t)\|<\eta \qquad \forall t\geq t_0+T(\delta_a,\eta) $$

Here, we let $$ x_{t_0}(\theta)=x(t_0+\theta) $$ for $$ \theta\in[-\tau(t_0),0] $$ denote the delayed state function at time $$ t_0 $$. The norm $$ \|x_{t_0}\|_{\mathcal{C}} $$ of this function is defined as the maximal value of the vector norm assumed by the state over the delayed time interval, given by:

 * $$ \|x_{t_0}\|_{\mathcal{C}}:=\max_{\theta\in[-\tau(t_0),0]} \|x(t_0+\theta)\| $$

Obtaining a feasible point for the LMI, this result can be proven using a Lyapunov-Krasovkii functional:

\begin{align} &V(t,x_t)=x^T(t)Px(t) + \int_{t-\tau(t)}^t x^T(s)Qx(s) ds\\ \end{align} $$

Notably, if matrices $$ P>0,Q>0 $$ prove feasibility of the LMI for the pair $$ (A,A_1) $$, these same matrices will also prove feasibility of the LMI for the pair $$ (A,-A_1) $$. As such, feasibility of this LMI proves uniform asymptotic stability of both systems:

\begin{align}

\dot{x}(t) &= Ax(t) \pm A_1x(t-\tau(t)) & t&\geq t_0, & 0&\leq\tau(t) \leq h & \dot{\tau}(t)&\leq d<1

\end{align} $$

Moreover, since the result is independent of the value of the delay, it will also hold for a delay $$ \tau(t)\equiv 0 $$. Hence, if the LMI is feasible, the matrices $$ A\pm A_1 $$ will be Hurwitz.

Implementation
An example of the implementation of this LMI in Matlab is provided on the following site:

 * https://github.com/djagt/LMI_Codes/blob/main/stblty_cTDS_SlowVarying.m

Note that this implementation requires packages for YALMIP with solver mosek, though a different solver can also be implemented.

Related LMIs

 * - Delay-dependent stability LMI for continuous-time TDS

 * - Stability LMI for delayed discrete-time system


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User:Margav06/sandbox/Click here to continue/Time-Delay Systems/LMI for Robust Stability of Retarded Differential Equation with Norm-Bounded Uncertainty

This page describes an LMI for stability analysis of an uncertain continuous-time system with a time-varying delay. In particular, a delay-independent condition is provided to test uniform asymptotic stability of a retarded differential equation with uncertain matrices through feasibility of an LMI. The system under consideration pertains a single discrete delay, with the extent of the delay at any time bounded by some known value. The matrices describing the system are assumed to be uncertain, with the norm of the uncertainty bounded by a value of one. In addition, the delay is assumed to vary only slowly in time, with a temporal derivative bounded by a value less than one. Solving the LMI for a particular value of this bound, uniform asymptotic stability can be shown for any time-delay satisfying this bound, independent of the value of the uncertainty function.

The System
The system under consideration is one of the form:

\begin{align}

\dot{x}(t) &= (A+H\Delta(t) E)x(t) + (A_1+H\Delta(t) E_1)x(t-\tau(t)) & t&\geq t_0, & 0&\leq\tau(t) \leq h, & \dot{\tau}(t)&\leq d<1

\end{align} $$

In this description, $$ A $$ and $$ A_1 $$ are matrices in $$ \mathbb{R}^{n\times n} $$. The variable $$ \tau(t) $$ denotes a delay in the state at time $$ t\geq t_0 $$, assuming a value no greater than some $$ h\in\mathbb{R}_+ $$. Moreover, we assume that the function $$ \tau(t) $$ is differentiable at any time, with the derivative bounded by some value $$ d<1 $$, assuring the delay to be slowly-varying in time. The uncertainty $$ \Delta(t)\in\mathbb{R}^{r_1\times r_2} $$ is also allowed to vary in time, but at any time $$ t\geq t_0 $$ must satisfy the inequality:

\begin{align}

\Delta^T(t)\Delta(t) &\leq I

\end{align} $$

The uncertainty affects the system through matrices $$ H\in\mathbb{R}^{n\times r_1} $$ and $$ E,E_1\in\mathbb{R}^{r_2\times n} $$, which are constant in time and assumed to be known.

The Data
To determine stability of the system, the following parameters must be known:

$$ \begin{align} A,A_1&\in\mathbb{R}^{n\times n} \\ H&\in\mathbb{R}^{n\times r_1} \\ E,E_1&\in\mathbb{R}^{r_2\times n} \\ d&\in [0,1) \end{align} $$

The Optimization Problem
Based on the provided data, uniform asymptotic stability can be determined by testing feasibility of the following LMI:

The LMI: Delay-Independent Robust Uniform Asymptotic Stability for Continuous-Time TDS

\begin{align} &\text{Find}:\\ &\qquad \epsilon\in\mathbb{R}, \quad P, Q \in\mathbb{R}^{n\times n}\\ &\text{such that:}\\ &\qquad \epsilon>0,\quad P>0,\quad Q>0  \\ &\qquad \begin{bmatrix} A^T P+PA+Q & PA_1 & PH & \epsilon E^T \\ A_1^TP & -(1-d)Q & 0 & \epsilon E_1^T \\ H^T P & 0 & -\epsilon I & 0 \\ \epsilon E & \epsilon E_1 & 0 & -\epsilon I \end{bmatrix}<0 \\ \end{align}

$$

Conclusion:
If the presented LMI is feasible, the system will be uniformly asymptotically stable for any delay function $$ \tau(t) $$ satisfying $$ \dot{\tau}(t)\leq d<1 $$, and any uncertainty $$ \Delta(t) $$ satisfying $$ \Delta^T(t)\Delta(t)\leq I $$. That is, independent of the values of the delays $$ \tau(t) $$, uncertainties $$ \Delta(t) $$, and the starting time $$ t_0\in\mathbb{R} $$:

 * For any real number $$ \epsilon>0 $$, there exists a real number $$ \delta>0 $$ such that:

 * $$ \|x_{t_0}\|_{\mathcal{C}}<\delta \quad \Rightarrow \quad \|x(t)\|<\epsilon \qquad \forall t\geq t_0 $$

 * There exists a real number $$ \delta_a>0 $$ such that for any real number $$ \eta>0 $$, there exists a time $$ T(\delta_a,\eta) $$ such that:

 * $$ \|x_{t_0}\|_{\mathcal{C}}<\delta_a \quad \Rightarrow \quad \|x(t)\|<\eta \qquad \forall t\geq t_0+T(\delta_a,\eta) $$

Here, we let $$ x_{t_0}(\theta)=x(t_0+\theta) $$ for $$ \theta\in[-\tau(t_0),0] $$ denote the delayed state function at time $$ t_0 $$. The norm $$ \|x_{t_0}\|_{\mathcal{C}} $$ of this function is defined as the maximal value of the vector norm assumed by the state over the delayed time interval, given by:

 * $$ \|x_{t_0}\|_{\mathcal{C}}:=\max_{\theta\in[-\tau(t_0),0]} \|x(t_0+\theta)\| $$

The proof of this result relies on the fact that the following inequality holds for any value $$ \epsilon>0 $$ and constant matrices $$ \alpha,\beta $$ of appropriate dimensions:

 * $$ \alpha \Delta(t)\beta + \beta^T \Delta^T(t)\alpha^T \leq \epsilon^{-1} \alpha\alpha^T + \epsilon\beta^T \beta $$

Using this inequality with $$ \alpha^T=\begin{bmatrix} H^TP & 0\end{bmatrix} $$ and $$ \beta = \begin{bmatrix} E & E_1 \end{bmatrix} $$, the described LMI can then be derived from that presented in, corresponding to a situation without uncertainty.

Implementation
An example of the implementation of this LMI in Matlab is provided on the following site:

 * https://github.com/djagt/LMI_Codes/blob/main/Rstblty_cTDS_SlowVarying.m

Note that this implementation requires packages for YALMIP with solver mosek, though a different solver can also be implemented.

Related LMIs

 * - Stability LMI for continuous-time RDE with slowly-varying delay without uncertainty

 * - LMI for quadratic stability of continuous-time system with norm-bounded uncertainty

 * - Stability LMI for delayed discrete-time system


text:
User:Margav06/sandbox/Click here to continue/Time-Delay Systems/Bounded Real Lemma under Slowly-Varying Delay

This page describes a bounded real lemma for a continuous-time system with a time-varying delay. In particular, a condition is provided to obtain a bound on the $$ L_2 $$-gain of a retarded differential system through feasibility of an LMI. The system under consideration pertains a single discrete delay, with the extent of the delay at any time bounded by some known value. This delay is only present in the state, with no direct delay in the effects of exogenous inputs on the state. In addition, the delay is assumed to vary only slowly in time, with a temporal derivative bounded by a value less than one, although results can also be attained if no bound is known. Solving the LMI for a particular value of the bound, while minimizing a scalar variable, an upper limit on the $$ L_2 $$-gain of the system can be shown for any time-delay satisfying this bound.

