text:
LMIs in Control/pages/Optimal Output Feedback H2 LMI

Optimal Output Feedback $$H_2$$ LMI
Similar to state feedback, output feedback is necessary when information about the output is not known. Often techniques such as Kalman filtering are implemented to tackle this problem. The method below, however, does not use a filtering technique and instead uses a combination of LMI constraints to perform the output feedback as well as find the minimal bound on the $$H_2$$ norm of the system. $$H_2$$ is often done using more classical tools such as Riccati equations. More recently LMI techniques have been created to solve problems such as full state feedback or output feedback as seen below.

The System
The system is represented using the 9-matrix notation shown below.
 * $$\begin{bmatrix} \dot{x}\\z\\y \end{bmatrix}=\begin{bmatrix} A & B_{1} & B_{2}\\ C_{1} & D_{11} & D_{12}\\ C_{2} & D_{21} & D_{22} \end{bmatrix} \begin{bmatrix} x\\w\\u \end{bmatrix}

$$ where $$x(t)\in \R^n$$ is the state, $$z(t)\in \R^p$$ is the regulated output, $$y(t)\in \R^q$$ is the sensed output, $$w(t)\in \R^r$$ is the exogenous input, and $$u(t)\in \R^m$$ is the actuator input, at any $$t\in \R$$.

The Data
$$A$$, $$B_{1}$$, $$B_{2}$$, $$C_{1}$$, $$C_{2}$$, $$D_{11}$$, $$D_{12}$$, $$D_{21}$$, $$D_{22}$$ are known.

The LMI: Optimal Output Feedback $$H_2$$ Control LMI
The following are equivalent.

1) There exists a $$\hat{K}=\begin{bmatrix} A_K & B_K \\ C_K & D_K \end{bmatrix}$$ such that $$||S(K,P)||_{H_2}<\gamma$$

2) There exists $$X_1$$, $$Y_1$$, $$Z$$, $$A_n$$, $$B_n$$, $$C_n$$, $$D_n$$ such that

\begin{bmatrix} AY_{1}+Y_{1}A^{\text{T}}+B_{2}C_{n}+C_{n}B_{2}^{\text{T}} & *^{\text{T}} & *^{\text{T}} \\ A^{\text{T}}+A_{n}+(B_{2}D_{n}C_{2})^{\text{T}} & X_{1}A+A^{\text{T}}+B_{n}C_{2}+C_{2}^{\text{T}}B_{n}^{\text{T}} & *^{\text{T}}\\ (B_{1}+B_{2}D_{n}D_{21})^{\text{T}} & (X_{1}B_{1} + B_{n}D_{21})^{\text{T}} & -\gamma I \\ \end{bmatrix} < 0 $$

\begin{bmatrix} Y_{1}  & *^T & *^T \\ I & X_{1} & *^T\\ C_{1}Y_{1}+D_{12}C_{n} & C_{1}+D_{12}D_{n}C_{2} & Z \end{bmatrix} > 0 $$

D_{11} + D_{12}D_{n}D_{21} = 0 $$

\text{trace}(Z) < \gamma^{2} $$

Conclusion:
The above LMI determines the the upper bound $$\gamma$$ on the H2 norm. In addition to this the controller $$\hat{K}(A_K,B_K,C_K,D_K)$$ can also be recovered.
 * $$ D_K=(I+D_{K2}D_{22})^{-1}D_{K2}$$
 * $$ B_K=B_{K2}(I+D_{22}D_K)$$
 * $$ C_K=(I+D_{K}D_{22})C_{K2}$$
 * $$ A_K=A_{K2}-B_K(I+D_{22}D_K)^{-1}D_{22}C_K$$

where,
 * $$\begin{bmatrix} A_{K2} & B_{K2} \\ C_{K2} & D_{K2} \end{bmatrix}= {\begin{bmatrix} X_{2} & X_{1}B_2 \\ 0 & I \end{bmatrix}}^{-1} \left[\begin{bmatrix} A_n & B_n \\ C_n & D_n \end{bmatrix}-\begin{bmatrix} X_1AY_1 & 0 \\ 0 & 0 \end{bmatrix}\right] {\begin{bmatrix} Y_2^T & 0 \\ C_2Y_1 & I \end{bmatrix}}^{-1}

$$ for any full-rank $$X_2$$ and $$Y_2$$ such that
 * $$\begin{bmatrix} X_1 & X_2 \\ X_2^T & X_3 \end{bmatrix}= {\begin{bmatrix} Y_1 & Y_2 B_2 \\ Y_2^T & Y_3\end{bmatrix}}^{-1}

$$.

Implementation
This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/OF_H2.m

Related LMIs
Optimal Output Feedback Hinf


text:
Talk:Issues in Interdisciplinarity 2019-20/Evidence in Religious Experience

A lot of work was done in a shared Google Document as well as in the Wikibook itself.

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


text:
LMIs in Control/pages/Mixed H2 Hinf optimal state feedback control

Mixed $$H_\infty/H_2$$ Optimal State Feedback Control

The Optimization Problem
This Optimization problem involves the same process used on the full-feedback control design; however, instead of optimizing the full output-feedback design of the Optimal output-feedback control design. This is done by defining the 9-matrix plant as such: $$ A \in \mathbb{R}^{m \times m} $$, $$ B_1 \in \mathbb{R}^{m \times n} $$, $$ B_2 \in \mathbb{R}^{m \times p} $$, $$ C_1 \in \mathbb{R}^{n \times m} $$, $$ D_{11} \in \mathbb{R}^{n \times n} $$, $$ D_{12} \in \mathbb{R}^{n \times p} $$, $$ C_2 = I $$, $$ D_{21} = 0 $$, and $$ D_{22} = 0 $$. Using this type of optimization allows for stacking of optimization LMIs in order to achieve the controller synthesis for both $$H_\infty $$ and $$ H_2$$.

The Data
The data is dependent on the type the state-space representation of the 9-matrix plant; therefore the following must be known for this LMI to be calculated: $$ A \in \mathbb{R}^{m \times m} $$, $$ B_1 \in \mathbb{R}^{m \times n} $$, $$ B_2 \in \mathbb{R}^{m \times p} $$, $$ C_1 \in \mathbb{R}^{n \times m} $$, $$ D_{11} \in \mathbb{R}^{n \times n} $$, $$ D_{12} \in \mathbb{R}^{n \times p} $$, $$ C_2 = I $$, $$ D_{21} = 0 $$, and $$ D_{22} = 0 $$.

The LMI: Mixed $$H_\infty/H_2$$ Optimal State Feedback Control
There exists the scalars $$ \gamma_1 $$, $$ \gamma_2 $$, along with the matrices $$ X > 0 $$, $$ W $$ and $$ Z $$ where:

\begin{align}

||S(P,K(0,0,0,F))||^{2}_{H_\infty} + ||S(P,K(0,0,0,F))||^{2}_{H_2} \leq \gamma_1 + \gamma_2 \\    \\     \begin{bmatrix} X A^T + Z^T B^{T}_2 + A X + B_2 Z & B_1 & X C^{T}_1 + Z^{T} D^{T}_{12} \\ B1^{T}_1 & -I & D^{T}_{11} \\ C_1 X + D_{12} Z & D_{11} & -\gamma_1 I    \end{bmatrix}&<0\\ \\    \\     trace W < \gamma_2 \\    \\     A X + X A^T + B_2 Z + Z' B_2'+B_1 B^{T}_{1} < 0 \\    \\     \begin{bmatrix} X & (C_1 X + D_{12} Z)^{T} \\ C_1 X + D_{12} Z & W \\ \end{bmatrix}&>0\\

\end{align}$$

Where $$ F = Z X^{-1} $$ is the controller matrix.

Conclusion:
The results from this LMI give a controller that is a mixed optimization of both an $$ H_\infty $$, and $$ H_2 $$ optimization.

Implementation

 * Hinf/H2 Mixed state-feedback optimization example


text:
LMIs in Control/pages/quadratic polytopic h2 optimal state feedback control

Quadratic Polytopic Full State Feedback Optimal $$H_2$$ Control
For a system having polytopic uncertainties, Full State Feedback is a control technique that attempts to place the system's closed-loop system poles in specified locations based on performance specifications given, such as requiring stability or bounding the overshoot of the output. By minimizing the $$H_2$$ norm of this system we are minimizing the effect noise has on the system as part of the performance specifications.

