text:
LMIs in Control/Algebraic Riccati Inequality

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

The System

\begin{align} \dot x(t)&=Ax(t)+Bu(t)\\ \end{align}$$

The Data
Following matrices are needed as Inputs:.

 * $$ A,B,N$$.

The Optimization Problem
In control systems theory, many analysis and design problems are closely related to Riccati algebraic equations or inequalities. Find

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

\begin{align} P>0, \\Q>0, \\ R>0.\\ \text{ The Algebraic Riccati inequality is given by}\;\\ A^T P+PA - (PB+N^T)R^{-1}(B^TP+N)+Q & \geq 0\\ \\ \text{ can be written using the Schur complement lemma as}\;\\ \begin{bmatrix} A^TP+PA+Q && PB+N^T\\ \star && R\\ \end{bmatrix}& \geq 0 \end{align}$

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

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

 * 1) REDIRECT []- CODE

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


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LMIs in Control/pages/systemzeros through feedthrough

Let's say we have a transfer function defined as a ratio of two polynomials: $$ \begin{align} H(s) = \frac{N(s)}{D(s)} \\ \end{align}$$ Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting $$ N(s) = 0 $$ and solving for s.The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Similarly, the system zeros are either real or appear in complex conjugate pairs. In the case of system zeros without feedthrough, we take the assumption that $$ D=0 $$.

The System
Consider a continuous-time LTI system, $$ G $$, with minimal statespace representation $$ (A,B,C,0) $$

\begin{align} \dot x(t) = Ax(t) + Bu(t)\\ y(t) = Cx(t) \end{align} $$

The Data
The matrices:

\begin{align} A \in \mathbb{R}^{n \times n} \\ M \in \mathbb{R}^{n \times q} \\ N \in \mathbb{R}^{q \times n} \\ \end{align}$$

The LMI: System Zeros without feedthrough
The transmission zeros of $$ G(s) = C(sI-A)^-1 B $$ are the eigenvalues of $$ NAM $$, where $$ N \in \mathbb{R}^{q \times n}, M \in \mathbb{R}^{n \times q}, CM = 0, NB = 0, NM = 1. $$. Therefore, $$ G(s) $$ is a minimum phase if and only if there exists $$ P \in \mathbb{S}^{q} $$, where $$ P>0 $$ such that

\begin{align} PNAM + M^T A^T N^T P <0 \end{align}$$

Conclusion:
If P exists, it ensures non-minimum phase. Eigenvalues of NAM then gives the zeros of the system.

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


text:
LMIs in Control/pages/systemzeroswithout feedthrough

Let's say we have a transfer function defined as a ratio of two polynomials: $$ \begin{align} H(s) = \frac{N(s)}{D(s)} \\ \end{align}$$ Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting $$ N(s) = 0 $$ and solving for s.The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Similarly, the system zeros are either real or appear in complex conjugate pairs. In the case of system zeros without feedthrough, we take the assumption that $$ D=0 $$.

The System
Consider a continuous-time LTI system, $$ G $$, with minimal statespace representation $$ (A,B,C,0) $$

\begin{align} \dot x(t) = Ax(t) + Bu(t)\\ y(t) = Cx(t) \end{align} $$

The Data
The matrices:

\begin{align} A \in \mathbb{R}^{n \times n} \\ M \in \mathbb{R}^{n \times q} \\ N \in \mathbb{R}^{q \times n} \\ \end{align}$$

The LMI: System Zeros without feedthrough
The transmission zeros of $$ G(s) = C(sI-A)^-1 B $$ are the eigenvalues of $$ NAM $$, where $$ N \in \mathbb{R}^{q \times n}, M \in \mathbb{R}^{n \times q}, CM = 0, NB = 0, NM = 1. $$. Therefore, $$ G(s) $$ is a minimum phase if and only if there exists $$ P \in \mathbb{S}^{q} $$, where $$ P>0 $$ such that

\begin{align} PNAM + M^T A^T N^T P <0 \end{align}$$

Conclusion:
If P exists, it ensures non-minimum phase. Eigenvalues of NAM then gives the zeros of the system.

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


text:
LMIs in Control/page/systemzeros with feedthrough

Let's say we have a transfer function defined as a ratio of two polynomials: $$ \begin{align} H(s) = \frac{N(s)}{D(s)} \\ \end{align}$$ Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting $$ N(s) = 0 $$ and solving for s.The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Similarly, the system zeros are either real or appear in complex conjugate pairs. In the case of system zeros with feedthrough, we take $$ D $$ as full rank.

The System
Consider a continuous-time LTI system, $$ G $$, with minimal statespace representation $$ (A,B,C,D) $$

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

The Data
The matrices needed as inputs are:

\begin{align} A \in \mathbb{R}^{n \times n} \\ B \in \mathbb{R}^{n \times m} \\ N \in \mathbb{R}^{p \times n} \\ \end{align}$$ In this case, $$ m \leq p $$

The LMI: System Zeros with feedthrough
The transmission zeros of $$ G(s) = C(sI-A)^-1 B + D $$ are the eigenvalues of $$ A-B(D^T D)^{-1} D^T C $$. Therefore, $$ G(s) $$ is a minimum phase if and only if there exists $$ P \in \mathbb{S}^{q} $$, where $$ P>0 $$ such that

\begin{align} P(A-B(D^T D)^{-1} D^T C) + (A-B(D^T D)^{-1} D^T C)^T P < 0 \\ \end{align}$$

Conclusion:
If P exists, it ensures non-minimum phase. Eigenvalues of $$ A-B(D^T D)^{-1} D^T C $$ then gives the zeros of the system.

Related LMIs
LMIs_in_Controls/pages/systemzeroswithoutfeedthrough

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


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LMIs in Control/pages/H2 Strip Region

Insensitive Strip Region Design with Minimum $$ H_2 $$ Gain

When designing controllers with insensitive region conditions, the aim is to place the closed-loop poles of the system in a particular region defined by its inner boundary. These regions are specified based on their insensitivity to perturbations to the system parameter matrices.

One type of such design is the Insensitive Strip Region Design. In this section, building upon that, optimization problems will be provided that ensure that the conditions for insensitive strip region design are satisfied with some bounds on the $$ H_2 $$ gain of the closed-loop system.

The System
A state-space representation of a linear system as given below:

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

where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$ and $$u(t)\in \R^r$$ are the system state, output, and the input vector respectively.

The Data
To solve the design optimization problem, the linear system matrices A,B,C are required. Furthermore, to define the strip region on the eigenvalue-space, two parameters $$ \gamma_1 $$ and $$ \gamma_2 $$ are required.

The Optimization Problem
The problem of designing an $$ H_2 $$ optimal controller that results in the closed loop system insensitive to a certain strip region involves two sub-problems:

 * Finding a control gain $$ K $$ such that: $$ \left\| K \right\|_2 < \gamma $$.
 * The conditions for insensitive strip region design for the closed-loop system, as provided in the section Insensitive Strip Region Design are fulfilled.
 * The optimization goal is to minimize $$ \gamma $$ such that above two hold.

The LMI: $$ H_2 $$ Optimal Control Design for Insensitive Strip Region
The problem above has a solution if and only if the following optimization problem has a solution $$ (K, \gamma) $$:

$$ \begin{align} \text{min   } & \gamma \\ \text{s.t.  } & \begin{bmatrix} - \gamma I & K \\ K^\top & - \gamma I \end{bmatrix} < 0 \\ & 2 \gamma_1 I < (A+BKC)^\top + (A+BKC) < 2 \gamma_2 I \end{align} $$

Conclusion:
By using the design problem provided here, an optimal $$ H_2 $$ controller is designed to make the closed-loop system robust to perturbations in the system matrices.

Implementation
To solve the optimization problem with LMI presented here, 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/H2_Strip_example.m

Related LMIs
Insensitive Strip Region Design

title: LMIs in Control/pages/H2 Strip Region


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LMIs in Control/pages/H2 Optimal Observer

State observer is a system that provides estimates of internal states of a given real system, from measurements of the inputs and outputs of the real system.The goal of $$ H_2$$ -optimal state estimation is to design an observer that minimizes the $$ H_2 $$ norm of the closed-loop transfer matrix from w to z. Kalman filter is a form of Optimal Observer.

The System
Consider the continuous-time generalized plant $$ P $$ with state-space realization

\begin{align} \dot x&=Ax+B_1w(t),\\ y&=C_2x+D_{21}w\\ \end{align}$$

The Data
The matrices needed as input are $$ A,B,C,D $$.

The Optimization Problem
The task is to design an observer of the following form:

\begin{align} \dot{\hat{x}}=A\hat{x} + L(y - \hat{y}),\\ \hat{y} = C_2\hat{x}\\ \end{align}$$

The LMI: $$ H_2 $$ Optimal Observer
LMIs in the variables $$ P, G, Z, \nu $$ are given by:

\begin{align}

\begin{bmatrix} PA+ A^TP-GC_2-{C_2}^TG^T && PB_1-GD_{21}\\ \star && -1 \end{bmatrix} <0\\ trZ < \nu \end{align}$$

Conclusion:
The $$ H_2$$ -optimal observer gain is recovered by $$ L = P^{-1}G $$ and the $$ H_2 $$ norm of T(s) is $$ \mu = \sqrt {\nu} $$

Implementation
https://github.com/Ricky-10/coding107/blob/master/H2%20Optimal%20Observer

= External Links =

A list of references documenting and validating the LMI.

 * LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
 * LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
 * LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
 * https://onlinelibrary.wiley.com/doi/abs/10.1002/rnc.1310


text:
LMIs in Control/pages/HInf Optimal Observer

$$ H\infin $$-Optimal observers yield  robust  estimates  of  some  or  all  internal  plant states  by  processing  measurement  data. Robust observers are increasingly  demanded in industry  as they  may provide state  and  parameter  estimates  for  monitoring  and  diagnosis purposes even in the presence of large disturbances such as noise etc. It is there where Kalman filters may tend to fail. State observer is a system that provides estimates of internal states of a given real system, from measurements of the inputs and outputs of the real system. The goal of $$ H_\infin$$ -optimal state estimation is to design an observer that minimizes the $$ H_\infin $$ norm of the closed-loop transfer matrix from w to z.

The System
Consider the continuous-time generalized plant $$ P $$ with state-space realization

\begin{align} \dot x&=Ax+B_1w,\\ y&=C_2x+D_{21}w\\ \end{align}$$

The Data
The matrices needed as input are $$ A,B_1,B_2,C_2,D_{21},D_{11} $$.

The Optimization Problem
The observer gain $$ L $$ is to be designed such that the $$ H\infin $$ of the transfer matrix from w to z, given by

\begin{align} T(s) = C_1(s1-(A-LC_2))^{-1}(B_1-LD_{21})+D_{11}\\ \end{align} $$ is minimized. The form of the observer would be:

\begin{align} \dot{\hat{x}}=A\hat{x} + L(y - \hat{y}),\\ \hat{y} = C_2\hat{x}\\ \end{align}$$

The LMI: $$ H_\infin $$ Optimal Observer
The $$H\infin $$-optimal observer gain is synthesized by solving for $$ P \in \mathbb{S} ^{n_x}, G \in \mathbb{R} ^{n_x \times n_y} $$, and $$ \gamma \in \mathbb{R}_{>0} $$ that minimize $$ \zeta(\gamma) = \gamma $$ subject to $$ P>0 $$ and

\begin{align}

\begin{bmatrix} PA+ A^TP-GC_2-{C_2}^TG^T && PB_1-GD_{21} && C_1^T\\ \star && -\gamma1 && {D_{11}}^T\\ \star && \star && -\gamma1 \end{bmatrix} <0\\ \end{align}$$

Conclusion:
The $$ H_\infin$$ -optimal observer gain is recovered by $$ L = P^{-1}G $$ and the $$ H_\infin $$ norm of T(s) is $$ \gamma$$.