The System
The system under consideration is one of the form:

\begin{align}

\dot{x}(t) &= Ax(t) + A_1x(t-\tau(t)) + B_0 w(t) & t&\geq t_0, & 0&\leq\tau(t) \leq h, & \dot{\tau}(t)&\leq d<1 \\ z(t) &= C_0x(t) + C_{1}x(t-\tau(t))

\end{align} $$

In this description, $$ A $$ and $$ A_1 $$ are constant matrices in $$ \mathbb{R}^{n\times n} $$. In addition, $$ B_0 $$ is a constant matrix in $$ \mathbb{R}^{n\times n_w}$$, and $$ C_0,C_{1} $$ are constant matrices in $$ \mathbb{R}^{n_z\times n}$$ where $$ n_w,n_z\in\mathbb{N} $$ denote the number of exogenous inputs and regulated outputs respectively. The variable $$ \tau(t) $$ denotes a delay in the state at time $$ t\geq t_0 $$, assuming a value no greater than some $$ h\in\mathbb{R}_+ $$. Moreover, we assume that the function $$ \tau(t) $$ is differentiable at any time, with the derivative bounded by some value $$ d<1 $$, assuring the delay to be slowly-varying in time.

The Data
To obtain a bound on the $$ L_2 $$-gain of the system, the following parameters must be known:

$$ \begin{align} A,A_1&\in\mathbb{R}^{n\times n} \\ B_0&\in\mathbb{R}^{n\times n_w} \\ C_0,C_1&\in\mathbb{R}^{n_z\times n} \\ h&\in \mathbb{R}_+ \\ d&\in [0,1) \end{align} $$

The Optimization Problem
Based on the provided data, we can obtain a bound on the $$ L_2$$-gain of the system by testing feasibility of an LMI. In particular, the bounded real lemma states that if the LMI presented below is feasible for some $$ \gamma>0 $$, the $$ L_2 $$-gain of the system is less than or equal to this $$ \gamma $$. To attain a bound that is as small as possible, we minimize the value of $$ \gamma $$ while solving the LMI:

The LMI: L2-gain for TDS with Slowly-Varying Delay

\begin{align} &\text{Solve}:\\ &\qquad\min \gamma \\ &\text{such that there exist}:\\ &\qquad P,P_2,P_3,R,S,S_{12},Q\in\mathbb{R}^{n\times n}\\ &\text{for which}:\\ &\qquad P>0,\quad R>0,\quad S>0,\quad Q>0  \\ &\qquad \begin{bmatrix} \begin{array}{c c c c | c c} \Phi_{11} & \Phi_{12} & S_{12} & R-S_{12}+P_2^T A_1 & P_2^T B_0 & C_0^T \\ \end{array} \end{bmatrix}<0 \\ &\text{where}:\\ &\qquad \Phi_{11} = A^T P_2 + P_2^T A + S + Q -R\\ &\qquad \Phi_{12} = P-P_2^T+A^T P_3 \\ &\qquad \Phi_{22} = -P_3-P_3^T +h^2 R \end{align}
 * & \Phi_{22} & 0 & P_3^T A_1 & P_3^T B_0 & 0\\
 * & * & -S-R & R-S_{12}^T & 0 & 0 \\
 * & * & * & -2R - S_{12} - S_{12}^T - (1-d)Q & 0 & C_1^T \\\hline
 * & * & * & * & -\gamma^2 I & 0 \\
 * & * & * & * & * & -I

$$

In this notation, the symbols $$ * $$ are used to indicate appropriate matrices to assure the overall matrix is symmetric.

Conclusion:
If the presented LMI is feasible for some $$ \gamma $$, the system is internally stable, and will have an $$ L_2$$-gain less than $$ \gamma $$. That is, independent of the values of the delays $$ \tau(t) $$:

 * $$ \|z\|_{L_2} < \gamma \|w\|_{L_2} $$

It should be noted that this result is conservative. That is, even when minimizing the value of $$ \gamma $$, there is no guarantee that the bound obtained on the $$ L_2 $$-gain is sharp.

In a scenario where no bound $$ d $$ on the change in the delay is known, the above LMI can still be used to obtain a bound on the $$ L_2 $$-gain. In particular, setting $$ Q=0 $$ in the above LMI, a bound can be attained independent of the value of the derivative of the delay.

Implementation
An example of the implementation of this LMI in Matlab is provided on the following site:

 * https://github.com/djagt/LMI_Codes/blob/main/L2gain_cTDS.m

Note that this implementation requires packages for YALMIP with solver mosek, though a different solver can also be implemented.

Related LMIs

 * - Bounded real lemma for continuous-time system without delay

 * - Bounded real lemma for discrete-time system without delay

 * - Stability LMI for continuous-time RDE with slowly-varying delay


text:
User:Margav06/sandbox/Click here to continue/Time-Delay Systems/LMI for L2-Optimal State-Feedback Control under Time-Varying Input Delay

This page describes a method for constructing a full-state-feedback controller for a continuous-time system with a time-varying input delay. In particular, a condition is provided to obtain a bound on the $$ L_2 $$-gain of closed-loop system under time-varying delay through feasibility of an LMI. The system under consideration pertains a single discrete delay in the actuator input, with the extent of the delay at any time bounded by some known value. Moreover, the delay is assumed to vary only slowly in time, with a temporal derivative bounded by a value less than one, although results may also be attained if no bound is known. Solving the LMI for a particular value of the bound, while minimizing a scalar variable, an upper limit on the $$ L_2 $$-gain of the system can be shown for any time-delay satisfying this bound.

The System
The system under consideration is one of the form:

\begin{align}

\dot{x}(t) &= Ax(t) + B_2u(t-\tau(t)) + B_1 w(t) & t&\geq t_0, & 0&\leq\tau(t) \leq h, & \dot{\tau}(t)&\leq d<1 \\ z(t) &= C_1x(t) + D_{12}u(t-\tau(t))

\end{align} $$

In this description, $$ A $$ and $$ A_1 $$ are constant matrices in $$ \mathbb{R}^{n\times n} $$. In addition, $$ B_1 $$ is a constant matrix in $$ \mathbb{R}^{n\times n_w}$$, and $$ B_2 $$ is a constant matrix in $$ \mathbb{R}^{n\times n_u}$$, where $$ n_w,n_u\in\mathbb{N} $$ denote the number of exogenous and actuator inputs respectively. Finally, $$ C_{1} $$ and $$ D_{12} $$ are constant matrices in $$ \mathbb{R}^{n_z\times n}$$ and $$ \mathbb{R}^{n_z\times n_u}$$ respectively, where $$ n_z\in\mathbb{N} $$ denotes the number of regulated outputs. The variable $$ \tau(t) $$ denotes a delay in the actuator input at time $$ t\geq t_0 $$, assuming a value no greater than some $$ h\in\mathbb{R}_+ $$. Moreover, we assume that the function $$ \tau(t) $$ is differentiable at any time, with the derivative bounded by some value $$ d<1 $$, assuring the delay to be slowly-varying in time.

The Data
To construct an $$ L_2 $$-optimal controller of the system, the following parameters must be known:

$$ \begin{align} A&\in\mathbb{R}^{n\times n} \\ B_1&\in\mathbb{R}^{n\times n_w} \\ B_2&\in\mathbb{R}^{n\times n_u} \\ C_1&\in\mathbb{R}^{n_z\times n} \\ D_{12}&\in\mathbb{R}^{n_z\times n_u} \\ h&\in \mathbb{R}_+ \\ d&\in [0,1) \end{align} $$

In addition to these parameters, a tuning scalar $$ \epsilon>0 $$ is also implemented in the LMI.