The System
Consider System with following state-space representation.

\begin{align} \dot x(t)&=Ax(t)+B_1 q(t) + B_2 w(t)\\ p(t)& = C_1 x(t) + D_{11} q(t) + D_{12} w(t)\\ z(t)& = C_2 x(t) + D_{21} q(t) + D_{22} w(t)\\ \end{align}$$

where $$x\in\mathbb{R}^{m}$$, $$q\in\mathbb{R}^{n}$$ , $$w\in\mathbb{R}^{g}$$, $$A\in\mathbb{R}^{mxm}$$, $$B_1\in\mathbb{R}^{mxn}$$, $$B_2\in\mathbb{R}^{mxg}$$, $$p\in\mathbb{R}^{p}$$ , $$C_1\in\mathbb{R}^{pxm}$$, $$D_{11}\in\mathbb{R}^{pxn}$$,  $$D_{12}\in\mathbb{R}^{pxg}$$, $$z\in\mathbb{R}^{s}$$, $$C_2\in\mathbb{R}^{sxm}$$, $$D_{21}\in\mathbb{R}^{sxn}$$ ,  $$D_{22}\in\mathbb{R}^{sxg}$$ for any $$t\in\mathbb{R}$$.

Add uncertainty to system matrices

A, B_1, B_2, C_1, C_2, D_{11}, D_{12} $$

New state-space representation

\begin{align} \dot x(t)&=(A+A_i)x(t)+(B_1+B_i) q(t) + (B_2+B_i) w(t)\\ p(t)& = (C_1+C_i) x(t) + (D_{11}+D_i) q(t) + (D_{12}+D_i) w(t)\\ z(t)& = C_2 x(t) + D_{21} q(t) + D_{22} w(t)\\ \end{align}$$

The Data
The matrices necessary for this LMI are

The Optimization Problem:
Recall the closed-loop in state feedback is: $$ S(P,K) = $$

\begin{align} \begin{bmatrix} A + B_{22}F&&B_1\\C_1 + D_{12}F&& D_{11}\end{bmatrix}\\ \end{align}$$ This problem can be formulated as $$ H_2 $$ optimal state-feedback, where K is a controller gain matrix.

The LMI: An LMI for Quadratic Polytopic $$ H_2 $$ Optimal
State-Feedback Control $$ ||S(P(\Delta),K(0,0,0,F))||_{H_2} \leq \gamma $$ $$ X > 0 $$

\begin{align} \begin{bmatrix} AX + B_2Z + XA^T + Z^TB^T_{2} &&B_1 \\B_1^T&&-I \end{bmatrix} +

\begin{bmatrix} A_iX + B_{2,i}Z + XA^T_i + Z^TB^T_{2,I} &&B_{1,i} \\B_{1,i}^T&&0 \end{bmatrix} < 0 \quad i = 1,......,k

\end{align}$$

\begin{align} \begin{bmatrix} X && (C_1X+D_{12}Z)^T \\C_1X+D_{12}Z && W \end{bmatrix} +

\begin{bmatrix} 0 && (C_{1,i}X+D_{12,i}Z)^T \\C_{1,i}X+D_{12,i}Z&&0 \end{bmatrix} > 0 \quad i = 1,......,k

\end{align}$$

\begin{align} \\TraceW < \gamma \end{align}$$

Conclusion:
The $$ H_2 $$ Optimal State-Feedback Controller is recovered by $$ F = ZX^{-1} $$

Implementation:
https://github.com/JalpeshBhadra/LMI/blob/master/H2_optimal_statefeedback_controller.m

Related LMIs
$ H_2 $ Optimal State-Feedback Controller


text: Category:Subject:University College London projects/all books


text:
LMIs in Control/pages/Quadratic polytopic stabilization

A Quadratic Polytopic Stabilization Controller Synthesis can be done using this LMI, requiring the information about $$A$$,$$\Delta_{A(t)} $$ ,$$B$$ and $$\Delta_{B(t)} $$ matrices.

The System

\begin{align} \dot x(t)&=Ax(t)+Bu(t),\\ x(0)&=x_0, \end{align}$$ where $$x(t)\in \R^n$$, $$u(t)\in \R^m$$, at any $$t\in \R$$. The system consist of uncertainties of the following form

\begin{align} \Delta_{A(t)} = A_1 \delta_1 (t)+....+A_k \delta_k (t)\\ \Delta_{B(t)} = B_1 \delta_1 (t)+....+B_k \delta_k (t)\\ \end{align}$$

where $$x\in\mathbb{R}^{m}$$,$$u\in\mathbb{R}^{n}$$,$$A\in\mathbb{R}^{mxm}$$ and $$B\in\mathbb{R}^{mxn}$$

The Data
The matrices necessary for this LMI are $$A$$,$$\Delta_{A(t)} \, ie \, A_i $$ ,$$B$$ and $$\Delta_{B(t)} \, ie \, B_i $$

The Optimization and LMI:LMI for Controller Synthesis using the theorem of Polytopic Quadratic Stability
There exists a K such that

\begin{align} \dot x(t)&=(A+\Delta_A+(B+\Delta_B)K)x(t)\\ \end{align}$$ is quadratically stable for $$ (\Delta_A,\Delta_B) \in C_0 ((A_1,B_2),...,(A_k,B_k)) $$ if and only if there exists some P>0 and Z such that

\begin{align} (A+A_i)P + P(A+A_i)^T + (B+B_i)Z+Z^T(B+B_i)^T < 0 \quad for \quad i = 1,...k. \end{align}$$

Conclusion:
The Controller gain matrix is extracted as $$ K = ZP^{-1} $$ Note that here the controller doesn't depend on $$ \Delta $$
 * If you want K to depend on $$ \Delta $$, the problem is harder.
 * But this would require sensing $$ \Delta $$ in real-time.

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

Related LMIs
Quadratic Polytopic $H_{\infty}$ Controller Quadratic Polytopic $H_{2}$ Controller


text:
LMIs in Control/pages/H stabilization

$$\mathbb{H}_{(\alpha,\beta)}$$-Stabilization

There are a wide variety of control design problems that are addressed in a wide variety of different ways. One of the most important control design problem is that of state feedback stabilization. One such state feedback problem, which will be the main focus of this article, is that of $$\mathbb{H}_{(\alpha,\beta)}$$-Stabilization, a form of $$\mathbb{D}$$-Stabilization where the real components are located on the left-half of the complex plane.

The System
For this particular problem, suppose that we were given a linear system in the form of:

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

where $$x\in\mathbb{R}^n$$ and $$u\in\mathbb{R}^r$$. Then the LMI for determining the $$\mathbb{H}_{(\alpha,\beta)}$$-stabilization case would be obtained as described below.

The Data
In order to obtain the LMI, we need the following 2 matrices: $$A\text{ and }B$$.

The Optimization Problem
Suppose - for the linear system given above - we were asked to design a state-feedback control law where $$u=Kx$$ such that the closed-loop system:

\begin{align} \dot x&=(A+BK)x\\ \end{align}$$ is $$\mathbb{H}_{(\alpha,\beta)}$$ stable, then the system would be stabilized as follows.

The LMI: $$\mathbb{H}_{(\alpha,\beta)}$$-Stabilization
From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a matrix $$W$$ and a symmetric matrix $$P>0$$ that satisfy the following:

\begin{align} \begin{cases} AP+P{A^T}+BW+{W^T}{B^T}+2{\alpha}P &<0\\ -AP-P{A^T}-BW-{W^T}{B^T}-2{\beta}P &<0 \end{cases} \end{align}$$

Conclusion:
Given the resulting controller matrix $$K=WX^{-1}$$, it can be observed that the matrix is $$\mathbb{H}_{(\alpha,\beta)}$$-stable.

Implementation

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

Related LMIs

 * ../D stabilization/ - Equivalent LMI for $$\mathbb{D}_{(q,r)}$$-stabilization.

title: LMIs in Control/pages/H stabilization


text:
LMIs in Control/pages/D stabilization

$$\mathbb{D}_{(q,r)}$$-Stabilization

There are a wide variety of control design problems that are addressed in a wide variety of different ways. One of the most important control design problem is that of state feedback stabilization. One such state feedback problem, which will be the main focus of this article, is that of $$\mathbb{D}_{(q,r)}$$-Stabilization, a form of $$\mathbb{D}$$-Stabilization where the closed-loop poles are located on the left-half of the complex plane.