Implementation
Link to the MATLAB code designing $$ H\infin $$- Optimal Observer

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

= External Links =

 * LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
 * LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
 * LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
 *
 *


text:
LMIs in Control/pages/Mixed H2 HInf Optimal Observer

The goal of mixed $$ H_2-H\infin $$-optimal state estimation is to design an observer that minimizes the $$ H_2 $$ norm of the closed-loop transfer matrix from $$ w_1 $$ to $$ z_1 $$, while ensuring that the $$ H\infin $$ norm of the closed-loop transfer matrix from $$ w_2 $$ to $$ z_2 $$ is below a specified bound.

The System
Consider the continuous-time generalized plant $$ P $$ with state-space realization

\begin{align} \dot x&=Ax+B_{1,1}w_1 +B_{1,2}w_2,\\ y&=C_2x+D_{21,1}w_1+D_{21,1}w_2\\ \end{align}$$

where it is assumed that $$ (A, C_2) $$ is detectable.

The Data
The matrices needed as input are $$ A,B_1,B_2,C_2,D_{21},D_{11} $$.

The Optimization Problem
The observer gain L is to be designed to minimize the $$ H_2 $$ norm of the closed-loop transfer matrix $$ T_{11}(s) $$ from the exogenous input $$ w_1 $$ to the performance output $$ z_1 $$ while ensuring the $$ H\infin $$ norm of the closed-loop transfer matrix $$ T_{22}(s) $$ from the exogenous input $$ w_2 $$ to the performance output $$ z_2 $$ is less than $$ \gamma_d $$, where

\begin{align} T_{11}(s) = C_{1,1}(s1-(A-LC_2))^{-1}(B_{1,1}-LD_{21,1})\\ T_{22}(s) = C_{1,2}(s1-(A-LC_2))^{-1}(B_{1,2}-LD_{21,2})+D_{11,22} \end{align} $$

is minimized. The form of the observer would be:

\begin{align} \dot{\hat{x}}=A\hat{x} + L(y - \hat{y}),\\ \hat{y} = C_2\hat{x}\\ \end{align}$$

is to be designed, where $$ L \in \mathbb{R}^{n_x \times n_y} $$ is the observer gain.

The LMI: $$ H_\infin $$ Optimal Observer
The mixed $$H_2 -H\infin $$-optimal observer gain is synthesized by solving for $$ P \in \mathbb{S} ^{n_x}, G \in \mathbb{R} ^{n_x \times n_y} $$, and $$ \nu \in \mathbb{R}_{>0} $$ that minimize $$ \zeta(\nu) = \nu $$ subject to $$ P>0, Z>0 $$,

\begin{align} \begin{bmatrix} PA+ A^TP-GC_2-{C_2}^TG^T && PB_1-GD_{21}\\ \star && -1 \end{bmatrix} <0\\

\begin{bmatrix} PA+ A^TP-GC_2-{C_2}^TG^T && PB_1-GD_{21} && C_1\\ \star && -\gamma1 && {D_{11}}^T\\ \star && \star && -\gamma1 \end{bmatrix} <0\\ \begin{bmatrix} P && C{_{1,1}}^T \\ \star && Z\end{bmatrix} >0\\ trZ < \nu \end{align}$$

Conclusion:
The mixed $$ H_2- H_\infin$$ -optimal observer gain is recovered by $$ L = P^{-1}G $$, the $$ H_2 $$ norm of $$ T_{11}(s) $$ is less than $$ \mu = \sqrt {\nu} $$ and the $$ H_\infin $$ norm of T(s) is less than $$ \gamma_d$$.

Implementation
Link to the MATLAB code designing $$ H_2- H\infin $$- Optimal Observer

Code $ H_2-H\infin $ Optimal Observer

Related LMIs

 * $ H_2 $ Optimal observer
 * $ H\infin $ Optimal observer


text:
LMIs in Control/pages/H2 Optimal Filter

Optimal filtering is a means of adaptive extraction of a weak desired signal in the presence of noise and interfering signals. Optimal filters normally are free from stability problems. There are simple operational checks on an optimal filter when it is being used that indicate whether it is operating correctly. Optimal filters are probably easier to make adaptive to parameter changes than suboptimal filters.The goal of optimal filtering is to design a filter that acts on the output $$ z $$ of the generalized plant and optimizes the transfer matrix from w to the filtered output.

The System:
Consider the continuous-time generalized LTI plant with minimal states-space realization

\begin{align} \dot x &=Ax + B_1w\\ z &=C_1x + D_{11}w,\\ y &=C_2x + D_{21}w, \end{align}$$

where it is assumed that $$ A $$ is Hurwitz.

The Data
The matrices needed as inputs are $$ A, B_1, C_2, D_{21} $$.

The Optimization Problem:
An $$ H_2 $$-optimal filter is designed to minimize the $$ H_2 $$ norm of $$ \tilde{P}(s) $$ in following equation.

\begin{align} \tilde{P}(s) = \tilde{C}_1 (sI - \tilde{A})^-1 \tilde{B}_1 + \tilde{D}_{11},\\ \text{where} \\ \tilde{A} = \begin{bmatrix} A && 0\\ B_fC_2 && A_f\end{bmatrix}&<0\\ \tilde{B}_1 = \begin{bmatrix} B_1\\ B_fD_{21}\end{bmatrix}&<0\\ \tilde{C}_1 = \begin{bmatrix} C_1-D_fC_2 -C_f\end{bmatrix}&<0\\ \tilde{D}_{11} = D_{11}-D_fD_{21}\\ \end{align} $$

To ensure that $$ \tilde{P}(s) $$ has a finite $$ H_2 $$ norm, it is required that $$ D_f = D_{11} $$, which results in $$ \tilde{D}_{11}=D_{11}-D_f=0 $$

The LMI: $$ H_2 $$- Optimal filter
Solve for $$ A_n \in \mathbb{R}^{n_x \times n_x}, B_n \in \mathbb{R}^{n_x \times n_y}, C_f \in \mathbb{R}^{n_x \times n_x} $$, $$ X,Y \in \mathbb{S}^{n_x}, Z \in \mathbb{S}^{n_z} $$ and $$ \nu \in \mathbb{R}_{>0} $$ that minimize $$ \zeta(\nu) =\nu $$ subject to $$ X >0, Y >0, Z >0 $$.

\begin{align}

\begin{bmatrix} YA + A^TY+B_nC_2 && A_n+C_2^TB_n^T+A^TX && YB_1+B_nD_{21}\\ \star && A_n+A_n^T && XB_1+B_nD_{21}\\ \star && \star && -1\end{bmatrix}&<0\\ \begin{bmatrix} -Z && C_1-D_fC_2 && -C_f\\ \star && -Y && -X\\ \star && \star && -X \end{bmatrix}&<0\\ Y-X >0 \\ trZ < /nu \end{align}$$

Conclusion:
The filter is recovered by $$ A_f=X^{-1}A_n $$ and $$ B_f=X^{-1}B_n $$.

Implementation
MATLAB code of $ H_2 $ Optimal filter

title: LMIs in Control/pages/H2 Optimal Filter


text:
LMIs in Control/pages/HInf Optimal Filter

Optimal filtering is a means of adaptive extraction of a weak desired signal in the presence of noise and interfering signals. The goal of optimal filtering is to design a filter that acts on the output $$ z $$ of the generalized plant and optimizes the transfer matrix from w to the filtered output.

The System:
Consider the continuous-time generalized LTI plant with minimal states-space realization

\begin{align} \dot x &=Ax + B_1w\\ z &=C_1x + D_{11}w,\\ y &=C_2x + D_{21}w, \end{align}$$

where it is assumed that $$ A $$ is Hurwitz.

The Data
The matrices needed as inputs are $$ A, B_1, C_2, C_1, ,D_{11},D_{21} $$.

The Optimization Problem:
An $$ H\infin $$-optimal filter is designed to minimize the $$ H_\infin$$ norm of $$ \tilde{P}(s) $$ in following equation.

\begin{align} \tilde{P}(s) = \tilde{C}_1 (sI - \tilde{A})^-1 \tilde{B}_1 + \tilde{D}_{11},\\ \text{where} \\ \tilde{A} = \begin{bmatrix} A && 0\\ B_fC_2 && A_f\end{bmatrix}&<0\\ \tilde{B}_1 = \begin{bmatrix} B_1\\ B_fD_{21}\end{bmatrix}&<0\\ \tilde{C}_1 = \begin{bmatrix} C_1-D_fC_2 -C_f\end{bmatrix}&<0\\ \tilde{D}_{11} = D_{11}-D_fD_{21}\\ \end{align} $$

The LMI: $$ H_\infin $$- Optimal filter
Solve for $$ A_n \in \mathbb{R}^{n_x \times n_x}, B_n \in \mathbb{R}^{n_x \times n_y}, C_f \in \mathbb{R}^{n_x \times n_x} $$, $$ X,Y \in \mathbb{S}^{n_x}$$ and $$ \nu \in \mathbb{R}_{>0} $$ that minimize $$ \zeta(\nu) =\nu $$ subject to $$ X >0, Y >0 $$.

\begin{align}

\begin{bmatrix} YA + A^TY+B_nC_2 && A_n+C_2^TB_n^T+A^TX && YB_1+B_nD_{21} && {C_1}^T-{C_2}^T{D_f}^T\\ \star && A_n+A_n^T && XB_1+B_nD_{21} && -{C_f}^T\\ \star && \star && -\gamma I && {D_11}^T-{D_21}^T{D_f}^T\\ \star && \star && \star && -\gamma I \end{bmatrix}&<0\\ \\ Y-X >0 \\ \end{align}$$

Conclusion:
The filter is recovered by $$ A_f=X^{-1}A_n $$ and $$ B_f=X^{-1}B_n $$.

Implementation

 * https://github.com/Ricky-10/coding107/blob/master/Hinfinity%20Optimal%20filter- $$ H\infin $$- Optimal Filter


text:
LMIs in Control/pages/Minimum Singular Value of a Complex Matrix

Minimum Singular Value of a Complex Matrix

The System
Consider $$ A \in \mathbb{C}^{n \times m} $$ as well as $$ \gamma $$. A minimum singular value of a matrix $$ A $$ is greater than $$ \gamma $$ if and only if $$ AA^H > \gamma^2I $$ or $$ A^H A > \gamma^2I $$, where $$ A^H $$ is the conjugate transpose or Hermitian transpose of the matrix $$ A $$. the inequality used depends on the size of matrix $$ A $$.

The Data
The matrix $$ A $$ is the only data required.

The LMI: Minimum Singular Value of a Complex Matrix
The following LMIs can be constructed depending on the size of $$ A $$:

if $$ A \in \mathbb{C}^{n \times m} $$, where $$ n \leq m $$, then:

\begin{align} \bar{\sigma}(A) > \gamma\\ A A^{H} > \gamma^2 I \end{align}$$

Else if $$ n \geq m $$, then:

\begin{align} \bar{\sigma}(A) > \gamma\\ A^{H} A > \gamma^2 I \end{align}$$

Conclusion:
The results from this LMI will give the maximum complex value of the matrix $$ A $$:

\begin{align} \bar{\sigma}(A) > \gamma \end{align}$$

This answer can also be proven using the following solution. Note that this solution only works if the matrix $$ A $$ is a square, invertible matrix: $$ \sigma_{min} = 1/||A^{-1}||_2 $$.