The Optimization Problem
Based on the provided data, we can construct an $$ L_2$$-optimal full-state-feedback controller of the system by testing feasibility of an LMI. In particular, we note that if the LMI presented below is feasible for some $$ \gamma>0 $$ and matrices $$ \bar{P}_2^{-1}>0 $$ and $$ Y $$, implementing the state-feedback $$ u(t)=Kx(t) $$ with $$ K=Y\bar{P}_2^{-1} $$, the $$ L_2 $$-gain of the closed-loop system will be less than or equal to $$ \gamma $$. To attain a bound that is as small as possible, we minimize the value of $$ \gamma $$ while solving the LMI:

The LMI: L2-Optimal Full-State-Feedback for TDS with Slowly-Varying Input Delay

\begin{align} &\text{Solve}:\\ &\qquad\min \gamma \\ &\text{such that there exist}:\\ &\qquad \bar{P},\bar{P}_2,\bar{R},\bar{S},\bar{S}_{12},\bar{Q}\in\mathbb{R}^{n\times n}, \quad Y\in\mathbb{R}^{n_u\times n}\\ &\text{for which}:\\ &\qquad \bar{P}>0,\quad \bar{P}_2>0,\quad \bar{R}>0,\quad \bar{S}>0  \\ &\qquad \begin{bmatrix} \begin{array}{c c c c | c c} \bar{\Phi}_{11} & \bar{\Phi}_{12} & \bar{S}_{12} & B_2Y+\bar{R}-\bar{S}_{12} & B_1 & \bar{P}_2^T C_1^T \\ \end{array} \end{bmatrix}<0 \\ &\text{where}:\\ &\qquad \Phi_{11} = A \bar{P}_2 + \bar{P}_2^T A^T + \bar{S} + \bar{Q} -\bar{R}\\ &\qquad \Phi_{12} = \bar{P}-\bar{P}_2+\epsilon \bar{P}_2^T A^T \\ &\qquad \Phi_{22} = -\epsilon \bar{P}_2-\epsilon\bar{P}_2^T + h^2 R \end{align}
 * & \bar{\Phi}_{22} & 0 & \epsilon B_2Y & \epsilon B_1 & 0\\
 * & * & -\bar{S}-\bar{R} & \bar{R}-\bar{S}_{12}^T & 0 & 0 \\
 * & * & * & -(1-d)\bar{Q}-2\bar{R}+\bar{S}_{12}+\bar{S}_{12}^T & 0 & Y^T D_{12}^T \\\hline
 * & * & * & * & -\gamma^2 I & 0 \\
 * & * & * & * & * & -I

$$

In this notation, the symbols $$ * $$ are used to indicate appropriate matrices to assure the overall matrix is symmetric.

Conclusion:
If the presented LMI is feasible for some $$ \gamma,Y,\bar{P}_2 x(t) $$, implementing the full-state-feedback controller $$ u(t)=Kx(t)=Y\bar{P}_2^{-1} $$, the closed-loop system will be asymptotically stable, and will have an $$ L_2$$-gain less than $$ \gamma $$. That is, independent of the values of the delays $$ \tau(t) $$, the system:

\begin{align} \dot{x}(t)&=Ax(t)+B_2Kx(t-\tau(t))+B_1 w(t) \\ z(t) &= C_1x(t)+D_{12}Kx(t-\tau(t)) \end{align} $$

with:

 * $$ \|z\|_{L_2} < \gamma \|w\|_{L_2}

\begin{align} K=Y\bar{P}_2^{-1} \end{align} $$

will satisfy:

 * $$ \|z\|_{L_2} < \gamma \|w\|_{L_2} $$

Here we note that $$ \bar{P}_2^{-1}x(t) $$ is guaranteed to exist as $$ P_2 $$ is positive definite, and thus nonsingular.

It should be noted that the obtained result is conservative. That is, even when minimizing the value of $$ \gamma $$, there is no guarantee that the bound obtained on the $$ L_2 $$-gain is sharp, meaning that the actual $$ L_2 $$-gain of the closed-loop can be (significantly) smaller than $$ \gamma $$.

In a scenario where no bound $$ d $$ on the change in the delay is known, or this bound is greater than one, the above LMI may still be used to construct a controller. In particular, if the presented LMI is feasible with $$ \bar{Q}=0 $$, the closed-loop system imposing $$ u(t)=Kx(t)=Y\bar{P}_2^{-1} $$ will be internally exponentially stable with an $$ L_2 $$-gain less than $$ \gamma $$ independent of the value of $$ \dot{\tau}(t) $$.

Implementation
An example of the implementation of this LMI in Matlab is provided on the following site:

 * https://github.com/djagt/LMI_Codes/blob/main/L2_OptStateFdbck_cTDS.m

Note that this implementation requires packages for YALMIP with solver mosek, though a different solver can also be implemented.

Related LMIs

 * - Bounded real lemma for continuous-time system with slowly-varying delay

 * - LMI for Hinf-optimal full-state-feedback control in a non-delayed continuous-time system

 * - LMI for Hinf-optimal output-feedback control in a non-delayed continuous-time system


text:
User:Margav06/sandbox/Click here to continue/Time-Delay Systems/Stability Condition for Discrete-Time TDS

This page describes an LMI for stability analysis of a discrete-time system with a time-varying delay. In particular, a delay-dependent condition is provided to test asymptotic stability of a discrete-delay system through feasibility of an LMI. The system under consideration pertains a single discrete delay, with the extent of the delay at any time bounded by some known value. Solving the LMI for different values of this bound, a limit on the delay can be attained for which the system remains asymptotically stable.

The System
The system under consideration is one of the form:

\begin{align}

x(k+1) &= Ax(k) + A_1x(k-\tau_k) & k&\in\mathbb{Z}_+, & \tau_k &\in\mathbb{N}, & 0&\leq\tau_k \leq h

\end{align} $$

In this description, $$ A $$ and $$ A_1 $$ are matrices in $$ \mathbb{R}^{n\times n} $$. The variable $$ \tau_k $$ denotes a delay in the state at discrete time $$ k $$, assuming a value no greater than some $$ h\in\mathbb{Z}_+ $$.

The Data
To determine stability of the system, the following parameters must be known:

$$ \begin{align} A&\in\mathbb{R}^{n\times n} \\ A_1&\in\mathbb{R}^{n\times n} \\ h&\in\mathbb{Z}_+ \end{align} $$

The Optimization Problem
Based on the provided data, asymptotic stability can be determined by testing feasibility of the following LMI:

The LMI: Asymptotic Stability for Discrete-Time TDS

\begin{align} &\text{Find}:\\ &\qquad P, S, R, S_{12}, P_2, P_3 \in\mathbb{R}^{n\times n}\\ &\text{such that:}\\ &\qquad P>0,\quad S>0,\quad R>0,  \\ &\qquad \begin{bmatrix} \Phi_{11} & \Phi_{12} & S_{12} & R-S_{12}+P_2^T A_1 \\ * & \Phi_{22} & 0 & P_3^T A_1\\ * & * & -(S+R) & R-S_{12}^T \\ * & * & * & -2R + S_{12} + S_{12}^T \end{bmatrix}<0 \\ &\text{where:}\\ &\qquad \Phi_{11} = (A^T-I)P_2 + P_2^T(A-I) + S-R\\ &\qquad \Phi_{12} = P-P_2^T+(A^T-I)P_3 \\ &\qquad \Phi_{22} = -P_3-P_3^T + P+h^2 R \end{align}

$$

In this notation, the symbols $$ * $$ are used to indicate appropriate matrices to assure the overall matrix is symmetric.

Conclusion:
If the presented LMI is feasible, the system will be asymptotically stable for any sequence $$ \tau_k $$ of delays within the interval $$ [0,h] $$. That is, independent of the values of the delays $$ \tau_k\in[0,h] $$ at any time:

 * For any real number $$ \epsilon>0 $$, there exists a real number $$ \delta>0 $$ such that:

 * $$ \|x(0)\|<\delta \quad \Rightarrow \quad \|x(k)\|<\epsilon \qquad \forall k\in\mathbb{N} $$

 * $$ \lim_{k\rightarrow\infty} x(k)=0 $$

Obtaining a feasible point for the LMI, this result can be proven using a Lyapunov-Krasovkii functional:

\begin{align} &V(x_k)=V_P(k)+V_S(k)+V_R(k)\\ \end{align} $$

where:

\begin{align} &V_P(k)=x^T(k)Px(k),  \\ &V_S(k)=\sum_{j=k-h}^{k-1} x^T(j)Sx(j), \\ &V_R(k)=h\sum_{m=-h}^{-1}\sum_{j=k+m}^{k-1}[x(j+1)-x(j)]^T R[x(j+1)-x(j)] \\ \end{align}

$$

Implementation
An example of the implementation of this LMI in Matlab is provided on the following site:

 * https://github.com/djagt/LMI_Codes/blob/main/D_TDS_DC.m

Note that this implementation requires packages for YALMIP with solver mosek, though a different solver can also be implemented.