The System
For this particular problem, suppose that we were given a linear system in the form of:

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

where $$x\in\mathbb{R}^n$$, $$u\in\mathbb{R}^r$$, and $$\rho$$ represents either the differential operator (in the continuous-time case) or the one-step forward operator (for the discrete-time system case). Then the LMI for determining the $$\mathbb{D}_{(q,r)}$$-stabilization case would be obtained as described below.

The Data
In order to obtain the LMI, we need the following 2 matrices: $$A\text{ and }B$$.

The Optimization Problem
Suppose - for the linear system given above - we were asked to design a state-feedback control law where $$u=Kx$$ such that the closed-loop system:

\begin{align} \rho x&=(A+BK)x\\ \end{align}$$ is $$\mathbb{D}_{(q,r)}$$-stable, then the system would be stabilized as follows.

The LMI: $$\mathbb{D}_{(q,r)}$$-Stabilization
From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a matrix $$W$$ and a symmetric matrix $$P$$ that satisfies the following:

\begin{align} \begin{bmatrix} -rP&&qP+AP+BW\\qP+PA^{T}+{W^T}{B^T}&&-rP \end{bmatrix} <0 \end{align}$$

Conclusion:
Given the resulting controller matrix $$K=WP^{-1}$$, it can be observed that the matrix is $$\mathbb{D}_{(q,r)}$$-stable.

Implementation

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

Related LMIs

 * ../H stabilization/ - Equivalent LMI for $$\mathbb{H}_{(\alpha,\beta)}$$-stabilization.
 * ../Continuous Time D-Stability Controller/ - LMI for deriving a Controller using D-Stability.
 * ../Continuous Time D-Stability Observer/ - LMI for deriving an Observer using D-Stability.

title: LMIs in Control/pages/D stabilization


text:
Talk:History of wireless telegraphy and broadcasting in Australia/Topical/Publications/Australian Radio History/Qld AM Stations

"with an increase to 180 lines on 136 metres" Unclear if lines means channels or metres mean meter band or meter distance.

--76.95.239.212 (discuss) 06:04, 6 December 2019 (UTC)


text:
LMIs in Control/pages/Quadratic Schur Stabilization

LMI for Quadratic Schur Stabilization A discrete-time system is said to be stable if all roots of its characteristic equation lie in the open unit disk. This provides a condition for the stability of discrete-time linear systems with polytopic uncertainties and a linear time-invariant system with this property is called a Schur stable system.

The System
Consider discrete time system

\begin{align} x_{k+1}=Ax_k+Bu_k,\\ \end{align}$$ where $$x_k\in \R^n$$, $$u_k\in \R^m$$, at any $$t\in \R$$. The system consist of uncertainties of the following form

\begin{align} \Delta_{A(t)} = A_1 \delta_1 (t)+....+A_k \delta_k (t)\\ \Delta_{B(t)} = B_1 \delta_1 (t)+....+B_k \delta_k (t)\\ \end{align}$$

where $$x\in\mathbb{R}^{m}$$,$$u\in\mathbb{R}^{n}$$,$$A\in\mathbb{R}^{mxm}$$ and $$B\in\mathbb{R}^{mxn}$$

The Data
The matrices necessary for this LMI are $$A$$,$$\Delta_{A(t)} \, ie \, A_i $$ ,$$B$$ and $$\Delta_{B(t)} \, ie \, B_i $$

The LMI:
There exists some X > 0 and Z such that

\begin{align} \begin{bmatrix} X && AX+BZ \\ (AX+BZ)^T && X \end{bmatrix} +

\begin{bmatrix} 0 && A_iX + B_iZ \\(A_iX + B_iZ)^T&&0 \end{bmatrix} > 0 \quad i = 1,......,k

\end{align}$$

The Optimization Problem
The optimization problem is to find a matrix $$\begin{align} K \in \mathbb{R}^{r\times n} \end{align}$$ such that:

$$\begin{align} ||A + BK||_{2} < \gamma \end{align}$$

According to the definition of the spectral norms of matrices, this condition becomes equivalent to:

$$\begin{align} (A + BK)^{T}(A + BK) < \gamma^{2}I \end{align}$$

Using the Lemma 1.2 in LMI in Control Systems Analysis, Design and Applications (page 14), the aforementioned inequality can be converted into:

$$\begin{align} \begin{bmatrix} -\gamma I & (A + BK) \\ (A + BK)^{T} & -\gamma I\end{bmatrix} < 0\\ \end{align}$$

Conclusion:
The Controller gain matrix is extracted as $$ F = ZX^{-1} $$ $$ u_k = Fx_k $$

\begin{align} x_{k+1}=Ax_k+Bu_k,\\ \quad\quad\quad = Ax_k + BFx_k\\ \quad\quad = (A+BF)x_k \end{align}$$

It follows that the trajectories of the closed-loop system (A+BK) are stable for any $$ \, \Delta \, \in \, C_0(\Delta_1,...,\Delta_k) $$

Implementation
https://github.com/JalpeshBhadra/LMI/blob/master/quadratic_schur_stabilization.m

Related LMIs
Schur Complement Schur Stabilization


text:
LMIs in Control/pages/Quadratic DStabilization

Continuous-Time D-Stability Controller

This LMI will let you place poles at a specific location based on system performance like rising time, settling time and percent overshoot, while also ensuring the stability of the system.

The System
Suppose we were given the continuous-time system

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

whose stability was not known, and where $$A\in\mathbb{R}^{mxm}$$, $$B\in\mathbb{R}^{mxn}$$, $$C\in\mathbb{R}^{pxm}$$, and $$D\in\mathbb{R}^{qxn}$$ for any $$t\in\mathbb{R}$$.

Adding uncertainty to the system

\begin{align} \dot x(t)&=(A+A_i)x(t)+(B+B_i)u(t)\\ \end{align}$$

The Data
In order to properly define the acceptable region of the poles in the complex plane, we need the following pieces of data:

 * matrices $$A$$, $$B$$, $$A_i$$, $$B_i$$
 * rise time ($$t_r$$)
 * settling time ($$t_s$$)
 * percent overshoot ($$M_p$$)

Having these pieces of information will now help us in formulating the optimization problem.

The Optimization Problem
Using the data given above, we can now define our optimization problem. In order to do that, we have to first define the acceptable region in the complex plane that the poles can lie on using the following inequality constraints:

Rise Time: $$\omega_n{\leq}{1.8 \over t_r}$$

Settling Time: $$\sigma{\leq}{-4.6 \over t_s}$$

Percent Overshoot: $$\sigma{\leq}{-ln({M_p}) \over {\pi}}|{\omega_d}|$$

Assume that $$z$$ is the complex pole location, then:

\begin{align} {\omega_n}^2=\|z\|^2&=z^{*} z\\ {\omega_d}=Im{z}&={(z-z^{*}) \over 2}\\ {\sigma}=Re{z}&={(z+z^{*}) \over 2} \end{align}$$

This then allows us to modify our inequality constraints as:

Rise Time: $$z^{*} z-{1.8^2 \over {t_r}^2}{\leq}0$$

Settling Time: $${(z+z^{*}) \over 2}+{4.6 \over t_s}{\leq}0$$

Percent Overshoot: $$z-z^{*}+{{\pi} \over ln({M_p})}|z+z^{*}|{\leq}0$$

which not only allows us to map the relationship between complex pole locations and inequality constraints but it also now allows us to easily formulate our LMIs for this problem.