Implementation

 * Minimum Singular Value Example

Related LMIs

 * Maximum singular value of a complex matrix


text:
LMIs in Control/pages/H2 Disk Region

Insensitive Disk Region Design with Minimum $$ H_2 $$ Gain

Apart from the design for the insensitive strip region with minimum $ H_2 $ gain, another type of such design is the Insensitive Disk Region Design. In this section, optimization problems will be provided that ensure that the conditions for insensitive disk region design are satisfied with some bounds on the $$ H_2 $$ gain of the closed-loop system.

The System
A state-space representation of a linear system as given below:

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

where $$x\in \R^n$$, $$y\in \R^m$$ and $$u\in \R^r$$ are the system state, output, and the input vector respectively. $$ \rho $$ represents the differential operation for continuous time systems, or the one-step shift forward operator for discrete time case.

The Data
To solve the design optimization problem, the linear system matrices A,B,C are required. Furthermore, to define the disk region on the eigenvalue-space, its radius $$ \gamma_0 $$ is required.

The Optimization Problem
The problem of designing an $$ H_2 $$ optimal controller that results in the closed loop system insensitive to a certain disk region involves two sub-problems:

 * Finding a control gain $$ K $$ such that: $$ \left\| K \right\|_2 < \gamma $$.
 * The conditions for insensitive disk region design for the closed-loop system, as provided in the section Insensitive Disk Region Design are fulfilled.
 * The optimization goal is to minimize $$ \gamma $$ such that above two hold.

The LMI: $$ H_2 $$ Optimal Control Design for Insensitive Disk Region
The problem above has a solution if and only if the following optimization problem has a solution $$ (K, \gamma) $$:

$$ \begin{align} \text{min   } & \gamma \\ \text{s.t.  } & \begin{bmatrix} - \gamma I & K \\ K^\top & - \gamma I \end{bmatrix} < 0 \\ & \begin{bmatrix} -\gamma_0 I & A+ BKC + qI \\ (A+ BKC + qI)^\top & -\gamma_0 I \end{bmatrix} < 0 \end{align} $$

Conclusion:
By using the design problem provided here, an optimal $$ H_2 $$ controller is designed to make the closed-loop system robust to perturbations in the system matrices.

Implementation
To solve the optimization problem with LMI presented here, 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/H2_Disk_example.m

Related LMIs
Insensitive Disk Region Design

title: LMIs in Control/pages/H2 Disk Region


text:
LMIs in Control/Controller Synthesis/Design For Insensitive Region

 * Generic Insensitive Strip Region Design
 * Generic Insensitive Disk Region Design
 * Design for Insensitive Strip Region
 * Design for Insensitive Disk Region


text:
LMIs in Control/pages/Minimum Gain Lemma

Minimum Gain Lemma

The System
Suppose there is a continuous-time LTI system where one must develop a controller for such system that is unstable. The minimum gain of a system would be obtained by taking the infimum of ratio between norms of the outputs and inputs over all nonzero inputs. With the Large Gain theorem expressing that if such an unstable system contains a finite value for the minimum gain of the system which is also non-zero, then any controller is capable of stabilizing the closed-loop feedback system, so long as the the controller also has a large minimum gain.

With this theorem led to the development of the Minimum Gain Lemma where applicable analysis can determine whether a closed loop system achieves a non-zero minimum gain value despite being inherently unstable in the open-loop case. By having an LTI system where the system $$ G $$ corresponds to the following matrices $$ A \in \mathbb{R}^{m \times n} $$, $$ B \in \mathbb{R}^{n \times m} $$, $$ C \in \mathbb{R}^{p \times n} $$, and $$ D \in \mathbb{R}^{p \times m} $$.

The Data
A system of the plant $$ G $$ will be required with the following matrices: $$ A \in \mathbb{R}^{m \times n} $$, $$ B \in \mathbb{R}^{n \times m} $$, $$ C \in \mathbb{R}^{p \times n} $$, and $$ D \in \mathbb{R}^{p \times m} $$.

The LMI: Minimum Gain Lemma
The following two LMI's are equivalent, with the same variable:

Suppose there exists $$ P \in \mathbb{S}^n $$, and $$ \upsilon \in \mathbb{R}_{\geq 0} $$, Where $$ P \geq 0 $$ and:

\begin{align}

\begin{bmatrix} P A + A^T P - C^T C & P B - C^T D \\ (P B - C^T D)^T & \upsilon^2 I - D^T D \end{bmatrix}&\leq 0

\end{align}$$

Or (Obtained using Schur compliment):

\begin{align}

\begin{bmatrix} P A + A^T P - C^T C & P B - C^T D & 0 \\ (P B - C^T D)^T & -D^T D & \upsilon I \\ 0 & \upsilon I & -I \end{bmatrix}&\leq 0

\end{align}$$

Conclusion:
Solving this LMI will give the minimum gain from the LTI system. This minimum gain can then be used if the system has been made stable based on the value of $$ \upsilon $$ obtained from the optimization.

Implementation

 * Minimum Gain Lemma Example


text:
LMIs in Control/pages/Modified Minimum Gain Lemma

Modified Minimum Gain Lemma

The System
Suppose there is a continuous-time LTI system where one must develop a controller for such system that is unstable. The minimum gain of a system would be obtained by taking the infimum of ratio between norms of the outputs and inputs over all nonzero inputs. With the Large Gain theorem expressing that if such an unstable system contains a finite value for the minimum gain of the system which is also non-zero, then any controller is capable of stabilizing the closed-loop feedback system, so long as the the controller also has a large minimum gain.

With this theorem led to the development of the Minimum Gain Lemma where applicable analysis can determine whether a closed loop system achieves a non-zero minimum gain value despite being inherently unstable in the open-loop case. By having an LTI system where the system $$ G $$ corresponds to the following matrices $$ A \in \mathbb{R}^{m \times n} $$, $$ B \in \mathbb{R}^{n \times m} $$, $$ C \in \mathbb{R}^{p \times n} $$, and $$ D \in \mathbb{R}^{p \times m} $$.

Essentially the same concept as the Minimum Gain Lemma; however, this modification ensure that the system being optimized is also Lyapunov stable. This Essentially means that the variable $$ Q $$ must ensure that the system is stable to this degree, otherwise the system itself is infeasible.

The Data
A system of the plant $$ G $$ will be required with the following matrices: $$ A \in \mathbb{R}^{m \times n} $$, $$ B \in \mathbb{R}^{n \times m} $$, $$ C \in \mathbb{R}^{p \times n} $$, and $$ D \in \mathbb{R}^{p \times m} $$.

The LMI: Modified Minimum Gain Lemma
The following two LMI's are equivalent, with the same variable:

Suppose there exists $$ Q \in \mathbb{S}^n $$, and $$ \upsilon \in \mathbb{R}_{\geq 0} $$, where $$ Q > 0 $$ such that:

\begin{align}

\begin{bmatrix} A Q + Q A^T & B - Q C^T D \\ (B - Q C^T D)^T & \upsilon^2 I - D^T D \end{bmatrix}&\leq 0

\end{align}$$

Or (Obtained using Schur compliment):

\begin{align}

\begin{bmatrix} A Q + Q A^T & B - Q C^T D & 0 \\ (B - Q C^T D)^T & -D^T D & \upsilon I \\ 0 & \upsilon I & -I \end{bmatrix}&\leq 0

\end{align}$$

Conclusion:
Solving this LMI will give the minimum gain from the LTI system. This minimum gain can then be used if the system has been made stable based on the value of $$ \upsilon $$ obtained from the optimization. This system will also prove that the plant is Lyapunov stable.

Implementation

 * Minimum Gain Lemma Example


text:
LMIs in Control/Stability Analysis/Continuous Time/D-Stability

 * Continuous Time D-Stability Observer


text:
LMIs in Control/Controller Synthesis/Continuous Time/D-Stability

 * Continuous Time D-Stability Controller


text:
LMIs in Control/Stability Analysis/Continuous Time/Time-Delay Systems

 * Delay Dependent Time-Delay Stabilization
 * Delay Independent Time-Delay Stabilization


text:
LMIs in Control/pages/Conic Sector Lemma

Conic Sector Lemma

For general input-output systems, sector conditions are formulated to verify or enforce the feedback stability. One of these sector conditions is the conic sector lemma, and the problem that designs the feedback controller is the conic sector theorem.

The System
Consider a square, contiuous-time linear time-invariant (LTI) system, $$\mathcal{G} : \mathcal{L}_{2e} \rightarrow \mathcal{L}_{2e}$$, with minimal state-space relization $$(A, B, C, D)$$, where $$A \in \R^{n\times n}, B \in \R^{n\times m}, C \in \R^{m\times n},$$ and $$ D \in \mathcal{R}^{m \times m} $$. The state-space representation is:

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

where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$ and $$u(t)\in \R^m$$ are the system state, output, and the input vector respectively.

The Data
The system coefficient matrices $$ (A,B,C,D) $$ are required. Optionally, the parameters to define a cone, either in the form of $$ [a,b] $$ where $$ a,b \in \R, a < b $$ or a radius $$ r \in \R_+ $$ and ceter $$ c \in \R $$.

The Feasibility LMI
The system $$ \mathcal{G} $$ is inside the given cone $$[a,b]$$ if the following is feasible:

\begin{align} \text{Find: } & P \\ \text{subj. to: } & P > 0 \\ & \begin{bmatrix} PA+A^\top P+C^\top C & PB - \frac{a+b}{2} C^\top + C^\top D \\ (PB - \frac{a+b}{2} C^\top + C^\top D)^\top & D^\top D - \frac{a+b}{2}(D+D^\top) + abI \end{bmatrix} \leq 0. \end{align} $$

The above LMI can be used to also determine the cone parameters by setting $$ a $$ as a variable along with the condition $$ a < b $$, and use the bisection method to find $$ b $$.

If the given cone is represented by a center $$ c $$ and radius $$ r $$, then the following feasibility problem can be evaluated to check if $$ \mathcal{G} $$ is inside the given cone:

\begin{align} \text{Find: } & P \\ \text{subj. to: } & P > 0 \\ & \begin{bmatrix} PA+A^\top P+C^\top C & PB - c C^\top + C^\top D \\ (PB - cC^\top + C^\top D)^\top & D^\top D - c(D+D^\top) + (c^2-r^2)I \end{bmatrix} \leq 0. \end{align} $$

In order to also find the cone parameters, substituting $$ \gamma = r^2 $$ as a decision variable with additional constraint $$ \gamma \geq 0 $$ and then solving for $$ c $$ via the bisection method will give the cone in which the system $$ \mathcal{G} $$ resides if the problem is feasible.

Conclusion:
The aforementioned LMIs can be utilized to either check if $$ \mathcal{G} $$ is in the specified cone or not, or can be used to check the stability of $$ \mathcal{G} $$ by finding if a feasible cone can be obtained that encloses $$ \mathcal{G} $$. An important point to note here is that the Conic Sector Lemma is a special case of the KYP Lemma for QSR dissipative systems with:

$$ Q = -1, S = frac{a+b}{2}I = cI, R = -abI = (r^2 - c^2)I $$.

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/Conic_sector_example.m

Related LMIs
Exterior Conic Sector Lemma.

KYP Lemma


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User:Dijellzaajdini/sandbox

Dijellza Ajdini

(d-yell zuh I-D Knee) was born on August 14th 2003. She is Albanian-American singer, songwriter, and dancer. Born in Chapel Hill North Carolina, Ajdini stared off by singing in school with her school chorus since the 4th grade. In the year of 2017 she released her first single called With You, at the time she 13 years young when released. As the time went on she made her debut album named "Scandalous" featuring the single from Scandalous called "I'll Be Waiting" FT. Ben The Realest who is her 20 year old brother.