Related LMIs

 * TDSDC – Delay-dependent stability LMI for continuous-time TDS

 * LMI for Stability of Retarded Differential Equation with Slowly-Varying Delay – Delay-independent stability LMI for continuous-time TDS

 * Discrete Time Lyapunov Stability – Stability LMI for non-delayed discrete-time system


text:
User:Margav06/sandbox/Click here to continue/Preface

Preface

This book will discuss Linear Matrix Inequalities in Control Systems. The recent introduction of Linear Matrix Inequality (LMI) methods in control has dramatically expanded the types and complexity of the systems we can control.

For example, consider the problems: Gain Scheduling for Missile Attitude Control (A switched system); Control of Robots over Noisy Communication Channels (sampled-data systems); Remote Control of Spacecraft Attitude (a delayed system); Behavioral Therapy (A system with binary inputs); or self-driving vehicles (a case of decentralized control). None of these systems can be studied using classical root-locus or PID control methods. Rather, advances in these fields have been made possible through the increased power and flexibility created by the LMI (optimization-based) approach to control.

The objective of this book is to let readers be able to use LMI solvers to synthesize optimal or suboptimal controllers and estimators for multiple classes of state-space systems. This book will require prior knowledge of linear algebra, integral and differential calculus, and at least some exposure to ordinary differential equations. Some background in control systems is also preferrable. In addition, a prior knowledge of integral transforms, specifically the Laplace transforms will be very beneficial. Access to MATLAB is also required, including the robust control toolbox. A guide on the installation and usage of software used in LMI and Optimization is given in this book. Use the links below for the reference on Control systems and transformation of Matrices.


text:
Hacking/Attack/Penetration

Examples

 * to impose usage of weaker protection. For example attack.
 * vulnerabilities
 * Crack
 * Cracking of wireless networks
 * by theft of a
 * s:, ,
 * to crack
 * s:, ,
 * to crack

Stages of related to  penetration stage:

 * 3. Delivery - Techniques resulting in the transmission of a weaponized object to the targeted environment.
 * 4. - Techniques aimed at the manipulation of people to perform unsafe actions.
 * 6. Persistence - Any access, action or change to a system that gives an attacker persistent presence on the system.
 * 7. - Techniques an attacker may specifically use for evading detection or avoiding other defenses.
 * 8. Command & control - Techniques that allow attackers to communicate with controlled systems within a target network.
 * 11. - The result of techniques that provide an attacker with higher permissions on a system or network.
 * 12. Execution - Techniques that result in execution of attacker-controlled code on a local or remote system.
 * 13. Credential access - Techniques resulting in the access of, or control over, system, service or domain credentials.
 * 14. Lateral movement - Techniques that enable an adversary to horizontally access and control other remote systems.

Tools:
 * https://docs.rapid7.com/metasploit/listeners
 * https://www.bettercap.org/modules/ethernet/spoofers/
 * https://www.bettercap.org/modules/ethernet/proxies/
 * bettercap net.fuzz

See also:

title: Hacking/Attack/Penetration


text:
Skaneateles Conservation Area/Invasive species/Juncaceae

Plantae: Tracheophyta: Liliopsida: Poales: Juncaceae

 * }

 * Luzula luzuloides (White Wood-rush) was collected at Hemlock Lake in Ontario County under the name Juncoides campestre in 1915 (GBIF record)

 * Luzula luzuloides (White Woodrush) Research Grade observation at Scarborough in Westchester County in 2021. (iNaturalist observation)


text:
Lentis/Ecological Implications of Commercial Marine Fishing

Overview
Trends in many marine populations today mirror those of previous mass extinction events. While most attention is focused on climate change, ocean warming, and ocean acidification, today, the greatest declines of life in the ocean are from overfishing. It is predicted that there has been an 84% decline in large marine animals from historical baselines, including nearly a decline of 95% in sea turtles, 90% in coastal birds, and 70% in whales. A number of coastal birds, the North Atlantic Right Whale, and all sea turtles species found in US waters are listed under the Endangered Species Act, one of the most popular bipartisan conservation laws. Overfishing and climate change will send our oceans past an ecological tipping point without intervention. This may include species extinctions and irreversible change of benthic (seafloor) habitats that critically support marine biodiversity.

Certain fishing methods, especially bottom trawling, are contributing to marine animal declines and making it harder for populations to recover due to habitat destruction. Recovery of seafloor after trawling disturbances takes years; often, these areas never are able to fully recover before being fished again. The combination of both direct and indirect human impact on marine ecosystems, especially benthic areas, is why management is becoming increasingly pertinent.

Rise of Overfishing
While the rise of overfishing is often attributed to have started around the mid-1900s, some will say that it can go back to the 1800s and possibly beyond that with commercialization attributed as a factor of more fishing over time. Around 1870, 300,000 tons of herring was being caught every year which is about the same as the allowable yearly catch in modern times. In the mid 1900s, more efficient and modern techniques began to develop through the evolution of technology and development of sophisticated methods. Consumers became used to having access to a wide selection of fish products which created a cause for more fishing. Although, since 1989 when 90 million metric tons of catch was taken from the ocean, yields have stagnated as a result of overfishing. According to the United Nations Food and Agriculture Organization (FAO), 34.2% of the fish stocks from the world’s fisheries had become overfished in 2017.

Fishery Subsidies
According the FAO, a fishery subsidies can be defined as "government actions or inactions that are specific to the fisheries industry and that modifies - by increasing or decreasing - the potential profits by the industry in the short-, medium- or long-term". Subsidies related to fishing can date back as far as the 1600s and throughout have been generally used to settle disputes related to fishing territories. In the mid 1900s, the United States government began to help more in expanding the fish market which included allowing foreign access to fish in the waters of the United states in exchange for allowing more foreign market access to the United States. Today, the estimated global costs of fisheries is $15 billion-$35 billion. Rashid Sumaila and Daniel Pauly of the University of British Columbia found that over $150 million are paid to deep sea fisheries that would have losses of $50 million otherwise. Much of this money is used for fuel in order to be able to do bottom trawling, a fishing technique that is very destructive. The dangers of overfishing and threats to sustainability that subsidies provide has been met with protest. 300 scientists around the world have called on the World Trade Organization (WTO) to ban these harmful subsidiaries.

Methods of Commercial Fishing
There are many different methods which commercial fisheries use to catch fish. Different methods are more effective for certain fish species and environments than others. Similarly, certain methods are more conducive to large scale fishing operations while others are more accessible to smaller, individual fisheries.

Bottom Trawl
Bottom Trawl, also called Otter Trawl, consists of a net being weighted so that it falls to the sea bed. That net is then dragged across the bottom to collect the fish in its path. The speed at which the net is pulled prevents the fish which it catches from escaping. The size of the mesh in the net dictates which fish are caught and which can escape. The rationale behind this is that fish below a certain size provide no commercial benefit and thus can be let go. There are many variations of bottom trawling which differ in factors such as the type of net used or the number of boats pulling the net. A notable variation of the bottom trawl is called Pulse or Electrical Trawling. To conduct pulse trawling, an electric shock is sent through the trawl net which shocks the fish out of the seabed making them easier to catch.

Purse Seine
Purse seining is accomplished by locating a school of fish and wrapping a net around the school, preventing escape. The net is then pulled together in the same fashion as a draw-string bag and dragged aboard the fishing boat with the fish inside. The biggest nets can be thousands of meters long, up to 250 meters deep and catch up to 400 tons of fish. Frequently a speed boat is used to wrap the net around schools of fish of this scale.

Gill Net
The Gill Net is set up to effectively be a wall filled with holes. Fish swim into the net and are caught by their gills, preventing escape. Some gill nets are fixed in a particular location, while others, referred to as Drift Nets, move with the tides and currents. The tangle net functions similarly and catches fish which become tangled in the netting. The benefit to fisheries of gill nets and tangle nets is that they require much smaller motor boats, and are a more accessible fishing method for smaller scale or less developed fisheries.

Longline
Longlines are long fishing lines with closely spaced hooks. Long lines can extend up to thirty miles long to maximize the number of fish caught. These lines are baited with live fish, and then reeled in after a day to collect the catch.

Additional Fishing Methods
There are many additional commercial fishing methods which are less notable due to lower rates of utilization or smaller environmental impact. One such method is Dredging which is similar to bottom trawling, with the exception that a metal rake is used rather than a net in order to collect shellfish. Midwater or Pelagic Trawling is similar to bottom trawling, but the net is dragged through water not along the bottom. Methods such as Lobster and Crab Pots are also used to catch said species.