The LMI: An LMI for Quadratic D-Stabilization
Suppose there exists $$ X > 0 $$ and $$ Z $$ such that

\begin{align} \begin{bmatrix} -rP && AP+BZ \\ (AP+BZ)^T && -rP \end{bmatrix} +

\begin{bmatrix} 0 && A_iP + B_iZ \\(A_iP + B_iZ)^T&&0 \end{bmatrix} < 0 \\

AP+BZ+(AP+BZ)^T + A_iP+B_iZ+(A_iP+B_iZ)^T +2\alpha P&<0, and\\

\end{align}$$

\begin{align} \begin{bmatrix} AP+BZ+(AP+BZ)^T && c(AP+BZ-(AP+BZ)^T) \\ c((AP+BZ)^T -(AP+BZ)) && AP+BZ+(AP+BZ)^T\end{bmatrix} +

\begin{bmatrix} A_iP+B_iZ+(A_iP+B_iZ)^T && c(A_iP+B_iZ-(A_iP+B_iZ)^T) \\c((A_iP+B_iZ)^T -(A_iP+B_iZ))&& A_iP+B_iZ+(A_iP+B_iZ)^T \end{bmatrix} < 0 \\

\end{align}$$

for $$ i = 1,...,k $$

Conclusion:
Given the resulting controller $$K=ZP^{-1}$$, we can now determine that the pole locations $$z\in\mathbb{C}$$ of $$A(\Delta)+B(\Delta)K$$ satisfies the inequality constraints $$|x|{\leq}r$$, $$Re(x){\leq}-{\alpha}$$ and $$z+z^{*}{\leq}-c|z-z^{*}|$$ for all $$ \,  \Delta \, \in \, C_0(\Delta_1,...,\Delta_k) $$

Implementation
The implementation of this LMI requires Yalmip and Sedumi https://github.com/JalpeshBhadra/LMI/blob/master/quadraticDstabilization.m

Related LMIs

 * ../Continuous Time D-Stability Observer/ - Equivalent D-stability LMI for a continuous-time observer.


text:
LMIs in Control/pages/NI-Lemma

Positive real systems are often related to systems involving energy dissipation. But the standard Positive real theory will not be helpful in establishing closed-loop stability. However transfer functions of systems with a degree more than one can be satisfied with the negative imaginary conditions for all frequency values and such systems are called "systems with negative imaginary frequency response" or "negative imaginary systems".

The System
Consider a square continuous time Linear Time invariant system, with the state space realization $$(A,B,C,D)$$

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

The Data
$$A\in\mathbb R^{n\times n}, B\in\mathbb R^{n\times m}, C\in\mathbb R^{m\times n}, D\in\mathbb S^m$$

The LMI: LMI for Negative Imaginary Lemma
According to the Lemma, The aforementioned system is negative imaginary under either of the following equivalent necessary and sufficient conditions

 * There exists a $$P$$ ∈ $$\begin{align}\mathbb{S}\end{align}$$n ,where $$P \geq 0$$, such that,

\begin{bmatrix} A^TP + PA & PB-A^TC^T \\ B^TP-CA & -(CB+B^TC^T) \end{bmatrix}$$$$\begin{align} \leq 0\end{align}$$

 * There exists a $$Q$$ ∈ $$\begin{align}\mathbb{S}\end{align}$$n ,where $$Q \geq 0$$, such that,

\begin{bmatrix} QA^T + AQ & B-QA^TC^T \\ B^T- CAQ & -(CB+B^TC^T) \end{bmatrix}$$$$\begin{align} \leq 0\end{align}$$

Conclusion
The system is strictly negative if det($$A$$) $$\neq$$ 0 and either of the above LMIs are feasible with resulting $$P>0$$ or $$Q>0$$

Implementation
A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/ygovada

Related LMIs
Positive Real Lemma

title: LMIs in Control/pages/NI-Lemma


text:
User:LGreg/sandbox/Approaches to Knowledge (LG seminar)/Group1/Power/Power of the Unconscious within Behavioral Economics

Behavioral economics
While decision making is unconsciously affected by emotions or external factors, we can perceive humans as unpredictable actors as opposed to classic economists assumption that humans are rational decision makers. Behavioural economists tend to anticipate and understand irrational decisions by dissecting the unconscious decision making process through a psychological lens, the aim being to understand how a consumer’s brain proceeds in order to adjust marketing strategies.

Unconscious consumption patterns
Accordingly, several patterns have been highlighted by behavioral economists through experiments. As per to David Brooks in The Social Animal, and Dan Ariely, individuals proceed to a choice architecture before taking decisions, relying on few principles. One of these principles notably illustrates that we make choices based on comparison. He also stated that ‘Every decision gets framed within a certain linguistic context’. Indeed, if the formula ‘Limited Units’ appears next to a product, the consumer is likely to buy more units than he would normally. Likewise, the environment in which we consume (odours, music...) is also highly influential. Therefore, depending on our unconscious emotional state, external factors can seriously influence if not control economic choices.

Overall, the concept of unconsciousness provides economists and market researchers with a new perspective to understand consumer behaviors, empowering them to better coordinate the market and the economy.


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LMIs in Control/pages/DCGain

The continuous-time DC gain is the transfer function value at the frequency s = 0.

The System
Consider a square continuous time Linear Time invariant system, with the state space realization $$(A,B,C,D)$$ and $$\gamma \in\mathbb{R}_{>0}$$

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

The Data
$$A\in\mathbb{R}^{n\times n}, B\in\mathbb{R}^{n\times m}, C\in\mathbb{R}^{p\times n}, D\in \mathbb{R}^{p\times m}$$

The LMI: LMI for DC Gain of a Transfer Matrix
The transfer matrix is given by$$G(s)=C(sI-A)^{-1}B+D$$ The DC Gain of the system is strictly less than $$\gamma$$ if the following LMIs are satisfied.

\begin{bmatrix} \gamma I & -CA^{-1}B+D \\ (-CA^{-1}B+D)' & \gamma I \end{bmatrix}$$$$\begin{align} > 0\end{align}$$ OR

\begin{bmatrix} \gamma I & -B^TA^{-T}C^T+DT \\ (-B^TA^{-T}C^T+DT)' & \gamma I \end{bmatrix}$$$$\begin{align} > 0\end{align}$$

Conclusion
The DC Gain of the continuous-time LTI system, whose state space realization is give by ($$A,B,C,D$$), is $$K = D - CA^{-1}B$$
 * Upon implementation we can see that the value of $$\gamma$$ obtained from the LMI approach and the value of $$K$$ obtained from the above formula are the same

Implementation
A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/LMI_wikibooks

title: LMIs in Control/pages/DCGain


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

Introduction
Opioids are used as painkillers, commonly prescribed in the US for patients suffering of extreme pain. When people talk about the opioid epidemic, they are referring to a variety of behaviours and events related to the abuse of prescription painkillers and/or Opium-derived illicit substances. The abuse of prescription Opioids (e.g. Hydrocodone, Oxycodone, Heroin, and et cetera), hospitalisations, overdoses, and deaths are all aspects of the Opioid epidemic currently affecting the 2.1 million Americans with an Opioid addiction. An average of 115 Americans die each day from Opioid-related overdose.

Power of opioids over the human body
It is first important to consider the power opioids have over the human brain. A scientific definition of addiction is "a psychological and physical inability to stop consuming a chemical, drug, activity, or substance, even though it is causing psychological and physical harm.”

This definition 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 dependancies and may negatively effect their entire lifestyle.

Opioids trigger the release of endorphins, ones brain's "feel-good" neurotransmitters. Endorphins muffle ones perception of pain and boost feelings of pleasure, creating a temporary but powerful sense of euphoria. When an opioid dose wears off, one may find themselves craving this euphoria. This is the first milestone on the path toward serious addiction.

When opioids bind to opioid receptors, they create a potassium conductance, which makes a neuron less likely to fire an action potential, and they inhibit calcium conductance, which makes a neuron less likely to release neurotransmitters. What effect these actions have depends on where the receptors are found. When opioid receptors on neurons in the spinal cord that carry information about pain are activated, it can inhibit the transmission of pain information to the brain.

Opioids can also act on descending pain modulatory systems to eliminate the pain. For example, by binding to receptors in an area of the brainstem called the periaqeuductal gray, opioids can prompt the inhibition of pain signalling in the spinal cord via a pathway that first synapses in the medulla.

The result of the biological effect of regular opioid abuse is physical dependency. In the article published by psychiatrists Thomas R. Kosten and Tony P. George, the power opioids have to catalyse a physical dependence is explained:

"Opioid tolerance, dependence, 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. Medications such as methadone, LAAM, buprenorphine, and naltrexone act on the same brain structures and processes as addictive opioids, but with protective or normalising effects. Despite the effectiveness of medications, they must be used in conjunction with appropriate psychosocial treatments."

Many individuals who find themselves physically addicted to opioids then seek rehabilitation. However, treatments are often expensive ("Well-known centers often cost up to $20,000 for a 30-day program" ), physically demanding, and long term. The treatment includes managing the physical pain associated with withdrawals as well as the psychological consequences of addiction. In addition, the relapses are frequent ; 40 to 60% of the patients in a drug addiction relapse.