Life and career

Dijellza Ajdini was born on August 14th 2003. Both her parents are Albanian from Presevo. Dijellza visits where she's from every summer to see her family. She has a older who is brother Ben Ajdini, a rapper, and songwriter. Her entire family are war refuges from the Yugoslavia war back in 99'.

As a kid, Ajdini would release song covers on youtube at the age of 8 and 9 years old. Song's as "I Knew You Were Trouble" By Taylor Swift and "Write Your Name" by Selena Gomez. Ajdini would also be in a every chorus class from her school. She also loved being in Musical Theater. Her favorite musicals are 'The Sound Of Music' and Hairspray.

At the age of 13 years old, she started pursing her music career. She has always wanted to create a soul R&B album with some pop. As years went on, Ajdini released her second single called Broken. She had released a music video for that song but the music video has been deleted the of the release.

2019–Present: Scandalous

Ajdini recored her first studio album a almost a 1 year period. It was released on May 22nd, 2019. Dijellza released the album on the 22nd due to the tragic event in Manchester UK involving the pop sensation of Ariana Grande. Ajdini released to support the people. She wanted to identify that positivity is all over the world. "Music is the power through positivity".

After releasing Scandalous she released her debut music video for "I'll Be Waiting". The original did not feature Ben The Realest. That song remix was a surprise to all of us as if she did not state that there was a remix and we were expecting the regular song. But we did not receive that.

She has talked about going on tour but has not confirmed any confirmations yet. But she has performed at the North Carolina Fillmore in Charlotte North Carolina!

Influences

Dijellza has spoken on who was inspired her music a lot since she was a child. Selena Gomez was her main idol growing up and still is to this day. Dijellza has said there is video footage of singing and dancing to Selena Gomez when she was 7 years old. Ajdini still loves Gomez to this day. She describes her as a "Strong wise intelligent woman".

title: User:Dijellzaajdini/sandbox


text:
LMIs in Control/pages/Discrete Time H2 Optimal Dynamic Output Feedback Control

Discrete-Time H2-Optimal Dynamic Output Feedback Control

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.

A Dynamic Output feedback controller is designed for a Discrete Time system, to minimize the H2 norm of the closed loop system with exogenous input $$w_{k}$$ and performance output $$z_{k}$$.

The System
Discrete-Time LTI System with state space realization $$(A_d,B_d,C_d,D_d)$$ $$ \begin{align} x_{k+1}&=A_dx_k + B_{d1}w_k + B_{d2}u_k\\ z_{k}&=C_{d1}x_k + D_{d11}w_k+ D_{d12}u_k\\ y_k &=C_{d2}x_k + D_{d21}w_k+ D_{d22}u_k\\ \end{align}$$

The Data
The matrices: System $$ (A_d,B_{d1},B_{d2},C_{d1},C_{d1},D_{d11},D_{d12},D_{d21},D_{d22}), X_1,Y_1,Z, ,X_2,Y_2 $$

Controller $$ (A_{dc},B_{dc},C_{dc},D_{dc}) $$

The Optimization Problem
The following feasibility problem should be optimized:

$$ \mu $$ is minimized while obeying the LMI constraints.

The LMI:
Discrete-Time H2-Optimal Full-State Feedback Control

The LMI formulation

H2 norm < $$\mu$$

$$ \begin{align} X_{1}, Y_{1}\in {S^{n_x}}; Z \in {S^{n_z}}; \mu \in {R_{>0}} \;\\ A_{dn} \in {R^{n_x*n_x}}; B_{dn} \in {R^{n_x*n_y}}; C_{dn} \in {R^{n_u*n_x}}; D_{dn} \in {R^{n_u*n_y}};\\ &X_{1}>0\\ &Y_{1}>0\\ &Z>0\\ \begin{bmatrix}X_{1} & 1 & X_{1}A_d+B_{dn}C_{d2} & A_{dn} & X_{1}B_{d1}+B_{dn}D_{d21}\\
 * & Y_{1} & A_d+B_{d2}D_{dn}C_{d2} & A_dY_{1}+B_{d2}C_{dn} & B_{d1}+B_{d2}D_{dn}D_{d21}\\
 * & * & X_{1} & 1 & 0\\
 * & * & * & Y_{1} & 0\\
 * & * & * & * & 1\end{bmatrix}&>0,\\

\begin{bmatrix}Z & C_{d1}+D_{d12}D_{dn}C_{d2} & C_{d1}Y_{1}^T+D_{d12}C_{dn}\\
 * & X_{1} & 1\\
 * & * & Y_{1}\end{bmatrix}&>0,\\

D_{d11}+D_{d12}D_{dn}D_{d21}=0\\

\begin{bmatrix}X_{1} & 1\\
 * & Y_{1}\end{bmatrix}&>0,\\

trZ<\mu ^2 \end{align}$$

The controller is recovered by

$$ \begin{align} &A_{dc} = A_{dk}-B_{dc}(1-D_{d22}D_{dc})^{-1}D_{d22}C_{dc}\\ &B_{dc} = B_{dk}(1-D_{d22}D_{dc})\\ &C_{dc} = (1-D_{dc}D_{d22})C_{dk}\\ &D_{dc} = 1+D_{dk}D_{d22})^{-1}D_{dk}\\ \end{align}$$

where, $$ \begin{align} \begin{bmatrix}A_{dk} & B_{dk}\\ C_{dk} & D_{dk}\end{bmatrix}&= \begin{bmatrix}X_{2} & X_{1}B_{d2}\\ 0 & 1\end{bmatrix}^{-1} (\begin{bmatrix}A_{dn} & B_{dn}\\ C_{dn} & D_{dn}\end{bmatrix} -\begin{bmatrix}X_{1}A_{d}Y_{1} & 0\\ 0 & 0\end{bmatrix}) \begin{bmatrix}Y_{2}^T & 0\\ C_{d2}Y_{1} & 1\end{bmatrix}^{-1}

\end{align}$$ and the matrixes $$X_2$$ and $$Y_2$$ satisfies $$X_2Y_2^T = 1 - X_{1}Y_{1}$$

Given $$X_1$$ and $$Y_1$$, the matrices $$X_2$$ and $$Y_2$$ can be found using a matrix decomposition, such as a LU decomposition or a Cholesky decomposition.

If $$D_{d11} = 0$$, $$D_{d12}$$ ≠ 0, and $$D_{d21}$$ ≠ 0, then it is often simplest to choose $$D_{dn} = 0$$ in order to satisfy the equality constraint

Conclusion:
The Discrete-Time H2-Optimal Dynamic Output feedback controller is the system $$ (A_{dc},B_{dc},C_{dc},D_{dc}) $$

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

Related LMIs
- Continuous Time H2 Optimal Dynamic Output Feedback Control


text:
LMIs in Control/pages/Discrete Time H∞ Optimal Dynamic Output Feedback Control

Discrete-Time H∞-Optimal Dynamic Output Feedback Control

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.

A Dynamic Output feedback controller is designed for a Discrete Time system, to minimize the H∞ norm of the closed loop system with exogenous input $$w_{k}$$ and performance output $$z_{k}$$.

The System
Discrete-Time LTI System with state space realization $$(A_d,B_d,C_d,D_d)$$ $$ \begin{align} x_{k+1}&=A_dx_k + B_{d1}w_k + B_{d2}u_k\\ z_{k}&=C_{d1}x_k + D_{d11}w_k+ D_{d12}u_k\\ y_k &=C_{d2}x_k + D_{d21}w_k+ D_{d22}u_k\\ \end{align}$$

The Data
The matrices: System $$ (A_d,B_{d1},B_{d2},C_{d1},C_{d1},D_{d11},D_{d12},D_{d21},D_{d22}), X_1,Y_1,Z, ,X_2,Y_2 $$

Controller $$ (A_{dc},B_{dc},C_{dc},D_{dc}) $$

The Optimization Problem
The following feasibility problem should be optimized:

$$ \gamma $$ is minimized while obeying the LMI constraints.

The LMI:
Discrete-Time H∞-Optimal Dynamic Output Feedback Control

The LMI formulation

H∞ norm < $$\gamma$$

$$ \begin{align} X_{1}, Y_{1}\in {S^{n_x}}; Z \in {S^{n_z}}; \gamma \in {R_{>0}} \;\\ A_{dn} \in {R^{n_x*n_x}}; B_{dn} \in {R^{n_x*n_y}}; C_{dn} \in {R^{n_u*n_x}}; D_{dn} \in {R^{n_u*n_y}};\\ &X_{1}>0\\ &Y_{1}>0\\ &Z>0\\ \begin{bmatrix}X_{1} & 1 & X_{1}A_d+B_{dn}C_{d2} & A_{dn} & X_{1}B_{d1}+B_{dn}D_{d21} & 0\\
 * & Y_{1} & A_d+B_{d2}D_{dn}C_{d2} & A_dY_{1}+B_{d2}C_{dn} & B_{d1}+B_{d2}D_{dn}D_{d21} & 0\\
 * & * & X_{1} & 1 & 0 & C_{d1}^T+C_{d2}^TD_{dn}^TD_{d12}^T\\
 * & * & * & Y_{1} & 0 & Y_{1}C_{d1}^T+C_{dn}^TD_{d12}^T\\
 * & * & * & * & -\gamma I & D_{d11}^T+D_{d21}^TD_{dn}^TD_{d12}^T\\
 * & * & * & * & * & -\gamma I\end{bmatrix}&>0,\\

\begin{bmatrix}X_{1} & 1\\
 * & Y_{1}\end{bmatrix}&>0,\\

\end{align}$$

The controller is recovered by

$$ \begin{align} &A_{dc} = A_{dk}-B_{dc}(1-D_{d22}D_{dc})^{-1}D_{d22}C_{dc}\\ &B_{dc} = B_{dk}(1-D_{dc}D_{d22})\\ &C_{dc} = (1-D_{dc}D_{d22})C_{dk}\\ &D_{dc} = 1+D_{dk}D_{d22})^{-1}D_{dk}\\ \end{align}$$

where, $$ \begin{align} \begin{bmatrix}A_{dk} & B_{dk}\\ C_{dk} & D_{dk}\end{bmatrix}&= \begin{bmatrix}X_{2} & X_{1}B_{d2}\\ 0 & 1\end{bmatrix}^{-1} (\begin{bmatrix}A_{dn} & B_{dn}\\ C_{dn} & D_{dn}\end{bmatrix} -\begin{bmatrix}X_{1}A_{d}Y_{1} & 0\\ 0 & 0\end{bmatrix}) \begin{bmatrix}Y_{2}^T & 0\\ C_{d2}Y_{1} & 1\end{bmatrix}^{-1}

\end{align}$$ and the matrixes $$X_2$$ and $$Y_2$$ satisfies $$X_2Y_2^T = 1 - X_{1}Y_{1}$$

Given $$X_1$$ and $$Y_1$$, the matrices $$X_2$$ and $$Y_2$$ can be found using a matrix decomposition, such as a LU decomposition or a Cholesky decomposition.

If $$D_{d11} = 0$$, $$D_{d12}$$ ≠ 0, and $$D_{d21}$$ ≠ 0, then it is often simplest to choose $$D_{dn} = 0$$ in order to satisfy the equality constraint

Conclusion:
The Discrete-Time H∞-Optimal Dynamic Output Feedback Controller is the system $$ (A_{dc},B_{dc},C_{dc},D_{dc}) $$

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

Related LMIs
- Continuous Time H∞ Optimal Dynamic Output Feedback Control


text:
LMIs in Control/pages/Discrete Time H∞ Optimal Full State Feedback Control

Discrete-Time H∞-Optimal Full-State Feedback Control

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.