Ecological Implications
Many marine species are facing multiple stressors with cumulative impacts. As direct exploitation of fish populations and bycatch of marine mammals, seabirds, and others continues to increase, the risk of severe biodiversity loss and extinction is increasing.

Benthic Habitat Loss
Benthic habitat loss is perhaps the most pronounced and obvious form of marine habitat modification. Typically, benthic habitats suffer damage from the use of invasive fishing procedures like bottom trawling. In recent years, roughly 1.3% of the ocean's seafloor surface was trawled every year. This equates to an area around half the land area of the United States. Benthic ecosystem disruption and loss of life from commercial fishing methods may not be considered bycatch despite the immediate causal connection and vast damage.

Extinction Risk and Biodiversity Loss
Our oceans are biodiversity hotspots. Biological and ecological processes in the marine environment also support life on land and maintain the biosphere. While the number of terrestrial species considered to be at risk of extinction is much larger than that of marine or anadromous species, declines in marine life are widespread and detrimental. Everything from top-level predators (e.g. cetaceans, sharks/rays, etc.) to the most fundamental component of marine food chains/ecosystems (phytoplankton) have experienced unprecedented declines from anthropogenic stressors.

Bycatch and the consequences of biodiversity loss extends far beyond the submerged world. Seabirds have been known to get entangled in fishing gear - especially longlines.

Greenhouse Gas Emissions
Trawling has the largest associated greenhouse gas emissions of any commercial fishing method. Through the disturbance of the sediments settled on the seafloor, trawling releases massive quantities of naturally sequestered carbon back into the atmosphere, worsening global warming and ocean acidification. The 1.47 billion tonnes of carbon dioxide released from trawling sediment disturbance is greater than the aviation sector, respectively.

Participants
There are numerous participants involved in the acts of commercial fishing, both in regards to supporting and discouraging the acts.

Fisheries / Fish Trade Organizations
Fisheries and fish trade organizations are primarily focused on gaining profits by utilizing effective commercial fishing methods. For example, In the Japanese town of Taiji, dolphin hunting is a common and controversial practice that has been a subject of criticism and debate. The captured dolphins are sold to aquariums or have their meat sold, sometimes fisherman kill dolphins in order to eliminate competition for the fish in the region. This tactic is another byproduct from the effects of overfishing. These fisheries usually use methods like bottom trawling to maximize the catch of fish regardless of its consequences.

Seafood / Aquaculture Companies
Seafood and aquaculture companies typically benefit from the sale of seafood with there being controversy surrounding the means by which they do so. This can be seen through organizations such as the National Oceanic and Atmospheric Association (NOAA) and seafood companies such as StarKist. Products like tuna are sold with "dolphin-safe" checks which are administered by organizations like NOAA that are supposed to assure consumers that dolphins were not harmed in the fishing for the product. However, this can be misleading as while some fisheries are monitored by third parties to assure that they are not harming dolphins, some fisheries can get verified through self declaration. Companies can use unsustainable fishing methods and still be granted a dolphin-safe tag, bycatch of other species are also often not accounted for. Greenpeace, a non-profit activist group, put StarKist in last place in a group of 20 tuna brands that were ranked by how ethical and sustainable they were. StarKist was given this rating due to the lack of transparency of fishing methods, and their association with destructive fisheries.

Governments
Governments can also be involved by allowing for underhanded business practices to happen without consequence. One notorious example of this is the violation of human rights of workers in Thai fisheries. Forced labor is a common occurrence with workers being subject to unfair treatment such as abuse and fraudulent deductions of workers' salaries. In 2015, the Thai government claimed that in an investigation of 474,334 fishery workers, not a single case of forced labor was identified which is contrary to reports made by current and former fisher workers.

Advocacy Groups
There are numerous advocacy groups giving their take on how to tackle the issue of overfishing and its ecological implications. Groups like Oceana are strong on advocating for laws to promote sustainable fishing. The idea of sustainable fishing has also seen criticism due to proposals of the idea lacking solid and constructive plans and for difficult implementations. Other groups such as World Day for the End of Fishing (WoDEF) advocate for the complete halt of fishing and the removal of aquaculture farms. Sea Shepherd, a conservation activism organization, is know for getting directly involved by physically intervening with illegal fishers by taking down ships.

Conclusion
The Commercial Fishing industry provides a good example of the challenges that arise from the presence of conflicting priorities and limitations between different groups. On the one hand there are the commercial fisheries who are trying to turn a profit and supply fish to meet the dietary demands of society. On the other hand are the ecosystems which are damaged and destroyed and the aquatic species which experience irreversible harm from the methods utilized by the fisheries.

Another observation which arises from the issues related to commercial fishing is the impact of technology and its role in society. In this case, as in innumerable others, the technologies in and of themselves are brilliant and achieve their goal of maximizing the number of fish caught. However, there are significant negative side effects of the technology to the environment and fish populations which must be considered.

An area which must be further analyzed is how these fishing methods and the priorities of the commercial fisheries can be balanced with the limitations and needs of the environment and in particular the oceanic ecosystems.


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Lentis/Protection Motivation Theory

Definition
Protection Motivation Theory (PMT) is a social cognition theory which attempts to explain motivation to respond to health threats ranging from COVID-19 to smoking. It is a framework used by researchers to evaluate the likelihood of individuals engaging in a protective behavior when faced with a general threat. For example, PMT seeks to explain what would motivate an individual to partake or refrain from getting a vaccine while in the midst of a global pandemic. PMT states that an individuals protection motivation is a product of two evaluations carried out by the individual: the threat appraisal and the coping appraisal.

Threat Appraisal
The threat appraisal is the evaluation of the degree to which an event has significant implications on an individual's wellbeing. It is composed of three components: the severity of the threat, the vulnerability to the threat, and the rewards of the behavior. The severity of the threat is the degree of harm from the threat. The vulnerability is the likelihood of experiencing harm. The rewards are the possible benefits from partaking in the behavior, and can be both intrinsic and extrinsic. For example, when faced with the threat of COVID-19, the severity of the threat would be the likelihood that contracting the disease would do harm to the individual, the vulnerability to the threat is the likelihood of contracting the disease, and the rewards might be sticking by political beliefs or increased mask sales.

Coping Appraisal
The coping appraisal is the evaluation of the capability to respond to the threat with the recommended preventative behavior. It is composed of three components: response efficacy, self efficacy, and the response costs. Response efficacy is the belief that a certain behavior will reduce the threat. Self efficacy is the belief that the individual has the ability to perform the behavior. Finally, response costs are the perceived costs of partaking in the preventative behavior. . In the COVID-19 example, response efficacy is the belief that that vaccines work, self efficacy is the belief that an individual has the ability to get a vaccine, and the response costs may be potential side effects

Health Sector Applications
PMT is frequently used as a cognitive framework in research studies to understand people's behaviors in response to threats. Researchers have sought to test the predictive value of PMT by comparing a prediction model to observed behavior; they have consequently diagnosed the primary factors (e.g., threat appraisal components and coping appraisal components) determinant of subject behavior. Identifying these factors advises policymaking groups such as government bodies, the CDC, boards of education, or advocacies on how they might influence individuals' behavior in similar situations (see choice architecture.) PMT was first developed and applied "to describe how individuals are motivated to react in a self-protective way towards a perceived health threat," but it is also relevant to any sector involving a perceived threat. For individuals to adapt their behavior they must perceive a high level of threat, vulnerability, behavioral efficacy, and self efficacy as well as a low cost. Representative case studies of PMT as a predictive model for categories of health-related behavior are outline below.

Exercise
Researchers examine "the utility of [PMT] in the prediction of exercise intentions and behavior in the year following hospitalization for coronary artery disease (CAD.) Based upon patient questionnaires measuring PMT's threat and coping appraisal, they find a good fit between structural equation modeling  and empirial exercise behavior data at 6 months post hospitalization, but less so after 12 months. Thus, they recommend "health education...concentrate on providing positive coping messages to enhance patients' confidence regarding exercise and their belief that exercise provides health benefits, as well as realistic information about disease severity."

Drugs and Alcohol
Researchers compare PMT constructs with "intention and tobacco use behavior" among a group of 440 high school boys in Noshahr, Iran. The "validity and reliability" of a questionnaire including "demographic information, the constructs of [PMT], and tobacco use behavior." Based on structural equation modeling they conclude that "PMT provides a useful framework for investigating factors of tobacco use among male students, [and] future tobacco prevention interventions should focus on increasing the vulnerability and fear, decreasing intrinsic reward, and improving self-efficacy to reduce tobacco use."