Lastly, as the addiction starts with prescriptions, the patient will seek more and more as the opioids' necessity gets more powerful. 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. This leads to death by overdose or even sicknesses like HIV from the injections.

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

While addiction statistics and scientific explanations have established the opioid epidemic as an issue that warrants government intervention, American pharmaceutical companies have been known to lobby state senators as well as congressmen in order to maintain high profits from opioid sales.

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 certain individuals in the House such as Paul Ryan receiving donations totalling approximately 230,000. 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 (ref). 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 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 and preserve their wealth.

Power of individuals over the opioids crisis
The second point worth concerning is the efforts artists were contributing to combat opioid epidemic. 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. The opioid addiction recovery helper, Nancy, 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 is.; Aside from art forms, social media is another powerful tool for combating opioid epidemic. For instance, the non-profit, state and donor supported organization pages primarily on facebook and instagram, got 250,000 visits in March, publishing contents related to research data and looking for reinforcing the social norm about opioid uses. The contents are created by students.; In the American music industry, Macklemore realised “Drug Dealer”, a new song and music video, referencing some celebrity death by having opioid. With that, he met President Obama to talk about this important issue. The official music video on Youtube has more than forty million views and seventeen thousand comments. The power of celebrity on combating opioid epidemic is definitely unstoppable.
 * Overall, people in the U.S. society are using their power (i.e. getting access to the public media or has their power as an artist) to combat opioid epidemic and it turns out the achieve really good results.


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English-Hanzi/Montenegro

Montenegro is the latest member to join NATO on June 5, 2017. So far, NATO has a total of 29 countries in Europe and North America.

黑山共和国是2017年6月5日加入北约的最新成员. 到目前为止，北约在欧洲和北美共有29个国家.

title: English-Hanzi/Montenegro


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User talk:46.226.205.126

Blocked
You cannot edit for 3 days due to disruptive editing. You are welcome to contribute constructively when editing privileges are restored. You may appeal this action with a reasonable argument by adding the text below. —Atcovi (Talk - Contribs) 15:39, 6 December 2019 (UTC)

title: User talk:46.226.205.126


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English-Hanzi/Albania

Albania is a country in southeastern Europe on the north coast of the Mediterranean Sea.

阿尔巴尼亚是欧洲东南部的一个国家，位于地中海的北海岸.

title: English-Hanzi/Albania


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

On the 19th of November, we met up to discuss the subject of our wikibook page. Finding it was difficult for us, as we had an array of ideas in mind. Our initial 3 subjects were 'the justification of colonization', using disciplines such as history, anthropology, economics, medecine (eugenics), religion ('mission civilisatrice') and litterature (Heart of Darkness by Joseph Conrad), 'marketing and the establishment of gender norms' with sociology, anthropology, economics and psychology, and finally 'mental health', with medicine, anthropology, epistemology and phenomenology. However, we quickly realized that these subjects were not the right fit for this assignment, and that it was going to be difficult to link them with issues.

We then thought about power and language, i.e. the power disparities language can create. Using the following disciplines: Despite how interesting this subject looked, it was too vague and we were struggling to find a common theme within it.
 * Linguistics : the establishment of koiné and creole languages + diglossia in post-colonial societies
 * Sociology : the importance of public speaking and knowing the vocabulary of the upper-class to be able to move up in the social structure.
 * Litterature : How female authors published their work with a male pseudonym to be taken seriously
 * Economic

Finally, Julie came up with the idea of the causes of homosexuality. Saumarkb (discuss • contribs) 03:07, 9 December 2019 (UTC)Amélie Clark

On the 1st of December, following clarifying that our topic was to be 'Evidence in the Causes of Homosexuality', we confirmed how we would contribute by assigning disciplines to each other. Amelie chose to write on 'Genetics', whilst I chose 'Behavioural Psychology' and Julie selected 'Endocrinology'. We all decided that we would then contribute equally to the fourth discipline in our Wikibook - Theology. Saumarkb (discuss • contribs) 03:07, 9 December 2019 (UTC)Oni Bhaumick

My first version of the "epigenetics" section was over 400 words. However, I read through some previous 'Evidence' wikibooks and realized that I was only summarizing and explaining the theories of the disciplines, and not actually looking at the way they used evidence. I decided to start over and cut my whole section. I did reuse some sentences in my second version, but found myself leaving many concepts that I had learned about out of the final version (which was disappointing because I had initially struggled to understand Saumarkb (discuss • contribs) 03:07, 9 December 2019 (UTC)Amélie

The first draft of the Endocrinology section being over 600 words long, I cut parts of it to reach the required word count limit. I decided to make away with the descriptions of the experiments referenced, as simply an analysis of the form of evidence and theory derived from it seemed sufficient, and the experiments themselves can be explored in the references section. -Julie Cavalera

As we were both struggling to find theological causes of homosexuality and meeting the word limit in our individual sections, we all decided to give up on the discipline of Theology. Saumarkb (discuss • contribs) 03:07, 9 December 2019 (UTC)Amélie

I wrote the basic structure of the introduction and conclusion. Saumarkb (discuss • contribs) 03:07, 9 December 2019 (UTC)Amélie

We initially decided that we would be focusing in the realm of Behavioural Psychology as one of the disciplines discussed. However, after research into this discipline and the main theory (Exotic Becomes Erotic) in this section, I decided (with consultation from the group) to change to Developmental Psychology. This rendered the bulk of my research into Behavioural Psychology and its viewpoint on our topic useless. However, I started looking into Developmental Psychology and found out the discipline's research methodologies and their flaws. Saumarkb (discuss • contribs) 03:07, 9 December 2019 (UTC)Oni Bhaumick

After reading our final few drafts for the Wikibook - I contributed to some changes in the 'conclusion' section and introduction. In the final section, I mentioned how different disciplines can take positivist or interpretivist approaches leading to evidence being presented in different ways. In the introduction, I added a sentence to reflect the prompt of our task. Saumarkb (discuss • contribs) 03:07, 9 December 2019 (UTC)Oni Bhaumick


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User:LBird BASc/sandbox/ATK/Seminar6/Power/Power in Language

Language is an aspect of humanity that has existed in human cultures for millenia. It is also an aspect of culture that perhaps holds the most power, considering how it is our means of communication and alters the way we think. Consequently, if a single entity has the power to manipulate a common language, it would hence have the power to control the minds of those who speak the language. This can be seen in cases where a single entity holds totalitarian authority over a large group of people, which is demonstrated both in literature and history.

In the novel 1984 by George Orwell, for example, the entity known as 'Big Brother', holding absolute authority in the fictional realm of 'Oceania', creates a new language called 'Newspeak', which, having limited vocabulary and grammar, is imposed to limit the freedom of thought within the population of Oceania. It also creates concepts such as 'thoughtcrime' to terrorise the population and prevent a potential uprising against the status quo. This diminishes the people's ability to communicate effectively and honestly, which solidifies the power of the status quo. It is therefore an established theory that the power in language is significant enough to alter thoughts in a group of people.

An example in the real world can be seen in cases where the colonisation of a country by another resulted in the restriction/elimination of the indigenous language, in attempts to gain control of the subject population. An example of this can be seen in Japan's expansion and dominance across South East Asia in the early 20th century. Once the Japanses gained political control over countries, such as in Korea, the use of the native language was greatly restricted and in some casese made illegal, instead being replaced by Japanese. This was done in order to establish the dominance of Japanese power, getting rid of a significant cultural element in language. This was successful to a large extent, as the native people would lose the ability to communicate in a manner that invoked a sense of nationalism and unity. This goes to show that language is a powerful tool that, if controlled and manipulated, could be a significant tool to establish power over a large group of people.


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English-Hanzi/Bulgaria

Bulgaria is a country located on the Balkan Peninsula in southeastern Europe.