A full state feedback controller design for a Discrete Time system, to minimize the H∞ norm of the closed loop system with exogenous input $$w_{k}$$ and performance output $$z_{k}$$.

The System
Discrete-Time LTI System with state space realization $$(A_d,B_d,C_d,D_d)$$ $$ \begin{align} x_{k+1}&=A_dx_k + B_{d1}w_k + B_{d2}u_k\\ z_{k}&=C_{d1}x_k + D_{d12}u_k\\ y_k &= x_k\\ \end{align}$$

The Data
The matrices: System $$ (A_d,B_{d1},B_{d2},C_{d1},D_{d12}), P, F_d $$.

The Optimization Problem
The following feasibility problem should be optimized:

$$ \gamma $$ is minimized while obeying the LMI constraints.

The LMI:
Discrete-Time H∞-Optimal Full-State Feedback Control

The LMI formulation

H∞ norm < $$\gamma$$

$$ \begin{align} P \in {S^{n_x}}; &F_d \in {R^{n_u*n_x}}; \gamma \in {R_{>0}} \;\\ &P>0\\ \begin{bmatrix}P & A_dP-B_{d2}F_d & B_{d1} & 0\\
 * & P & 0 & PC_{d1}^T-F_{d}^TD_{d12}^T\\
 * & * & -\gamma I & D_{d11}^T\\
 * & * & * & -\gamma I\end{bmatrix}&>0,\\

\end{align}$$

Conclusion:
The H∞-optimal full-state feedback gain is recovered by $$K_{d}=F_{d}P^{-1}$$

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

Related LMIs
- Continuous Time H∞ Optimal Full State Feedback Control


text:
User:Tobepriscille/sandbox/Approaches to Knowledge (LG seminar)/ Group 2/Evidence/Evidence in Art forgery

Evidence in Art forgery
by Priscille

Art forgery
It’s easy to prove the genuineness in fashion or in technology, there are certificate of authenticity. However, it turns to be harder in Art when the artist is already dead, and nobody knows how many sculptures or paintings he/she has left. Art forgery is the creating and selling of works of art which are falsely credited to other, usually famous artists. There are so many counterfeits in circulation in museums. So how to prove the authenticity of a work of art? A counterfeit in art is an imitation of a work of art which is not presented as a fake or an original masterpiece credited to an artist more famous with something he hasn’t done.

Anthropology
All societies don’t consider that copy an artwork is a counterfeit. Kwoma people (Aboriginal Australia) reckon that all art pieces are copies, some are just less similar to the original than others. This means that any art piece is an attempt to copy the nature. Therefore, artworks from tribal societies are not assigned to an individual artist but to a whole community. Unlike in the modern wester society, where no matter the beauty of a forgery, it is not valued as much as the original work because newness is the mark of creativity and creativity is one of the defining features of art. Something that is creative can be created only once.

Sciences
When Curators cannot determine whether a work of art is a deception and provenance is obscure, sciences can help but it’s as the other methods not absolutely reliable. One of the most used method is microscopy. By looking under a microscope, the investigator can find out whether a layer of paint has been added recently or checks if there are craquelures, a proof that the painting is old. Another way is to analyse the chemical compositions because some materials was not available for the artist at his time. There are lot of other methods like mass spectrometry or infrared reflectography but the more the techniques to foil the forgers become precise the more the forgers develop their own methods to swindle. However these methods are useful only for old artworks.


text:
LMIs in Control/pages/Discrete Time Mixed H2-H∞ Optimal Full State Feedback Control

Discrete-Time Mixed H2-H∞-Optimal Full-State Feedback Control

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.

A full-state feedback controller $$K = K_d \in {R^{n_u*n_x}}$$ (i.e., $$uk = K_dx_k$$) is to be designed to minimize the H2 norm of the closed loop transfer matrix $$T11(z)$$ from the exogenous input $$w_{1,k}$$ to the performance output $$z_{1,k}$$ while ensuring the H∞ norm of the closed-loop transfer matrix $$T22(z)$$ from the exogenous input $$w_{2,k}$$ to the performance output $$z_{2,k}$$ is less than $$\gamma_{d}$$.

The System
Discrete-Time LTI System with state space realization

$$ \begin{align}

x_{k+1}&=A_{d}x_{k}+B_{d1,1}w_{1,k}+B_{d1,2}w_{2,k}+B_{d2}u_{k},\\

\begin{bmatrix} Z_{1,k}\\ Z_{2,k}\end{bmatrix} &= \begin{bmatrix} C_{d1,1}\\ C_{d1,2}\end{bmatrix}x_{k}+ \begin{bmatrix} 0 & D_{d11,12}\\ D_{d11,21} & D_{d11,22}\end{bmatrix} \begin{bmatrix} w_{1,k}\\ w_{2,k}\end{bmatrix}+ \begin{bmatrix} D_{d12,1}\\ D_{d12,2}\end{bmatrix}u_{k}\\

y_k &= x_k\\ \end{align} $$

The Data
The matrices: System $$ (A_d,B_{d1,1},B_{d1,2},B_{d2},C_{d1,1},C_{d1,2},D_{d11,12},D_{d11,21},D_{d11,22},,D_{d12,1},D_{d12,2}), P, F_d $$.

The Optimization Problem
The following feasibility problem should be optimized:

Minimize the H2 norm of the closed loop transfer matrix $$T11(z)$$, while ensuring the H∞ norm of the closed-loop transfer matrix $$T22(z)$$ is less than $$\gamma_{d}$$, while obeying the LMI constraints.

The LMI:
Discrete-Time Mixed H2-H∞-Optimal Full-State Feedback Controller is synthesized by solving for $$P \in S^{n_{x}}, Z \in S^{n_{w}}, F_{d} \in R^{n_{u} * n_{x}}$$, and $$\mu \in R_{>0}$$ that minimize $$\mu$$ subject to $$P>0, Z>0$$

The LMI formulation

H∞ norm < $$\gamma_{d}$$

H2 norm < $$\mu$$

$$ \begin{align}

\begin{bmatrix}P & A_{d}P-B_{d2}F_{d} & B_{d1,1}\\
 * & P & 0\\
 * & * & 1\end{bmatrix}&>0\\

\begin{bmatrix}P & A_{d}P-B_{d2}F_{d} & B_{d1,2} & 0\\
 * & P & 0 & PC_{d1,2}^T-F_{d}^TD_{d12,2}^T\\
 * & * & \gamma_{d} I & D_{d11,22}^T\\
 * & * & * & \gamma_{d} I\end{bmatrix}&>0\\

\begin{bmatrix} Z & C_{d1,1}P-D_{d12,1}\\
 * & P\end{bmatrix}&>0\\

trace(Z) &< \mu^2

\end{align}$$

Conclusion:
The H2-optimal full-state feedback controller gain is recovered by $$K_{d}=F_{d}P^{-1}$$

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

Related LMIs
- Continuous Time Mixed H2-H∞ Optimal Full State Feedback Control


text:
User:Tobepriscille/sandbox/Approaches to Knowledge (LG seminar)/Group 2/History/History of Young adult fiction/

History of Young adult fiction
by Priscille

Introduction
The recognition of the child or adolescent is recent in our society. The convention on the rights of the child was enacted only in 1989. Although children books have always existed, nothing was written for adolescents and most of children books were fairy- tales before the 20th century. The main reason for the lack of young adult fictions is that adolescence is a relatively new idea, people used to think that children turn straight to adults. Apart from the acknowledgement of the adolescence as a stage of life, it is definitely the growth potential that enabled the genre to develop.

Definition of Young adult fiction
Young adult fiction (YA) is a genre of fiction designed for readers from 12 to 18 years. While it is targeted to adolescents, half of the readers are adults. YA novels are distinct from teen or new adult books which are respectively designed for a younger and an older audience. The settings of YA stories are only limited by the imagination and skill of the author, but they must portray an adolescent as the protagonist. YA books span the entire spectrum of fiction genres.


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LMIs in Control/Stability Analysis/Sector Conditions

 * 1) Conic Sector Lemma


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Issues in Interdisciplinarity 2019-20/Power in the opioids crisis in the US

Introduction
Opioids prescription painkiller commonly prescribed in the United States of America. The 'opioid epidemic' refers to large scale over prescription of and addiction to opium-derived prescription pills across america. The abuse of prescription Opioids such as Hydrocodone, Oxycodone, Heroin, and et cetera currently affects 2.1 million Americans, claiming the lives of over 115 people every day. The following article will explore different power structures within this issue and how the influence the opioid epidemic along with wider society. These power structures include:

 * 1) Biology: the power opioids have over the human body.
 * 2) Politics: the power pharmaceutical companies have over governments through lobbying.
 * 3) Law: the power governments have over laws influencing the opioid epidemic.
 * 4) The Arts: the influence popular art forms have had on the opioid epidemic.

Biology
A definition of addiction is "a psychological and physical inability to stop consuming a substance, even though it is causing psychological and physical harm.” This explains how physical addiction is representative of a power structure between drugs and the human brain, as physically addictive drugs cause human beings to develop dependencies and may negatively affect their entire lifestyle. Opioids trigger the release of endorphins, the brain's "feel-good" neurotransmitters. Endorphins muffle ones sensation of pain and boost feelings of pleasure, creating a temporary but powerful sense of euphoria. When an opioid high wears off, one may find themselves craving this euphoria.

Opioids can also act on descending pain modulatory systems to eliminate the pain. By binding to receptors in an area of the brainstem called the periaqueductal gray, opioids can prompt the inhibition of pain. The result of the biological effect of regular opioid abuse is physical dependency. In an article published by psychiatrists T. Kosten and T. George, the power opioids have to catalyse a physical dependence is explained: "Opioid tolerance and addiction are all manifestations of brain changes resulting from chronic opioid abuse. The opioid abuser’s struggle for recovery is in great part a struggle to overcome the effects of these changes." Many individuals who find themselves physically addicted to opioids then seek rehabilitation. However, treatments are expensive ("Well-known centers often cost up to $20,000 for a 30-day program" ). The treatment includes managing the physical pain associated with withdrawals as well as the psychological consequences of addiction. In addition, relapses are common; most of the patients in a drug addiction relapse, further indicating the power of opioids. Lastly, as the tolerance increases with addictions, patients may seek more powerful and cheaper, illicit opioids such as heroin. In consequence, after the "doctor-shopping" (visiting multiple doctors in order to obtain more prescriptions) phase, the addicts often turn to injecting as the tolerance grows.

Politics
Another power structure present in the United State’s opioid epidemic is the power pharmaceutical companies have over members of senate and federal agencies.

The pharmaceutical company as a collective, “which has about two lobbyists for every member of Congress, spent $130 million on influencing legislation in 2019. To put this in perspective, considering that gun control is another one of America’s pressing issues, lobbyists for pro-firearm rights donated a mere $8.1 million to influence legislation the same year. Furthermore, “nine out of 10 members of the House of Representatives and all but three of the US’s 100 senators have taken campaign contributions from pharmaceutical companies seeking to affect legislation on everything from the cost of drugs to how new medicines are approved”. With these statistics, it can be reasonably inferred that lobbying on behalf of pharmaceutical companies has swayed legislators to be more lax on the industry, perfectly demonstrating the power the industry has in politics, law making, and consequentially American society.