Sexually Transmitted Infections (STI)
To inform HIV prevention, researchers "explored youth's knowledge, attitudes and behaviors related to sexual risk taking through the lens of [PMT]." Focus groups among reservation-based American Indian youth suggest low perceived self-efficacy of condom use, HIV prevention knowledge, and vulnerability to STIs. They also identify factors such as alcohol, parental monitoring, and peer influence as related to sexual risk taking. Guided by the PMT framework, they submit that "future HIV prevention interventions should be delivered to gender-specific peer groups, include a parental component, teach sexual health education and communication skills, integrate substance-use prevention, and work to remove stigma around obtaining and using condoms."

Disease
Researchers evaluate the predictive value of PMT and the Big Five (B5) model of personality in regard to the SARS-CoV-2 pandemic (COVID-19). They measure "both adaptive (i.e., preventive) and maladaptive (i.e., avoidant behavior and wishful thinking) coping behavior" during the first wave of the pandemic among the general Polish population. They find "strong support for the PMT constructs' predictive value, especially with regard to pandemic-related adaptive behavior" and incremental validity of the PMD model over demographic variables and personality traits. Thus "to increase adherence to preventive measures during an epidemic, all the PMT constructs should be considered while persuasive communication to the public is formulated."

Other Sectors Applications
While PMT gained legitimacy in the health sector, it has since expanded into research in various other fields. In all sectors, researchers are using PMT to find connections between threats and public response, aimed at understanding this relationship to alter public response in the future. The same concepts of threat severity, vulnerability, and coping efficacy that were originally applied to medical decisions are being applied to understand the decisions and responses the public has to threats from all directions.

Cybersecurity
Researchers are evaluating if messages sent to subjects can generate a response that strengthens their cybersecurity habits while online. A recent study sent either threatening or coping messages to participants and evaluated participants responses through their online behaviors. Using the results, the researchers were able to draw conclusions on what kind of message generates the greatest response and can apply this information to nudge users towards behavior that will prevent theft, fraud, etc.

Natural Disasters
In the field of natural disaster preparedness, a large gap exists between those who are aware of an impending natural disaster and those who are prepared for it. Similar to the cybersecurity studies, researchers are using PMT to evaluate public service messaging to find what kind of message resonates the most with individuals. Further understanding of how to use PMT with messaging aims to achieve "safer processes and response behavior," when the population is faced with an impending disaster to ultimately protect human and animal life.

Climate Change
Researchers are also examining in the fields of climate change seeking fossil fuel reduction are using PMT to find any correlation between an intent to reduce fossil fuel use and PMT concepts such as severity of the threat and self efficacy. A study asking participants to rank their agreement with PMT constructs as the relate to climate change found that those who perceive climate change as a greater threat and that they had the ability to cope with this threat were most likely to change their fossil fuel habits.

Key Takeaways

 * Broader Application - Predictive models like PMT can be used to predict behavior and influence policy decision making (i.e. choice architecture), in any field. Several studies across various fields have concluded that PMT constructs can act as indicators of individuals' intentions to refrain or partake in a given behavior.
 * Connecting Sosh and Tech - Technical innovations require social strategies (like PMT) for meaningful implementation. The understanding of PMT constructs can act as a bridge between advancing technology and implementation by the public.

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Lentis/Antipiracy

Introduction
Antipiracy refers to the movement against the use and distribution of works protected by copyright without the permission of its copyright holder. Widespread access to file sharing on the Internet has made pirating rampant. An estimated 24% of all global bandwidth is used for digital piracy. Therefore, a significant amount of development for online content distribution is put towards preventing piracy. Countering piracy takes a multitude of forms. Some entities go after major distributors of pirated content. Others attempt to make the media is only accessible if the user is proven to be legitimate.

Piracy
Digital piracy is any form of illegal consumption of digital products such as software, entertainment, or other forms of copyrighted material. Piracy is extremely common, with an estimated 37% of all software on personal devices being pirated globally.

Motivations
As the internet grows, piracy too becomes more common. Theft is widely accepted as legally and mostly morally wrong, though piracy becomes a gray area. Many believe that digital piracy is more justifiable than other forms of theft due to a variety of factors, including physicality and direct observable loss. As digital piracy lacks an observable physical effect, people may be more inclined to ignore morality concerns.

People primarily pirate to save money. Piracy is perceived to be cost-saving, time-saving, and convenient. Many consumers would prefer to spend no money rather than purchasing a digital good, especially when those digital products come from larger corporations. Piracy is significantly more common in countries with lower economies. It’s estimated that over 80% of software was consistently duplicated in Indonesia from 2011 to 2017.

The internet grows and changes faster than laws can keep up, so consumers frequently are unaware of the legal repercussions from their actions. A study shows that people are less likely to continue pirating when they are educated on the consequences of their behavior. This study further suggests that copyright owners put forth an effort to educate consumers themselves on the legalities of piracy to decrease the theft of their material.

Torrents
Torrents are the most infamous approach to pirating media. Torrents leverage peer-to-peer networking for its efficient file sharing capabilities. Any downloader can simultaneously be an uploader, uploading downloaded portions of an ongoing download. This improves download speeds as multiple computers work together to download content. Public torrents have large catalogs of content for anyone to download, including music, movies, books, games, TV shows, and software. Notable public torrents are Pirate Bay, uTorrent, and BitTorrent. The major risk of public torrents is that they are ridden with viruses and malware. Private (invite-only) torrent sites, such as Torrent Day or PassThePopcorn, offer the safest torrenting experience by having far less viruses and malware. They too have the best catalogs, including popular or recently released movies and TV shows. Due to limited users on private torrents, they enforce a "ratio" to ensure rich catalogs. For each user, this number is the ratio between the amount of content uploaded versus the amount of content downloaded. An ideal ratio is 1.0 or higher, indicating the user gives back at least as much bandwidth as they use. If a user's ratio remains below a certain threshold for too long, the site will remove the user's registration.

Sites
Illegal sites are an option for pirated access to movies and TV shows. 123movies, Putlocker, Primewire, FMovies, WatchFree, and Vumoo are examples. Streaming sites like Nbabite, Nflbite, Streameast, Methstreams, and others provide pirated access to live sports. These sites are obvious alternatives to paying for subscriptions like NBA League Pass or Netflix. Like torrents, these sites pose security risks as malware and viruses are often embedded in the many advertisements appear. Ad-blockers are useful tools to use in conjunction with these sites.

Other Tools
Bypassers are browser extensions that bypass paywalls that prevent access to articles. Once downloaded, they can be "loaded as unpacked" in the developer mode of a browser to be used. They are commonly used for viewing news articles without a subscription. Some supported websites include The Atlantic, The New York Times, The Washington Post, Time Magazine, Bloomberg, Business Insider, Forbes, Discover Magazine, and many more. Media centers like Plex and Jellyfin are services that enable content to be hosted in a netflix-like interface. Some media centers, such as Plex, have robust APIs, enabling users to curate their libraries automatically with scripts. Media centers are not tools intended for piracy, but users may fill their libraries with pirated content for convenient viewing and sharing. Media managers like Sonarr and Radarr track and download shows and movies from torrent sites once available. File sharing tools like SyncThing and File.pizza allow for convenient one-off file transfers between computers.

Copyright Owners
Monetary damage is argued to be the most harmful effect of digital piracy, with an estimated loss of billions of dollars a year. In the film industry alone, monetary loss due to digital piracy is estimated to hit $52 billion. It can be discouraging for creators to continue making digital content when they assume it’ll just be pirated, thus halting innovation. Additionally, pirating affects smaller creators more heavily, who rely on the sale of their products to continue their work.

Digital Pirates
As money is at the root of piracy, most sites that host pirated content for “free” will either have ads to generate revenue, or have an ulterior motive with hidden malware in downloaded content. At least half of all pirated content is infected with malware. This is especially dangerous for those who are not technologically skilled and unable to rid their devices of the malware. Even if people are able to rid the viruses from their devices, it still affects the health of their device long-term.

Torrenting specifically introduces a more personal effect for both consumers and distributors, since IP addresses are shared. This can be tracked by law enforcement and lead to legal repercussions or by hackers and open up the device to attacks. Further, internet service providers that notice torrent activity may send warning letters or suspend the service.