保加利亚是位于东南欧巴尔干半岛上的国家.

title: English-Hanzi/Bulgaria


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LMIs in Control/Robustness/Continuous Time

 * Robust Stabilization of Second-Order Systems
 * Robust Stabilization of $ H{\infty} $ Optimal State Feedback Control


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LMIs in Control/Controller Synthesis/Continuous Time/Optimal State Feedback/H-infinity

 * Full State Feedback Optimal Control (Hinf)
 * LMI for L2-Optimal State-Feedback Control under Time-Varying Input Delay


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LMIs in Control/Controller Synthesis/Continuous Time/Optimal State Feedback/H-2

 * Full State Feedback Optimal Control (H2)


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LMIs in Control/Controller Synthesis/Continuous Time/Optimal Output Feedback/H-2

 * Optimal Output Feedback Control (H2)


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LMIs in Control/Controller Synthesis/Continuous Time/Optimal Output Feedback/H-infinity

 * Optimal Output Feedback Control (Hinf)


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LMIs in Control/Stability Analysis/Lure's Systems

 * Stability of Lure's Systems
 * L2 Gain of Lure's Systems
 * Output Energy Bound for Lure's Systems


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LMIs in Control/pages/Dissipativity of Systems

Dissipativity of Systems

The dissipativity of systems is associated with their supply function. In general, a linear system is dissipative if the accumulated sum (integration) of the supply function is non-negative over all the duration of $$ T \geq 0 $$.

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)+Du(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. A, B, C and D are system coefficient matrices of appropriate dimensions. The control input u is restricted to be a piece-wise continuous vector function defined of $$[0,\infty)$$.

The transfer function of such a system can be evaluated as:

\begin{align} G(s)=C(sI-A)^{-1}B+D \end{align}$$

For such a system, a general quadratic supply function is defined as:

\begin{align} s(u,y) &= \begin{bmatrix} y & u \end{bmatrix} Q \begin{bmatrix} y \\ u \end{bmatrix} \end{align}$$

where Q is a real symmetric matrix of (m+r) dimensions. Q need not be either symmetric positive or negative definite.

The Data
Number of states n, number of outputs m and number of control inputs r need to be known. Moreover, the system matrices A,B,C,D are also required to be known. The system should also be controllable.

The Feasibility LMI
The system $$ G(s) $$ defined can be evaluated to be dissipative with respect to a supply function $$ s(u,y) $$ iff there exist $$P \geq 0 $$ and a $$Q$$ (defining $$s(u,y)$$) such that the following is feasible:

\begin{align} \text{Find} \; & P, Q:\\ & P \geq 0 \\ \begin{bmatrix} A^\top P + PA & PB \\ B^\top P     & 0 \end{bmatrix} & - \begin{bmatrix} C & D \\ 0 & I \end{bmatrix}^\top Q \begin{bmatrix} C & D \\ 0 & I \end{bmatrix} \leq 0. \end{align}$$

Conclusion:
If there is a feasible solution to the aforementioned LMI, then there exists a supply function $$ s(u,y) $$ for which the system $$ G(s) $$ is dissipative. Since the assumption of the system being controllable is required for it to be dissipative, this check can be used of as a sufficient condition to check the controllability of the linear system, just like the feasibility for Lyapunov stability.

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/Dissipativity_example.m

Related LMIs
Continuous_Time_Lyapunov_Inequality


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LMIs in Control/pages/L2 gain of Lure systems

The System

\begin{align} \dot x(t)&=Ax(t)+B_pp(t)+B_ww(t),\\ z(t)&=C_zx(t)\\ p_i(t)&=\phi_i(q_i(t)), i=1,\dots, n_p\\ q&=C_qx,\\ 0 &\leq \sigma \phi_i(\sigma) \leq \sigma^2 \ \forall \sigma \in \mathbb{R} \end{align}$$

The Data
The matrices $$ A,B_p,B_w,C_q,C_z $$.

The Optimization Problem:
The following semi-definite problem should be solved.

\begin{align} &\min_{\{P \succ 0, \Lambda=diag(\lambda_1,\dots,\lambda_{n_p})\succeq 0,T=diag(\tau_1,\dots,\tau_{n_p}) \succeq 0\}} \gamma^2 \; \\ &\quad \quad \quad \quad \quad \quad \quad \quad  s.t. \quad \begin{bmatrix} A^\top P+PA+C^\top_zC_z & PB_p+A^\top C^\top_q \Lambda +C^\top_q T & PB_w \\ B^\top_p P + \Lambda C_qA+TC_q & \Lambda C_q B_p+B^\top_p C^\top_q \Lambda-2T & \Lambda C_q B_w \\ B^\top_wP & B^\top_w C^\top_q\Lambda & -\gamma^2I \end{bmatrix} \preceq 0\\ \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/12a7039f9e3d966e24b43fd58a3cce3725282ed2

Conclusion
The value function returns the square of the smallest provable upper bound on the $$ \mathcal{L}_2 $$ gain of the Lure's system.

Remark
The Lyapunov function which is used to proof is similar to the one for the systems with unknown parameters.


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LMIs in Control/pages/Projection Lemma

WIP, Description in progress

A condition for eliminating a variable in an LMI using orthogonal complements is presented.

Definition 1: Orthogonal Complements
Let $$A\in \mathbb{R}^{m\times n} $$, Then, $$M_a$$ is called a left orthogonal complement of $$A$$ if it satisfies

$$ M_aA=0, \quad \text{rank}(M_a)=m-\text{rank}(A) $$;

and $$N_a$$ is called a right orthogonal complement of $$A$$ if it satisfies

$$ AN_a=0, \quad \text{rank}(N_a)=n-\text{rank}(A) $$.

Using the definition of orthogonal complements, we have the following projection lemma:

Projection Lemma
Let $$P$$, $$Q$$ and $$H=H^T$$ be given matrices of appropriate dimensions, $$N_p$$ and $$N_q$$ be the right orthogonal complement of $$P$$ and $$Q$$, respectively. Then, there exists $$X$$ such that

$$H+P^TX^TQ+Q^TXP<0$$

if and only if

$$N_p^TP^T=0, \quad N_q^TQ^T=0 $$.

WIP, additional references to be added

title: LMIs in Control/pages/Projection Lemma


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LMIs in Control/pages/Output Energy Bounds for Lure System

The System

\begin{align} \dot x(t)&=Ax(t)+B_pp(t)+B_ww(t),\\ z(t)&=C_zx(t)\\ p_i(t)&=\phi_i(q_i(t)), i=1,\dots, n_p\\ q(t)&=C_qx(t),\\ 0 &\leq \sigma \phi_i(\sigma) \leq \sigma^2 \ \forall \sigma \in \mathbb{R} \end{align}$$

The Data
The matrices $$ A,B_p,B_w,C_q,C_z,x(0) $$.

The Optimization Problem:
The following optimization problem should be to find the tightest upper bound for the output energy of the above Lur'e system.

\begin{align} &\min_{P \succ 0,\Lambda=diag(\lambda_1,\dots,\lambda_{n_p})\succeq 0,T=diag(\tau_1,\dots,\tau_{n_p}) \succeq 0} x^\top(0)(P+C^\top_q \Lambda C_q)x(0)\\ & \quad \quad \quad \quad \quad \quad \quad \begin{bmatrix} A^\top P+PA & PB_p+A^\top C^\top_q \Lambda +C^\top_q T \\ B^\top_p P + \Lambda C_qA+TC_q & \Lambda C_q B_p+B^\top_p C^\top_q \Lambda-2T \end{bmatrix} \preceq 0\\ \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/blob/master/LMIs%20for%20Output%20Energy%20Bounds%20of%20Lure's%20Systems

Conclusion
The value function returns the the lowest bound for the energy function of the Lure's systems, i.e., $$ J=\int_{0}^{\infty} z^\top z \ dt $$ with initial conditions $$ x(0) $$.

Remark
The key step in the proof is to satisfy $$ \frac{d}{dt}V(x)+z^\top z \leq 0 $$, where $$ V(.)$$ is Lyapunov function in a special form.


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LMIs in Control/pages/Switched systems Hinf Optimization

Switched Systems $$H_\infty$$ Optimization

The Optimization Problem
This Optimization problem involves the use of the State-feedback plant design, with the difference of optimizing the system with a system matrix which "switches" in properties during optimization. This is similar to considering the system with variable uncertainty; an example of this would be polytopic uncertainty in the matrix $$ A $$. which ever matrix is switching states, there must be an optimization for both cases using the same variables for both.