Lobbying on behalf of the pharmaceutical industry also develops a power structure involving Federal agencies. A recent scandal involving Donald Trump’s nominee for Drug Czar Tom Marino in April 2016 revealed his role in forging legislation which would have made it “virtually impossible for the DEA (Drug Enforcement Agency) to freeze suspicious narcotic shipments from the companies”
 * Marino received a donation of nearly $100,000 from the pharmaceutical industry that same year. Furthermore, pharmaceutical companies engaged in unlawful activities have not faced sufficient punishment in the recent years. For example, pharmaceutical company “The McKesson Cooperation” were under investigation for providing millions of opioid pills to American pharmacies without sufficient due diligence in 2017, and while the DEA proposed a $1 billion fine for the company, Mckesson were only fined $150 million, an almost insignificant sum in comparison to the company’s $200 billion in revenue the same year. This is only one of multiple examples of how pharmaceutical companies have used their power to evade harsh legal penalties in order to preserve their wealth via continuing to supply opioids in excessive quantities to the American people.

Laws
Both federal and state administrations have tried to jam the opioid crisis by developing a pain policy. Indeed, many states have already adopted regulations laws to fight against opioid addictions. All states have adopted the Prescription Drug Monitoring Program, an exam that physicians must pass to prescribe opioids. However, the test is a handbook of 11 questions with the responses at the end.

The most powerful institution for taking decision concerning drugs is FDA. The Food and Drug Administration is a federal agency evaluating the safety and effects of every products related to health and food. In order to fight against prescription of opioids, the FDA created the TIRF-REMS program. A strict set of rules to make sure doctors prescribed opioids only to late stage cancer patients. Nevertheless, the FDA outsourced the program to the pharmaceutical companies and these companies hired McKesson, one of the world’s largest drug distributors to run it. Furthermore, Barack Obama enacted the Comprehensive Addiction and Recovery Act in 2016. It is intended to increase both the number of prevention and treatment programs. Nevertheless, he also signed into law the 21st Century Cures Act and the “Ensuring Patient Access and Effective Drug Enforcement Act” which reduce respectively the action of FDA and DEA.

Concerning the illegal side of the opioid crisis, the problem extends beyond the United States of America, as statistics indicate that China is the biggest Opioid supplier in the pharmaceutical industry out of the entire country. The Trump administration pushes for China to adapt proper laws but even if China has regulations, the government rarely enforced them since the output is not intended for Chinese people. However, as this issue affects the trade war with the US, China is working with the U.S law enforcement to find a solution.

Arts
Arts and storytelling have been successfully changing the public perception of opioid addiction. For instance, the “prescribed to Death” memorial is to represent, remember and memorize the victims of the opioid crisis. ; Moreover, arts and storytelling have been successfully changing the public perception of opioid addiction. Nancy, an opioid addiction recovery specialist, helped her client Ava to express her feelings about using opioids in her drawing, claiming that the patient cannot control their lives and is gradually disconnecting from reality as a result of her addiction. She hopes people can see their lives as what they are truly. Furthermore, In the American music industry, rapper Macklemore realised “Drug Dealer”, a new song and music video, referencing some celebrity deaths induced by opioids. With that, he met President Obama to talk about this important issue. The official music video on Youtube has more than 40 million views and seventeen thousand comments; thus proving the power structure of the media over society in raising awareness about the opioid epidemic.


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Cookbook:Kanom Piak Poon (Thai Coconut and Pandan Pudding)

Kanom piak poon is a thickened Thai sweet made of coconut, rice flour, and pandan.

Ingredients

 * 80 g rice flour
 * 40 g arrowroot starch
 * 200 g sugar
 * 600 ml limewater (saturated calcium hydroxide solution)
 * 200 ml fresh pandan juice
 * 200 ml of coconut milk
 * 1 teaspoon salt
 * 1 cup grated coconut meat

Procedure

 * 1) Combine the rice flour, arrowroot, and lime water. Stir in the sugar, water, pandan juice, coconut milk, and salt until the sugar dissolves. Strain this mixture into a brass pan or pot to remove any solids.
 * 2) Heat the pan over low heat. Stir the mixture continuously over low heat to keep it from sticking to the bottom of the pan.
 * 3) Cook until the mixture is glossy and thickened. This will take about 30–40 minutes.
 * 4) Pour the mixture into an oiled mold. Smooth the top, then let rest until cool.
 * 5) When the pudding is cooled, use a knife to cut into the desired shape. Serve with grated coconut.

Notes, tips, and variations

 * If the pot or pan starts to heat until the smoke rises, reduce the heat a little and stir the flour mixture quickly.


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History of wireless telegraphy and broadcasting in Australia/Topical/Biographies/Arthur Nelson Stephens

Arthur Nelson Stephens
Arthur Nelson Stephens was a prominent coastal radio operator, amateur operator, amateur broadcaster and amateur club administrator in the mid 1910s to mid 1920s and early 1930s, and ultimately senior state public servant. His station callsign was 4CG (4=Queensland, C,G allocated sequentially) and he operated from Brisbane, Queensland. A summary biography has not yet been prepared however the following resources have been assembled in preparation:
 * Transcriptions and Notes - Arthur Nelson Stephens - More than 50 transcriptions and notes
 * Photos, QSL cards and other graphics - Arthur Nelson Stephens - Nothing uploaded as yet


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History of wireless telegraphy and broadcasting in Australia/Topical/Biographies/Arthur Nelson Stephens/Notes

1917 02
Social page announcement of impending marriage of 4CG1 "From Near and Far. . . . Mr. and Mrs. Harry Bracker and their daughters, who have been residing in Toowoomba for the past twelve months, are returning to Brisbane about April. Miss Nellie Bracker's marriage to Mr. A. N. Stephens (engineer for the wireless station, Pinkenba) takes place shortly."

1952 06
4CG reminisces about his early days in wireless "SOUTHPORT ROTARY. Last night at the weekly Rotary Club dinner, at Theodore's Cafe, President Mal Burke reported that with a number of officers and Rotarians he had visited Beenleigh the previous evening when they met a number of citizens of that town who were desirous of forming a Rotary club. A couple of hours was spent in explaining the aims and objects of Rotary, and as the result of the visit it is more than likely, within the next couple of weeks, a move will be made for the formation of a Provisional Rotary Club at Beenleigh. The guest speaker of the evening was Mr. A. N. Stephens who gave a very interesting account of the introduction of wireless into Australia about 1911, by the then Prime Minister (Mr. Andrew Fisher), and his (the speaker's) association with wireless some 40 years ago. Mr. Stephens outlined the progress of wireless during the years, and at the end of his informative address he was accorded a hearty vote of thanks moved by Rotarian E. J. Gaven, and carried by acclamation."


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Talk:Issues in Interdisciplinarity 2019-20/The Issue of History in Wellbeing

15/11/2019
First group meeting

Attendance: Rachel and Rita

We brainstormed possible ideas and decided to work on personality due to our shared interest in the MBTI personality test.

Agreed to arrange for a second meeting next week and to go back and carry out more in-depth research on the topic.

19/11/2019
Second group meeting

Attendance: Charmaine, Rachel, Rita and Jason

Came up with a more specific topic idea: how different disciplines believe personalities are created and the way they collect evidence to support their claim

Came up with four different disciplines to focus on: Psychology, Anthropology, Biology and Philosophy

Divided the research based on disciplines:

Rachel: Psychology

Rita: Anthropology

Jason: Philosophy

Charmaine: Biology

Presented our idea to James during ATK tutorial and he commented that the topic might be too broad. We understood why he would think so and decided to go back and research other ideas and meet again during the week to present our findings

22/11/2019
Third group meeting

Attendance: Charmaine, Rita and Rachel

Rachel suggested that instead of looking at personalities, we could look at well-being in interdisciplinary research. We were still unsure if we should focus on Evidence or History and decided to carry out our research first and then decide

Decided to use the disciplines: Economics (Rachel), Anthropology (Rita), Philosophy (Jason), Psychology and Biology (Charmaine)

Also settled the structure of our wikibook:

Introduction

1 paragraph per discipline

Evaluation in relation to either Evidence/ History

Conclusion Conclusion

28/11/2019
Fourth group meeting:

Attendance: Rachel and Rita

Decided that based on the research we all found it would be best for us to focus on History instead of Evidence as we found a lot of ways we could explore how the definition of well-being developed over the course of each discipline’s history

Discussed the content of the introduction and conclusion

Agreed and informed Charmaine and Jason that we would finalise our individual discipline’s paragraphs by Sunday and meet up the next week in order to discuss the evaluation based on what we had found

Rita explained that her research for Anthropology wasn’t fruitful and decided to help Jason with Philosophy instead since she already had some ideas for it

1/12/2019
All of us uploaded our paragraphs

3/12/2019
Fifth meeting

Attendance: Charmaine, Rachel, Rita, Jason

Compared what we had found and crafted point for the evaluation, introduction and conclusion

Decided that Rachel would do the evaluation, Jason would do the conclusion and Rita would do the Introduction

Also decided that we needed to refine our paragraphs to put it on the wikibook

Jason and Rita combined their research for philosophy

5/12/2019
Rachel created the Wikipage

6/12/2019-8/12/2019
All of uploaded our paragraphs onto the Wikipage

Jason, Rachel and Rita attended ATK tutorial and presented our idea to James who approved it

Charmaine decided to combine Psychology and Biology under Neuroscience.

Rita and Rachel helped to refine Charmaine’s Neuroscience paragraph: moved part of it to Jason’s conclusion

Rita and Rachel helped to cut the words from 1500+ to 1192.

Charmaine proofread the wikipage as well as made the final overall edits of the language, sentence structures and phrasings of it.

Thanks to everyone for their work:)


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Talk:Lentis/Technology in the 2019 Hong Kong Protests

Edits at 16:31, 16:56 from 199.111.227.67 were by Michael (mis7we)


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User:LGreg/sandbox/Approaches to Knowledge (LG seminar)/Group 1/History/History of Art - Minnie and Lola

What is Historiography?
History as an issue is particularly relevant to Art History. Indeed, the way in which the arts are studied is changing significantly according to the socio-cultural paradigms of different periods. Thus Historiography studies "the writing of history, especially the writing of history based on the critical examination of sources, the selection of particular details from the authentic materials in those sources, and the synthesis of those details into a narrative that stands the test of critical examination. The term historiography also refers to the theory and history of historical writing" which means that the way artworks are studied over the years is itself the subject of study.

How and when did the History of European Art emerge?
The study of History of European Art began in the 3rd millenium BC, as a form of research into Paleolithic rock and cave painting. The discipline focuses its research into different time periods, including: Pre-History, Ancient Art, Medieval, Renaissance and Modern. Art History is studied by looking in detail at the techniques, processes and materials of the artwork, as well as the subject that is depicted.

How has it changed?
The purpose of this discipline would be to explain and male works of art understandable, according to their time. But a main conflict within the discipline is the idea that the proper technique used may not give the exact message of an artwork to the viewer. Indeed, some art historians say that History of Art is rather about the viewer's experience as opposed to the formal analysis.

In addition, philosophical question regarding the fundamentals of this discipline. What is "art" and what is not? Can an artwork be explained? Should an artwork be explained? So many questions that have challenged the discipline.

Kant's perspective about understanding art
According to Kant, it is not possible to fully undertsand the "genius" of the artist. In opposition to the scientist, the artist does not necessarly have a preconceived idea of what he will produce, nor is he able to explain it. Thus, this would make art history a discipline that tries to explain things that the creator himself could not explain. This calls into question his history of discipline explaining the works.

Freud's perspective about understanding art
For Freud, the artist usually turns away from reality or his desires are unfulfilled, which leads him to create art. The artist has a certain weakness to repress his desires, but a great ability to sublimate. Works of art are the imaginary satisfaction of unconscious desires, and art is this reserve organized during the transition from the principle of pleasure to the principle of reality: like a dream, art is a compromise. Thus, art history has a triple mode of investigation:

I- In a classic explaination of the work

II- In figures: psychoanalysis applied to the art

III- Social and historical condition: sociology applied to the art

In conclusion, History of Art engendered debates throughout over the years. Beyond the traditional description of the artwork, the discipline has had an history for being used for multiple studies (psychoanalysis, sociological...)