Alliance for Creativity and Entertainment
One group advocating for antipiracy is the Alliance for Creativity and Entertainment (ACE), founded in June 2017 by multiple entertainment and streaming companies, most notably HBO, Netflix, and Twentieth Century Fox. The group claims to support a "global ecosystem of artists, creators, and content proviers". Additionally, they claim to partner with law enforcement agencies and governments around the world to find illegal streaming websites and shut them down. In August 2021, they shut down a popular illegal streaming website called Flixanity, and in May 2021, they shut down the website 123movies.la, a Tunisian illegal streaming website that received an average of 30 million visits. Similar shutdowns are documented on ACE's website, occurring every few months.

Copyright Alliance
The Copyright Alliance advocates for antipiracy and claims to represent the interests of "authors, photographers, performers, artists, software developers, musicians, journalists, directors, songwriters, game designers, and many other individual creators", as well as support policies that promote copyright protection. Their recent news releases frequently issue statements on antipiracy laws passed by Congress and support court cases where copyright law was upheld, including participating in hearings such as the DMCA hearing in December 2020.

Governments
The federal government of the United States has piracy laws structured in such a way that the focus is much more on the copyright holder’s rights rather than the acts of digital piracy itself. In Article 1 Section 8 of the US Constitution, the founders laid out the power for the federal government “To promote the Progress of Science and useful Arts, by securing for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries”. Although this does protect the rights of intellectual property holders, it does not explicitly speak on what the government will do to protect those rights. The Strengthening Measures to Advance Rights Technologies (SMART) Copyright Act of 2022 is a bill which as of December 2022 is in the Senate Judiciary Committee. Proposed by Senator Thomas Tillis, it states, “A covered service provider shall use commercially reasonable efforts to accommodate and not interfere with designated technical measures that apply to that covered service provider”.

The European Union enacted the Digital Services Act in November of 2022, which functions as a list of new standards for how the European Commission believes the internet should function. In terms of illegal goods and services, which would include pirated content, it “imposes new mechanisms allowing users to flag illegal content online, and for platforms to cooperate with specialised ‘trusted flaggers’ to identify and remove illegal content”. It also imposes new rules regarding the tracing of sellers and their goods in order to find distributors of illegal goods more easily.

Nintendo
The video game company Nintendo has a long history fighting with digital pirates over their content. They now have antipiracy programs in 40 countries dedicated to securing their intellectual property. The organization works with governments and “aggressively pursues legal actions worldwide against those involved in Nintendo piracy”. These programs also work toward consumer awareness by conducting campaigns to notify the consumer about the illegal practices of piracy. Another facet of these programs is to lobby governments in their respective regions for stronger policy against digital piracy. One section states that they also work towards educating law enforcement in order to better protect their copyright. Although much of the information is reliable and true, some of the claims made in their frequently asked questions section have controversial legal standing. In one such case Nintendo claims that there is misinformation being spread online over the legality of making copies of games one already owns.

Defective by Design
Defective by Design is an initiative formed on May 24, 2006 by the Free Software Foundation that oppose the use Digital Rights Management (DRM) technology in the sale and distribution of digital media. Digital Rights Management (DRM) is a blanket term for technologies designed to enforce copyright protection on digital media and software. These technologies make it so that a given piece of software can't run properly without authentication from the copyright holder. Common forms of DRM include proprietary file formats, product keys and online authentication. The philosophy behind the opposition is that DRM, by nature, "creates a damaged good" and "prevents [the user] from doing what would be possible without it" In effect, DRM only serves give companies the power to control the media that their customers consume.

Low Income Demographics
Industry standard software, such as Adobe Photoshop or Microsoft Office, is often expensive. This creates a monetary barrier to entry for those entering these industries. Piracy grants a way around this barrier. Some software developers recognize this and adapted their model accordingly. Unity, a popular game engine, offers a Personal Plan that allows free use of the software as long as the revenue received from the game remains below a certain threshold. Shortly after adoption, the Unreal Engine, a major competitor, began to mimic that model.

Good Old Games (GOG)
GOG.com is a video game and film distribution platform that bans the use of DRM in what they sell. Based out of Poland, it was founded in a place where video game piracy was rampant. In order to combat this, they found ways to incentivize users to buy the game instead of pirating. Methods in the past included localizing the game for foreign audiences and updating the game for newer hardware.

Password Sharing
Data surveys show that upwards of 49% of people have admitted to password sharing in order to access paid content with percentages higher in younger demographics. In hopes to lower these large numbers, streaming services have started campaigns to remind users that password sharing is morally wrong and illegal. Additionally, measures such as two-factor authentication are now being implemented in testing phases by companies such as Netflix as a deterrent to make password sharing an extra step more cumbersome. The downside is that adding this extra step makes the service less satisfying for non-pirating users.

Watermarking
Watermarking is a technique used to discourage the use of proprietary property by allowing functionality without the ability to remove a quality final product. Watermarking is a new take on access control allowing potential consumers to become acquainted with a product before having to make a financial purchase. This technique is used heavily by software products and online media vendor sites.

The technique also serves as piracy detection known as Coded Anti-piracy. Modern technology can encode data into film or streaming content giving information on where the media is being presented without being obviously visible to the viewer or pirate consuming the content. This information could be the ID or location of a movie theatre showing a film or operating system of a device playing stolen content. The pirated content which can be easily analyzed by the producer who has the decoding filter to view the embedded watermarked data and form conclusions about where the content is being stolen from.

Pay-what's-fair Model
The pay-what's-fair model is a method which independent creators, including the rock band Radiohead, have used to limit piracy of their content. Instead of charging a fixed price to access the content, the creator will implement the model, which allows each consumer to pay however much they think the content is worth. In the case of Radiohead, the English rock band's album In Rainbows was released using the pay-what's-fair model in 2007. Radiohead made $3 million off of this model, with the average user paying $6 to buy the album. Additionally, their lead vocalist Thom Yorke claimed that Radiohead "made more money out of this record [In Rainbows] than out of all the other Radiohead albums put together”, which demonstrates the success that this model can have.

Analog Hole
The Analog Hole is a concept often used to posit that digital piracy is an inevitability. It describes the idea that no matter how you protect something from piracy, there will always be a "hole" in the form of analog recording. In other words, as long as a human can consume a piece of media, there will always exist a simple way to reproduce it. An example of this concept would be bootleg recordings of films and TV shows. As long as people can watch movies or TV shows, there can be little stopping someone from recording it and uploading it elsewhere.

Negative Performance Impact
DRM comes with additional processing on top of the software it is protecting. Sometimes, this can make the software perform significantly worse. Denuvo is a DRM software that is popular among AAA video game studios. The software itself is often criticized for being cumbersome and unreliable. In some cases, it made it impossible for certain users to access the games they own.

Reliability and Abuse
For large-scale content distribution platforms, such as YouTube, copyright protection systems are largely automated. It is not uncommon for these system to be abused or falsely flag legitimate users as pirates. UMG has garnered a reputation for abuse on multiple platforms for making false claims of copyright infringement. YouTube's systems has been criticized for its automatic system for identifying pirated content. Due to the nature of these systems, a creator's content can be taken down or completely demonetized with little ability for them to appeal these false claims.

Motives and Choice Architecture
The primary motive for pirating content is dominated by money. People do not want to pay for highly-priced subscription streaming services that often are not affordable in order to watch one specific source of media. However, it is important to note that in many cases a large number of these pirates are actual willing to pay for either legal or pirated content. These users are simply motivated by finding access to large selections of reasonably priced quality content.

Honor alone does not appear to be a strong enough factor to prevent pirate activity. Research indicates that up to 75% of pirates will admit that what they are doing is wrong. Consumers of pirated material are continuing to receive access to desired media for cheaper prices and are not facing consequences. Current crackdown on illegal pirate activity is mainly focusing on stopping the providers distributing the pirated material rather than the consumers. A bill titled the Protecting Lawful Streaming Act included as part of the Covid-19 Stimulus act now has illegal streamers facing heavy fines and jail time upwards of 10 years to create a larger deterrent. One programmer, Darryl Julius Polo, was sentenced to 57 months of jail and forced to give up $1 million for helping and starting streaming services Jetflicks and iStreamitAll which reportedly held databases individually larger than Netflix and Hulu.