This is first done by defining the 9-matrix plant as such: $$ A \in \mathbb{R}^{m \times m} $$, $$ B_1 \in \mathbb{R}^{m \times n} $$, $$ B_2 \in \mathbb{R}^{m \times p} $$, $$ C_1 \in \mathbb{R}^{n \times m} $$, $$ D_{11} \in \mathbb{R}^{n \times n} $$, $$ D_{12} \in \mathbb{R}^{n \times p} $$, $$ C_2 = I $$, $$ D_{21} = 0 $$, and $$ D_{22} = 0 $$. Using this type of optimization allows for stacking different LMI matrix states in order to achieve the controller synthesis for $$H_\infty $$.

The Data
The data is dependent on the type the state-space representation of the 9-matrix plant; therefore the following must be known for this LMI to be calculated: $$ A \in \mathbb{R}^{m \times m} $$, $$ B_1 \in \mathbb{R}^{m \times n} $$, $$ B_2 \in \mathbb{R}^{m \times p} $$, $$ C_1 \in \mathbb{R}^{n \times m} $$, $$ D_{11} \in \mathbb{R}^{n \times n} $$, $$ D_{12} \in \mathbb{R}^{n \times p} $$, $$ C_2 = I $$, $$ D_{21} = 0 $$, and $$ D_{22} = 0 $$. What must also be considered is which matrix or matrices will be "switched" during optimization. This can be denoted as $$ A(\delta) $$.

The LMI: Switched Systems $$H_\infty$$ Optimization
There exists a scalar $$ \gamma $$, along with the matrices $$ X > 0 $$ and $$ Z $$ where:

\begin{align}

||S(P,K(0,0,0,F))||_{H_\infty} \leq \gamma \\    \\     \begin{bmatrix} X A(\delta)^T + Z^T B^{T}_2 + A(\delta) X + B_2 Z & B_1 & X C^{T}_1 + Z^{T} D^{T}_{12} \\ B1^{T}_1 & -\gamma I & D^{T}_{11} \\ C_1 X + D_{12} Z & D_{11} & -\gamma I    \end{bmatrix}&<0\\

\end{align}$$

Where $$ F = Z X^{-1} $$ is the controller matrix. This also assumes that the only switching matrix is $$ A $$; however, other matrices can be switched in states in order for more robustness in the controller.

Conclusion:
The results from this LMI gives a controller gain that is an optimization of $$ H_\infty $$ for a switched system optimization.

Implementation

 * Hinf switched state-feedback optimization example


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Lentis/China’s Social Credit System

"“Allow the trustworthy to roam everywhere under heaven while making it hard for the discredited to take a single step” - Planning Outline for the Construction of a Social Credit System, 2014"

Introduction
The Chinese Social Credit System is a system in development by the Chinese government to assess the economic and social standing of its citizens and businesses. The Chinese government announced it in 2014 with the Planning Outline for the Construction of a Social Credit System with the program beginning in 2020.

The Chinese government aims to have this system create a public, quantitative measure of how various citizens and businesses are acting in accordance with the government's behavioral standards. The credit system is intended to protect the rights and interests of the citizens, and to provide common ethical standards for Chinese citizens to strive for. The system also aims to deter dishonest, unpatriotic, or criminal behavior by providing a reward and punishment system relative to a given individual or businesses social credit score.

Current Implementation
While the nationwide program has not formally begun, in the time since its announcement the People’s Bank of China partnered with 8 companies to implement pilot systems to get a sense of how the 2020 version of the system can be implemented successfully. At the forefront of these pilot programs is a large system called Sesame Credit owned by the Alibaba Group, a multinational conglomerate holding company. Each company handles separate cities and provinces to implement and test their respective private Social Credit Systems.

Sesame Credit
Sesame Credit is owned by the Alibaba Group and developed by the Ant Financial Services Group. It spans over five fields: credit history, online behavior, use of products and services, personal characteristics, and interactions and relationships with others. The score is modeled similarly to the American FICO credit score. The data is tracked by using Alibaba services, like Alipay, for monitoring. There are 300 million registered users and 37 million small businesses that use these services. They state that all of the data is encrypted and segregated and can only be accessed through the authorization of the subject.

Future Implementation
In 2020, the Chinese government plans to remove all of the pilot programs and make one unified system where all 1.7 billion citizens, including businesses, are stored in a centralized database to be scored and accredited based upon all of the programs previously created.

Functionality
Increasing and Decreasing Score Users' score varies from day to day increasing and decreasing for the smallest of actions to make sure they are acting in the form of a model citizen. A short example list for both include:

Increase Decrease
 * Donating to Charity
 * Blood Donations
 * Purchasing Diapers
 * Paying bills on time
 * Cleaning/upkeep business storefront
 * Taking care of your pet properly
 * Praising the government
 * Helping the poor
 * Illegally crossing the street
 * Smoking in prohibited areas
 * Missing Bill Payments
 * Improperly sorting disposables
 * Blocking passageways in highly congested areas (eg: airports)
 * Attending rallies/riots
 * Purchasing alcohol
 * Playing video games for extended periods
 * Not visiting elderly relatives often enough

Benefits and Punishments
Depending on social credit score, individuals may gain access to various benefits, or, in cases of extremely low scores, receive punishments. Benefits

 * Higher Exposure on dating applications compared to normal users
 * Rentals without deposits
 * Better interest rates at banks
 * Decreased heating/electricity bills
 * Priority patient in hospitals
 * Priority children’s school admissions
 * Faster work promotions
 * Tax Breaks
 * Free Starbucks Drinks

Punishments

 * Banned from planes/trains
 * Throttling of internet speeds
 * Banning of persons/children of persons from best schools
 * In 2018, a student’s acceptance into a Chinese university was revoked because of his father’s low social credit score
 * His father failed to pay $29,900 he owed to a bank
 * Denial of jobs
 * Inability to book hotels
 * Loss of pet-owning rights
 * Inability to purchase using WeChat
 * Public blacklist
 * Displayed in movie theater before movie starts
 * Billboard public when cross walk
 * Restricted public services

Cameras
Users are tracked through the credit system through not just their accounts, but also through surveillance. Currently, there are an estimated 200 million cameras installed in China, compared to 50 million in the United States, to keep watch over its citizens and that number is expected to increase to 626 million in 2020. Along with a vast camera system, China is also developing "cutting edge" cameras such as a 500 megapixel one that can produce clear images of human faces in high traffic areas. These cameras can be connected to an identification system that allows for facial recognition to find out what users are doing, and identify vehicles and people that break the law, such as not stopping at crosswalks are jaywalking which can result in deductions to their social credit score.

Effect
The surveillance system has allowed the police to arrest find and arrest 6,000 suspects in 2016. This system has also been used to follow not only citizens but also tourist and stores them into the police database also to watch out for disturbances.

Smart Glasses
There are also advanced smart glasses that will/can have access to the government/company database to use facial identification technology to analyze and identify citizens to help the police see citizen records and identify criminals.

Spy Birds
in 2018, China has also added a surveillance technology being bird drones equipped with an HD camera, GPS, an electric motor to flap its wings, and a flight control system.

Messaging Super app
China’s SuperApp, WeChat, is an app that allows users to be able to do almost anything from simple messaging to online and in-person payments, food delivery, ticket purchasing for movies and travel, and even games. The app was created by one of the largest companies in the world, Tencent, who was even one of the eight companies that the government contracted to help implement a private version of the social credit system. Tencent claims one billion daily active users and WeChat accounts for market penetration in China at a staggeringly 79% which makes it the largest application in China’s market.

In May of 2014, all types of messaging apps such as WeChat, are required to allow the government to monitor the activity and information of its users to remove malpractice of “rumors and information leading to violence, terrorism, and pornography.”

Chinese Reception
Social credit systems have been highly approved of in China. They are viewed as a technological improvement meant to benefit society and citizens are not especially concerned with their privacy as many are under the assumption that the government already is storing their data.

A study conducted at the University of Berlin surveyed over 2000 Chinese citizens on their social credit system involvement. 80% of the Chinese population has taken part in some sort of commercially available social credit system. 80% of the population also approves of using social credit systems while only 20% are either neutral or disapprove and distrust them. Systems run by the central government were used by less than 8% of users as commercially available systems currently are more developed, but will likely see more users in the future with improvement.

There is some variation across certain demographics. Individuals who have a higher income, live in urban areas, and have higher levels of education tend to show more approval of social credit systems. This could be attributed to the increase in opportunities and benefits that come from social credit involvement for individuals in these demographics.

It should be noted, however, that this study’s data was obtained through an online questionnaire that may have been under Chinese government supervision and could have influenced results.