(By Minnie and Lola)


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User talk:Melisa.Ozturk.00

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

2A02:C7F:C030:F800:DD3:CB78:D240:CD83 (discuss) 17:13, 8 December 2019 (UTC) Maïa

title: User talk:Melisa.Ozturk.00


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

Introduction and conclusion was written by all members of the group, typed by Viky and Patrick, respectively. Viky Klein (discuss • contribs) 17:25, 8 December 2019 (UTC)


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LMIs in Control/pages/H infinity Norm for Affine Parametric Varying Systems

The System

\begin{align} \dot x(t)&=Ax(t)+B_w w(t),\\ z(t)&=C_z(\theta)x(t)+D_{zw}(\theta)w(t), \end{align}$$ where $$ C_z $$ and $$ D_{zw}$$ depend affinity on parameter $$ \theta \in \mathbb{R}^p $$.

The Data
The matrices $$ A,B_w,C_z(.),D_{zw}(.) $$.

The Optimization Problem:
Solve the following semi-definite program

\begin{align} &\min_{\{P \succ 0,\gamma \geq 0\}} \gamma \\ &\quad  s.t. \begin{bmatrix} A^\top P +PA & PB_w \\ B^\top_wP &  -\gamma^2 I \end{bmatrix}+ \begin{bmatrix} C^\top_z(\theta) \\ D^\top_{zw}(\theta) \end{bmatrix}\begin{bmatrix} C_z(\theta) & D_{zw}(\theta) \end{bmatrix}\preceq 0. \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/5462bc1dc441bc298d50a2c35075e9466eba8355

Conclusion
The value function of the above semi-definite program returns the $$ \mathcal{H}_{\infty}$$ norm of the system.

Remark
It is assumed that $$ A $$ is stable and $$ (A,B_w) $$ is controllable and the semi-infinite convex constraint $$ \|H_{\theta}(j\omega)\| < \gamma $$ for all $$ \omega \in \mathbb{R}$$, is converted into a finite-dimensional convex LMI.


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LMIs in Control/pages/Entropy Bound for Affine Parametric Varying Systems

The System

\begin{align} \dot x(t)&=Ax(t)+B_w w(t),\\ z(t)&=C_z(\theta)x(t)+D_{zw}(\theta)w(t), \end{align}$$ where $$ C_z $$ and $$ D_{zw}$$ depend affinely on parameter $$ \theta \in \mathbb{R}^p $$.

The Data
The matrices $$ A,B_w,C_z(.),D_{zw}(.) $$.

The Optimization Problem:
Solve the following semi-definite program

\begin{align} &\min_{\{P \succ 0,\gamma^2,\lambda,\theta\}} \gamma^2 \\ &\quad  s.t.\quad D_{zw}(\theta)=0, \begin{bmatrix} A^\top P +PA & PB_w & C_z(\theta)^\top \\ B^\top_wP &  -\gamma^2 I & 0 \\ C_z(\theta) & 0 & -I \end{bmatrix}\preceq 0, \quad \rm{Tr}(B^\top_w P B_w) \leq \lambda. \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/02f31a2d7a22b2464dfe9212eb76409bda9439b1

Conclusion
The value function of the above semi-definite program returns a bound for $$ \gamma $$-entropy of the system, which is defined as

\begin{align} I_{\gamma}(H_{\theta}) \triangleq \begin{cases} \frac{-\gamma^2}{2\pi}\int_{-\infty}^{\infty}\log\det(I-\gamma^2 H_{\theta}(i\omega)H_{\theta}(i\omega)^*)d\omega, \quad \text{if} \ \|H_{\theta}\|_{\infty} < \gamma \\ \infty, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text{otherwise.}\end{cases} \end{align} $$

Remark
When it is finite, $$ I_{\gamma}(H_{\theta}) $$ is given by $$ \rm{Tr}(B^\top_wPB_w)$$ where $$ P$$, is asymmetric matrix with the smallest possible maximum singular value among all solutions of a Riccati equation.


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Talk:Advanced Mathematics for Engineers and Scientists/Details and Applications of Fourier Series

Inverse of a given function
The Lagrange Inversion Theorem provides a Taylor series for the inverse of a given function. Is there an analogous theorem using Fourier series? If so, it might be included. Regards, ... PeterEasthope (discuss • contribs) 19:12, 8 December 2019 (UTC)


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LMIs in Control/pages/Modified Exterior Conic Sector Lemma

The Concept
The conic sector theorem is a powerful input-output stability analysis tool, providing a fine balance between generality and simplicity of system characterisations that is conducive to practical stability analysis and robust controller synthesis.

The System
Consider a square, contiuous-time linear time-invariant (LTI) system, $$\mathcal{G} : \mathcal{L}_{2e} \rightarrow \mathcal{L}_{2e}$$, with minimal state-space relization (A, B, C, D), where $$\mathcal{E, A} \in \mathcal{R}^{n\times n}, \mathcal{B} \in \mathcal{R}^{n\times m}, \mathcal{C} \in \mathcal{R}^{p\times n},$$ and $$\mathcal{D} \in \mathcal{R}^{p\times m} $$.

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

The Data
The matrices The matrices $$ A,B,C $$ and $$D $$

LMI : Modified Exterior Conic Sector Lemma
The system $$\mathcal{G}$$ is in the exterior cone of radius r centered at c (i.e. $$\mathcal{G} \in $$exconer(c)), where $$ r \in \mathcal{R}_{>0} $$ and $$ \in \mathcal{R}$$, under either of the following sufficient conditions.
 * 1. There exists P $$ \in \mathcal{S}^{n}$$, where P $$\ge 0$$, such that

 * $$\begin{bmatrix}

PA + A^{T}P &  PB - C^{T}(D - CI)  \\ (PB - C^{T}(D - CI))^{T} &  r^{2}I - (D - cI)^{T}(D - cI) \end{bmatrix} \le 0.$$

 * Proof. The term $$ -C^{T}C $$ in the Actual Exterior Conic Sector Lemma makes the matrix inequality more neagtive definite.

Therefore,
 * $$\begin{bmatrix}

PA + A^{T}P -C^{T}C &  PB - C^{T}(D - CI)  \\ (PB - C^{T}(D - CI))^{T} &  r^{2}I - (D - cI)^{T}(D - cI) \end{bmatrix} \le \begin{bmatrix} PA + A^{T}P &  PB - C^{T}(D - CI)  \\ (PB - C^{T}(D - CI))^{T} &  r^{2}I - (D - cI)^{T}(D - cI) \end{bmatrix}$$
 * 2. There exists P $$ \in \mathcal{S}^{n}$$, where P $$\ge 0$$, such that

 * $$\begin{bmatrix}

PA + A^{T}P &  PB - C^{T}(D - CI) & 0  \\ (PB - C^{T}(D - CI))^{T} &  - (D - cI)^{T}(D - cI) & rI \\ 0 & (rI)^{T} & -I \end{bmatrix} \le 0.$$
 * Proof. Applying the Schur complement lemma to the $$ r^{2}I $$ terms in (1) gives (2).

Conclusion:
If there exist a positive definite $$P$$ matrix satisfying above LMIs then the system $$\mathcal{G}$$ is in the exterior cone of radius r centered at c.

Implementation
Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs
KYP Lemma State Space Stability Exterior Conic Sector Lemma


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LMIs in Control/pages/Dissipativity of Affine Parametric Varying Systems

The System

\begin{align} \dot x(t)&=Ax(t)+B_w w(t),\\ z(t)&=C_z(\theta)x(t)+D_{zw}(\theta)w(t), \end{align}$$ where $$ C_z $$ and $$ D_{zw}$$ depend affinely on parameter $$ \theta \in \mathbb{R}^p $$.

The Data
The matrices $$ A,B_w,C_z(.),D_{zw}(.) $$.

The Optimization Problem:
Solve the following semi-definite program

\begin{align} &\min_{\{P \succ 0,\gamma,\theta\}} \gamma \\ &\quad  s.t.\quad \begin{bmatrix} A^\top P +PA & PB_w- C_z(\theta)^\top \\ B^\top_wP-C_z(\theta) &  2\gamma I-D_{zw}(\theta)-D_{zw}(\theta)^\top \end{bmatrix}\preceq 0. \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/b6cd6b81f75be4a2052ba3fa76cad1a2f9c49caa

Conclusion
The dissipativity of $$ H_{\theta}$$ (see [Boyd,eq:6.59]) exceeds $$ {\gamma}$$ if and only if the above LMI holds and the value function returns the minimum provable dissipativity.

Remark
It is worth noticing that passivity corresponds to zero dissipativity.


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User:RhomaAndHerPerfectionary

''Hello, everyone! RhomaAndHerPerfectionary here!'' Just a part-time editor, and that's about it right now. xD

About Me
I'm a big fan of books. I love to read them and write them. When writing books, this website's Writing Adolescent Fiction/Character names page has helped me name characters within them.

Books I'm a Fan Of

 * Beautiful Creatures
 * the WARRIORS series (It's about cats)
 * the ZODIAC series
 * Harry Potter (more into the books than the movies)
 * the Survivors and Seekers series (Survivors is about dogs, Seekers is about bears)
 * the Shadowhunters series (never seen the movies or TV shows or whatever)
 * Redwall

and...

 * the Lunar Chronicles

Trivia

 * My WARRIORS name is Buzzardlight
 * I am a Virgo, so my device in the ZODIAC series would be a Perfectionary.

title: User:RhomaAndHerPerfectionary


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LMIs in Control/pages/LMI for H2/Hinf Polytopic Controller for Robot Arm.

The System:

\begin{align} \dot x(t)&=A x(t)+B_1 w(t)+B_2 u(t) \\ z(t)&=C_1 x(t)+D_{11} w(t)+D_{12} u(t)\\ y(t)&=C_2 x(t)+D_{21} w(t)+D_{22} u(t)\\ \dot x_K(t)&=A_K x_K(t)+B_K y(t)\\ u(t)&=C_K x_K (t)+D_K y(t)\\ \end{align} $$

The Optimization Problem:
Given a state space system of

\begin{align} \dot x(t)&=A x(t)+B_1 w(t)+B_2 u(t) \\ z(t)&=C_1 x(t)+D_{11} w(t)+D_{12} u(t)\\ y(t)&=C_2 x(t)+D_{21} w(t)+D_{22} u(t)\\ \dot x_K(t)=A_K x_K(t)+B_K y(t)\\ u(t)&=C_Kx_K(t)+D_Ky(t)\end{align}$$

where $$A_K,$$ ,$$B_K,$$,$$C_K,$$ and $$D_K,$$ form the K matrix as defined in below. This, therefore, means that the Regulator system can be re-written as:

\begin{bmatrix} \dot x(t)\\

z_1(t)\\z_2(t)\\

y(t) \end{bmatrix} = \begin{bmatrix} \begin{array}{c|c c|c} A&{B}&0&{B}\\ C&D&0&D\\ 0&0&0&I\\ C&D&I&D \end{array} \end{bmatrix} \begin{bmatrix} \dot x(t)\\ w_1(t)\\w_2(t)\\ u(t) \end{bmatrix} $$ With the above 9-matrix representation in mind, the we can now derive the controller needed for solving the problem, which in turn will be accomplished through the use of LMI's. Firstly, we will be taking our $$H_2$$/$$H_\infty$$state-feedback control and make some modifications to it. More specifically, since the focus is modeling for worst-case scenario of a given parameter, we will be modifying the LMI's such that the mixed $$H_2$$/$$H_\infty$$ controller is polytopic.