Conclusion
The success of the antipiracy movement will be mostly dependent on the quality of the choice architecture presented to consumers by companies and the artists they represent. Consumers will use the content that provides the most value to them with quality and price holding high priority and placing lesser regard on whether their action is legal or honorable. Users are currently not being held accountable for their actions and are not facing legal repercussions. Information is very widespread and hard to lock down and legal processes take time and money so piracy is a hard problem to stop. Technological methods can be used to make pirating harder and are worth improving, but companies should shed more focus on balancing the benefits of boosting profits while providing consumers reasonable opportunity and motive to not pirate content. Additional research into the antipiracy movement could further explore the psychology behind piracy or more details on economic harm.

title: Lentis/Antipiracy


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Lentis/The School to Prison Pipeline

The "School to Prison Pipeline" is a concept originating in the United States theorizing that the discrepancy in the incarceration rates of disadvantaged youth is due to national policies as well as practices within schools. This chapter explores the history of the School to Prison Pipeline as well as how modern advancements in technology have affected the concept.

What is the School to Prison Pipeline?
The School to Prison Pipeline refers to disciplinary policies and practices that disproportionately impact students from disadvantaged backgrounds. These policies tend to increase the likelihood of these students suffering academically, or becoming incarcerated later in life. Practices such as zero tolerance policies and placement of police officers in schools continue this pattern. In turn, this results in early exposure to law enforcement for many students, which can cause lasting negative effects.

Origins
The term “School to Prison Pipeline” was coined during President Nixon’s “War on Drugs” during the late 1970’s. The United States began making stricter drug laws resulting in a nationwide increase in incarcerations. In turn, schools began cracking down harder on minor rule violations via suspensions. Forcing students out of the classroom as punishment increases the likelihood of them ending up in the criminal justice system. This is amplified in underfunded school districts; where there is a lack of qualified teachers and the ability to properly discipline students. In 1994, the Gun-Free Schools Act was implemented nationally. It was the first national Zero Tolerance policy. Under the law, bringing a gun onto school property would result in a mandatory year-long suspension. With the first Zero Tolerance policy in place, the School to Prison Pipeline became a prominent national topic and focus for a variety of social groups across the country. These policies are still being used in modern-day schools and state districts as well as nationwide.

Overview
Many current policies within the United States have an effect on the performance of disadvantaged youth and children of color in schools. Two of the most influential policies in effect today are Zero Tolerance policies within schools as well as the implementation of School Resource Officers (SROs) in both primary and secondary schools. Both of these policies have solidified the role of institutional racism within education, perpetuating a cycle that has a net-negative effect on the ability of minority groups to pursue an education as stated by the National Education Association (NEA).

Zero Tolerance Policy
The NEA recently issued a policy stating that “Zero Tolerance policies mean school disciplinary policies that set predetermined consequences or punishments for specific offenses or rule infractions. Zero Tolerance policies forbid persons in positions of authority from exercising discretion or changing punishments to fit individual circumstances”. The use of Zero Tolerance policies within schools is founded on the principle that a strong law enforcement approach to minor crime will prevent more serious crime from occurring. With punishments generally resulting in suspension/expulsion no matter the infraction, students are often alienated from their peers and driven towards criminal activity in their time outside of structured learning environments.

Law Enforcement in Schools
School Resource Officers were originally added to schools in order to combat school shootings by having a law-enforcement presence within the school. However, this idea was poorly thought out as a large majority of school shooters are not unknown assailants and instead are a student of the target school. Not only are SROs ineffective for their intended purpose, but their presence is also very harmful to the students they “protect”. Students learn from an early age to fear the presence of law enforcement when they should have very little interaction with any part of the criminal system in their most formative years. Law enforcement in the United States especially has a well-documented history of incorrectly racially profiling Black and Latino students, a trend that has been continued by SROs.

Effectiveness of Discipline Policies
There is much debate as to whether these policies work or not. For example, the reasoning for the increased implementation of School Resource Officers in schools was to prevent school shootings and violent crimes. However, when comparing schools with and without School Resource Officers, there is little disparity when it comes to violent or drug-related incidents occurring. The only notable difference is a near four times increase in disorderly conduct arrests in schools with SROs. Also, since Nixon’s “War on Drugs”, out-of-school suspensions have more than doubled. Per research conducted by Shoshana Jarvis and Jason Okonofua of The University of California, “African-American students are nearly four times as likely to be suspended from school as Caucasians.” Studies have shown that not only does missing time in school negatively impact academic performance, but also increases the likelihood of the student going to jail in the future.

Influence of Technology
The technological support disparity has drastically increased during the COVID-19 pandemic and had even shown worrying effects in the years prior. Technological support disparity is the idea that with the recent focus to move a large portion of the average students learning online, many students a part of low-income districts or those who are a part of low-income families are being left behind their more privileged peers. Studies have shown that educational achievement differences are more closely aligned with family income than any other demographic factor, highlighting the importance of a child’s family income in their ability to participate in school and avoid the school-to-prison pipeline. An article from CNN stated that 70% of America’s teachers assigned homework to be completed online with 17% of teenage students being unable to complete assignments due to a lack of technological resources before the pandemic.

The COVID-19 pandemic predictably increased the amount of work assigned to students online, but it also perpetuated the school-to-prison pipeline in less obvious ways. With the introduction of zoom classes as the norm for education during the pandemic, teachers and SROs were given a direct view into a student’s life at home during class time whether or not the student consented. This breach of privacy created the opportunity for school authorities to police the events of their students’ households, often incorrectly leading to visits from Child Protective Services (CPS) and disrupting a child’s life outside of school. Zoom classes are also much more difficult to attend for students without WiFi access at home which results in attendance punishments disproportionately being given to low-income students.

A modern look at the effect of technology on the school to prison pipeline can be seen in the implementation of facial recognition software within schools. This surprising and unnecessary security feature has seen a surprising amount of support from technology security companies in recent years with companies such as RealNetworks even offering to upgrade their current security systems to include new facial recognition software for free for K-12 institutions. This software tracks a student’s biometric data for 7+ hours they attend school, a massive invasion of their privacy given that this data is often stored and could be vulnerable to leaks or hacks. Another concern with this technology is how ineffective it can be. Even ignoring instances when students may try and purposely trick the software, children are constantly growing and having their features change through puberty which would affect the accuracy of the facial recognition software. Along with student’s features changing, the databases of this software often come from police departments which are overpopulated with people of color. This discrepancy can lead to the students of color being biasedly targeted and discriminated against, furthering the disproportionate amount of minorities involved in the school to prison pipeline. With the consequences of a false match being possibly traumatic for an innocent student being falsely accused, it seems that the risks of implementing AI-based recognition software far outweigh the benefits.

Alternative Solutions
The institution of the “School to Prison Pipeline” has created a culture in which the youth of minority communities are expected to fail in the education system and be raised through the penal system. Alternative efforts must be pushed so perceptions around youth in minority communities regarding education and their future can change. Banning the use of suspensions and expulsions would minimize the number of time students have in encountering trouble in their communities or being exposed to bad influences. Keeping students in school would give faculty the opportunity to pinpoint issues in the class and work on creating a space where the student is able to effectively learn in a classroom setting. If schools decide to ban suspensions/expulsions, they must supplement this change with increased funding to pay for well-educated teachers and guidance counselors that are willing to spend extra time with students to help them succeed. The expectation of teachers, guidance counselors, and faculty is to foster a strong connection with students so students gain trust and believe faculty are willing to help them graduate with a high school diploma. One step in creating a strong connection between teachers and students is to implement restorative practices. According to McClure, restorative practices ”can dramatically improve the school climate and strengthen the social and emotional skills of young people and adults”. Introducing circles into classrooms would push for students to be vulnerable and explain how they feel in a positive way such that they are able to express themselves. This not only serves as a tool for teachers to guide students in any trouble they are having but also gives students autonomy over their feelings and is a way for students to take control of their education. With faculty spending more time on restorative practices and creating stronger relationships with students, there should be less of a need for school resource officers to be stationed in schools. According to Kendrick Washington, “the arrest rates for schools with SROs were 3.5 times the rate of those without SROs, and in some states, the arrest rates are as much as eight times the rate of schools without”. The concentration of these arrests is based on non-serious incidents such as tardiness, bad grades, and disorderly conduct like cursing. This leads to blurred lines between students learning how to behave in a school setting and society branding them as delinquents. School resources officers expose youth to the penal system earlier than schools without them and don’t give students the opportunity to live and learn.

title: Lentis/The School to Prison Pipeline


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User:Margav06/sandbox/Click here to continue/Applications of Non-Linear Systems

Applications of Non Linear systems

 * 1) /Lyapunov Stability of a System With Polynomial Dynamics/
 * 2) /Global Minimum of Polynomial via SOS Method/
 * 3) /Local Minimum of Polynomial via SOS Method/
 * 4) /Global Lyapunov Function Search for System with Polynomial Dynamics/