The LMI:
$$H_2$$/$$H_\infty$$ Polytopic Controller for Quadrotor with Robotic Arm.

Recall that from the 9-matrix framework, $$w_1(t)$$ and $$ {w_2}(t)$$ represent our process and sensor noises respectively and $$u(t)$$ represents our input channel. Suppose we were interested in modeling noise across all three of these channels. Then the best way to model uncertainty across all three cases would be modifying the $$D$$ matrix to $$D_i$$, where ($$i=1,..,k$$ parameters, $$D_i=nI$$, and $$n$$ is a constant noise value). This, in turn results in our $$D_{11}$$-$$D_{22}$$ matrices to be modifified to $$D_{11,i}$$-$$D_{22,i}$$

Using the LMI's given for optimal $$H_2$$/$$H_\infty$$-optimal state-feedback controller from Peet Lecture 11 as reference, our resulting polytopic LMI becomes:

$$ \min\limits_{\gamma_1,\gamma_2,X_1,Y_1,Z,A_n,B_n,C_n,D_n}$$$$ \gamma^2_1$$+$$\gamma^2_2$$

\begin{align} \begin{bmatrix} AA_i&AB^T_i&AC^T_i\\ AB_i&BB_i&BC^T_i\\ AC_i&BC_i&-I \end{bmatrix}&<0\\ \begin{bmatrix} AA_i&AB^T_i&AC^T_i&AD^T_i\\ AB_i&BB_i&BC^T_i&BD^T_i\\ AC_i&BC_i&-I&CD^T_i\\ AD_i&BD_i&CD_i&{-\gamma^2_{2}}I \end{bmatrix}&<0 \\ \begin{bmatrix} Y_1&I&AD^T_i\\ I&X_1&BD^T_i\\ AD_i&BD_i&Z \end{bmatrix}&>0 \\ \end{align}$$ CD=0

$$trace(Z)<\gamma^2_1$$

where i=1,..,k,$$||S(K,P)||_{H_2}$$&$$<\gamma_1$$ and $$||S(K,P)||_{H_\infty}<\gamma_2$$ and:

$$ \begin{align} AA_i=A Y_1+ Y_1 A^T + B_2 C_n+ C^T_n B^T_2\\ AB_i=A^T+A_n+[B_2 D_n C_2]^T\\ AC_i=[B_1+B_2 D_n D_{21,i}]^T\\ AD_i=C_1 Y_1+D_{12,i}C_n\\ BB_i=X_1 A+A^T X_1+B_n C_2+C^T_2 B^T_n\\ BC_i=[X_1 B_1+B_n D_{21,i}]^T\\ BD_i=C_1+D_{12,i} D_n C_2\\ CD_i=D_{11,i}+D_{12,i} D_n D_{21,i} \end{align} $$

After solving for both the optimal $$H_2$$ and $$H_\infty$$ gain ratios as well as $${X_1},{Y_1},Z,{A_n},{B_n},{C_n},{D_n}$$, we can then construct our worst-case scenario controller by setting our $$D$$ matrix (and consequently our $${D_{11}},{D_{12}},{D_{21}},{D_{22}}$$ matrices) to the highest $$n$$value. This results in the controller:
 * $$\begin{align}

K=\begin{bmatrix}\begin{array}{c|c}{A_K}&{B_K}\\ \hline {C_K}&{D_K}\\\end{array}\end{bmatrix}\end{align} $$ which is constructed by setting:

\begin{align} &{D_K}=(I+D_{K_2}D_{22})^{-1}{D_{K_2}}\\ &{B_K}={B_{K_2}}(I+D_{22}D_{K})\\ &{C_K}=(I-D_{K}D_{22}){C_{K_2}}\\ &{A_K}={A_{K_2}}-{B_K}(I-D_{22}D_{K})^{-1}D_{22}{C_K} \end{align} $$

where:

\begin{align} &{X_2}{Y^T_2}=I-{X_1}{Y_1}\\ &\begin{bmatrix}\begin{array}{c|c}{A_{K_2}}&{B_{K_2}}\\ \hline {C_{K_2}}&{D_{K_2}}\end{array}\end{bmatrix}= \begin{bmatrix}{X_2}&{X_1}{B_2}\\0&I\end{bmatrix}^{-1} \begin{bmatrix}{A_n}-{X_1}A{Y_1}&{B_n}\\{C_n}&{D_n}\end{bmatrix} \begin{bmatrix}Y^T_2&0\\{C_2}{Y_1}&I\end{bmatrix} \end{align} $$

Conclusion:
The LMI is feasible and the resulting controller is found to be stable under normal noise disturbances for all states.


text:
LMIs in Control/pages/Hankel Norm for Affine Parametric Varying Systems

The System

\begin{align} \dot x(t)&=Ax(t)+B_w w(t),\\ z(t)&=C_z(\theta)x(t)+D_{zw}(\theta)w(t), \end{align}$$ where $$ C_z $$ and $$ D_{zw}$$ depend affinely on parameter $$ \theta \in \mathbb{R}^p $$.

The Data
The matrices $$ A,B_w,C_z(.),D_{zw}(.) $$.

The Optimization Problem:
Solve the following semi-definite program

\begin{align} &\min_{\{Q \succeq 0,\gamma^2,\theta\}} \gamma^2 \\ &\quad  s.t.\quad D_{zw}(\theta)=0, \quad  A^\top Q +QA + C_z(\theta)C_z(\theta)\preceq 0,\quad \gamma^2 I-W^{1/2}_c Q W^{1/2}_c \succeq 0, \end{align}$$ where $$ W_c$$ is the controllability Gramian, i.e., $$ W_c \triangleq \int_{0}^{\infty} e^{At}B_wB^\top_w e^{A^\top t}dt $$.

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/0faedcdd9fba92bc27a318d80159c04a0b342f35

Conclusion
The Hanakel norm (i.e., the square root of the maximum eigenvalue) of $$ H_{\theta}$$ is less than $$ {\gamma}$$ if and only if the above LMI holds and the value function returns the maximum provable Hankel norm.

Remark
$$ D_{zw}$$ is assumed to be zero.


text:
Issues in Interdisciplinarity 2019-20/Power in Film Preferences

Introduction
This article discusses power within the interdisciplinary field of marketing, using research into individual film preferences as a case study. Marketing outlines and provides logical ground for the techniques that merchants use to sell their products. Specifically, social marketing is “the adaptation...of commercial marketing...as a means to induce behavioural change in a targeted audience on a temporary or permanent basis to achieve a social goal”, which implies the need for power, or the ability to produce an effect. Therefore, due to its natural dependency on power, marketing, as a discipline, can be better understood by analyzing the power dynamics within it through exploring people’s film preferences from different political, economic, and biological backgrounds.

Why Film Preferences
Companies and associations often depend on films and television programs to advocate their ideas or products. Since marketing involves “not selling just goods and services but were also offering solutions to customer problems”, it is granted the power to provide solutions, or at least placebos, to the problems that a particular cohort of society is facing. These solutions are often biased towards a particular political, economic, or biological stance depending on where the prevailing power of society is at the time. Therefore, revealing the power dynamics in marketing would be a crucial step in order to determine the effect of marketing on people’s aesthetic values.

Biological Power in Marketing
From a biological perspective, marketing is essentially about understanding human behaviour. Similar to how ecologists need to grasp behavioural patterns and drivers of the fauna and flora in a habitat in order to understand the entire ecosystem, marketers need to consider biological factors, such as age and gender to best understand the motivations of their target audiences.

In the context of film preferences, both age and gender are demographic core factors which determine individual preferences, and therefore affect how films will be marketed. With age segmentation, marketing can be aimed at specific life cycle stages: children, adolescents, middle-age, and seniors or be generation-based: baby boomers, millennials, generation Z. Gender segmentation is relatively more straightforward, mainly distinguishing between males and females. Demographic segmentation is paramount when marketing films, as research has shown that younger moviegoers typically prefer comedy, animation, and comic book film genres, while older moviegoers prefer dramas, documentaries, and classic films. Similarly, females are typically more inclined to watch romantic genres over males, while males prefer action genres over females. Therefore, embracing adaptable marketing is critical as members of different age/gender groups usually have distinctive thought processes and attitudes towards what appeals to them. However, it is important to note that the concept of film genres is linked to the legacy of imperialism, as the concept of a film having a single ‘genre’ predominantly exists for Western films, whereas non-Western films such as Bollywood films usually mesh several different genres into one.

Overall, since marketing itself is interdisciplinary, successful marketing requires more than just demographic segmentation through a biological perspective. An example of overreliance on gender segmentation was when Doritos nearly launched “Lady Doritos” targeted at women, which received immediate online backlash for being sexist. Hence, marketing strategies must also reflect other disciplines, such as politics and economics, where factors such as political attitudes and income of consumers are studied to better identify consumer realities.

Economic Power in Marketing
From an economic perspective, it is possible to segment the population into different income groups. In marketing, it is common to have different press as representatives of different socio-economic groups, as they themselves target specific groups. For example, the readers of Financial Times or The Times might be more interested in politics or finance, whereas readers of The Sun might be more interested in celebrities and videogames (making them more favourable targeting audience for film campaigns)

Different forms of media also help marketers target different economic groups more precisely. Since dramatic or documentary film genres appeal more to older and often wealthier members of the population, specific adverts may be placed in taxis to target audiences of higher income brackets who use taxis more frequently than lower income earners. A similar phenomenon was observed in China, with the marketing of the arthouse film Long Day’s Journey Into Night. It achieved tremendous box office success by targeting audiences on the app DouYin (Tik Tok in oversea markets), which is predominantly used by “Average Joe[s]”, with a campaign which promotes its romance.

Segmenting the population also makes price discrimination possible as different socio-economic groups have different levels of expenditure. The ideal situation for marketers is to persuade everyone to pay the maximum affordable price that they consider acceptable (consumer surplus). Price discrimination is employed to maximise revenue this way. Price discrimination recognises groups and sets different prices respectively, e.g. “college students are willing to pay less than yuppies”. In the case of the film, students, veterans, and seniors are usually granted lower price to attract attendance. This low-cost strategy is usually marketed as a concession to target the groups that have lower income, while the cost is still significantly lower.

Political Power in Marketing
One of the main ideas in marketing is to respect the norm, or the appropriate behavior, of the society, which is largely influenced by the government as “deviations from norms are punished by sanctions”. Therefore, “all films are political” because they will follow the norms of the targeted country, which supports a particular political view. For instance, after the Civil Rights Movements in the United States, many Civil Rights Melodrama are streamed on television, including the television program Any Day Now. This television program focuses on the two female protagonists, one from each race, in a family setting and uses their daily life to represent the United States during the movement. This program reflects the United States government’s will to “heal racism” through trivializing the matter of racial discrimination to “…the individual and the family rather than on social movements and institutions”. Therefore, the success of Any Day Now can be contributed partly to its “political correctness”. On the other hand, when a film fails to fit into the political norm, it is likely that it would be resisted by the government and will never make it to the screen in a particular country. Zhangke Jia’s A Touch of Sin is banned from screening in China because of its focus on violence done by unsatisfied labourers. It is banned because it portrays the negative aspects of the government system, which is against the social norm in China.

Conclusion
Ever since Lumiere’s first screening of Sortie d’usine, films have been regarded as objective means of presenting “living portraits”. However, when talking about mainstream films that aim to make it to the cinema, that objectivity no longer exists due to marketing. The interplay of power between different disciplines in marketing has shaped individual film preferences by appealing to different biological cohorts of the population through demographic segmentation, use of different advertising based on income segmentation and adaptation to the political climate of the target country. Therefore, as shown from viewpoints in biology, economics and politics, the understanding of marketing is not complete without looking at it interdisciplinarily.