United States
Public opinion of Chinese social credit systems in the United States has been disapproving. The media has labeled these systems as a dystopian and nightmarish, describing them as analogous to popular culture references such as George Orwell’s “1984”  and Black Mirror. Words like “rank”, “grade”, “judge” and other similar words are often found in articles discussing how the systems work. In 2015, the ACLU published an article demonstrating the fear of implementation and potential abuse of such a system in the United States.

In the US, these systems have been called “social credit systems”, carrying a connotation similar to that of the "credit score", assessing one's ability to repay debt. However, a more literal translation from Chinese may be “public trust system” which reflects the goals of the system as outlined by the Chinese Government in a 2014 document.

Uighur
Uighur, a Chinese Muslim group, are primarily under fire from the Chinese government. The government has been exposed to using facial recognition software, proxy by cameras, along with using DNA recognition to detain these citizens and place them in concentration camps.

Christians
The Christian community has had churches shut down for reasons such as having unlawful gatherings and being uncooperative in allowing the government to install cameras in their church. Pastors have also been jailed for times up to 15 years, and the government has a plan to promote Chinese Christianity which calls for a rewriting of the Bible.

Case Study: Opponent of the Government
Liu Hu, a previously largely popular detective in China, lost a case where he accused an official of extortion. The punishment was to publish a public apology and pay a fine. The court then asked for an extra fee, but when he refused, Hu was then blacklisted and now is under effective house arrest as he can not apply for a high-speed train to travel quickly anymore. His social media accounts, where he posted most of his journalism, have also been shut down.

Life Insurance
In February 2019, The New York Department of Financial Services announced that life insurance companies would be able to base premiums off of customers' social media posts. The goal of this was to streamline this process by replacing in-person lifestyle questionnaires with an automated algorithm.

Patronscan
Patronscan is a company that creates ID verification systems. Their products can record incidents that occur under a specific identity and puts those names in a database. Patronscan customers can then view this list and prevent individuals with a record from entering events requiring identification.

Netflix
Netflix takes user watch history data and attempts to recommend specific shows and movies that the user might be interested in.

Uber
Uber allows for drivers to rate their riders and makes it visible to other drivers which could influence riders’ ability to obtain a ride. In 2015 Uber announced that riders with a rating significantly below average could be banned from using Uber.

Conclusions
China has been developing a social credit system for a more efficient and harmonious society. This system will monitor all Chinese citizens and assign them a corresponding score that translates to certain benefits or punishments. Reception in China appears to be positive but has been negative in other countries, such as the United States. Yet, despite the disapproval, similar systems already exist outside of China to streamline certain processes. As China continues to proceed with its social credit plan, whether or not other countries follow suit will depend on their societal values. It is the balance between efficiency and privacy. Those willing to sacrifice privacy for a more streamlined society may be willing to take part in such a system, while those who value privacy more may not.

title: Lentis/China’s Social Credit System


text:
LMIs in Control/pages/Switched systems H2 Optimization

Switched Systems $$H_2$$ Optimization

The Optimization Problem
This Optimization problem involves the use of the State-feedback plant design, with the difference of optimizing the system with a system matrix which "switches" in properties during optimization. This is similar to considering the system with variable uncertainty; an example of this would be polytopic uncertainty in the matrix $$ A $$. which ever matrix is switching states, there must be an optimization for both cases using the same variables for both.

This is first done by defining the 9-matrix plant as such: $$ A \in \mathbb{R}^{m \times m} $$, $$ B_1 \in \mathbb{R}^{m \times n} $$, $$ B_2 \in \mathbb{R}^{m \times p} $$, $$ C_1 \in \mathbb{R}^{n \times m} $$, $$ D_{11} \in \mathbb{R}^{n \times n} $$, $$ D_{12} \in \mathbb{R}^{n \times p} $$, $$ C_2 = I $$, $$ D_{21} = 0 $$, and $$ D_{22} = 0 $$. Using this type of optimization allows for stacking different LMI matrix states in order to achieve the controller synthesis for $$H_\infty $$.

The Data
The data is dependent on the type the state-space representation of the 9-matrix plant; therefore the following must be known for this LMI to be calculated: $$ A \in \mathbb{R}^{m \times m} $$, $$ B_1 \in \mathbb{R}^{m \times n} $$, $$ B_2 \in \mathbb{R}^{m \times p} $$, $$ C_1 \in \mathbb{R}^{n \times m} $$, $$ D_{11} \in \mathbb{R}^{n \times n} $$, $$ D_{12} \in \mathbb{R}^{n \times p} $$, $$ C_2 = I $$, $$ D_{21} = 0 $$, and $$ D_{22} = 0 $$. What must also be considered is which matrix or matrices will be "switched" during optimization. This can be denoted as $$ A(\delta) $$.

The LMI: Switched Systems $$H_2$$ Optimization
There exists a scalar $$ \gamma $$, along with the matrices $$ X > 0 $$, $$ W $$, and $$ Z $$ where:

\begin{align}

||S(P,K(0,0,0,F))||^{2}_{H_2} \leq \gamma \\    \\     trace W < \gamma \\    \\     A X + X A^T + B_2 Z + Z' B_2'+B_1 B^{T}_{1} < 0 \\    \\     \begin{bmatrix} X & (C_1 X + D_{12} Z)^{T} \\ C_1 X + D_{12} Z & W \\ \end{bmatrix}&>0\\

\end{align}$$

Where $$ F = Z X^{-1} $$ is the controller matrix. This also assumes that the only switching matrix is $$ A $$; however, other matrices can be switched in states in order for more robustness in the controller.

Conclusion:
The results from this LMI gives a controller gain that is an optimization of $$ H_2 $$ for a switched system optimization.

Implementation

 * H2 switched state-feedback optimization example


text:
LMIs in Control/pages/Reciprocal Projection Lemma

WIP, Description in progress

Another condition for eliminating a variable in an LMI called reciprocal projection lemma is introduced.

Reciprocal Projection Lemma
For a given symmetric matricx $$\Phi\in \mathbb{S}^n$$, there exists a matrix $$S\in \mathbb{R}^{n\times n}$$ satisfying

$$\Phi +S^T+ S < 0$$

if and only if, for an arbitrarily fixed symmetric matrix $$P\in \mathbb{S}^n $$, there exist a matrix $$W\in \mathbb{R}^{n\times n}$$ satisfying

$$ \begin{bmatrix} \Phi + P - (W^T + W) & S^T + W^T \\ S + W & -P \end{bmatrix} < 0 $$.

WIP, additional references to be added


text:
LMIs in Control/pages/Discrete Time QSR

The Concept
In systems theory the concept of dissipativity was first introduced by Willems which describes dynamical systems by input-output properties. Considering a dynamical system described by its state $$ x(t) $$, its input $$u(t)$$ and its output $$y(t)$$, the input-output correlation is given a supply rate $$ w(u(t),y(t))$$. A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function $$ V(x(t))$$ such that $$V(0)=0$$, $$V(x(t))\ge 0 $$ and

 * $$ \dot{V}(x(t)) \le w(u(t),y(t))$$

As a special case of dissipativity, a system is said to be passive if the above dissipativity inequality holds with respect to the passivity supply rate $$ w(u(t),y(t)) = u(t)^Ty(t) $$.

The physical interpretation is that $$V(x)$$ is the energy stored in the system, whereas $$w(u(t),y(t))$$ is the energy that is supplied to the system.

This notion has a strong connection with Lyapunov stability, where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions.

Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear and nonlinear systems. Dissipative systems theory has been discussed by Vasile M. Popov, Jan Camiel Willems, D.J. Hill, and P. Moylan. In the case of linear invariant systems, this is known as positive real transfer functions, and a fundamental tool is the so-called Kalman–Yakubovich–Popov lemma which relates the state space and the frequency domain properties of positive real systems.Dissipative systems are still an active field of research in systems and control, due to their important applications.

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

 * $$x(k+1)=\mathcal{A}_{d}x(k)+\mathcal{B}_{d}u(k)$$
 * $$y(k)=\mathcal{C}_{d}x(k)+\mathcal{D}_{d}u(k), k=0,1... $$

The Data
The matrices $$ \mathcal{A}_{d},\mathcal{B}_{d},\mathcal{C}_{d} $$ and $$\mathcal{D}_{d}$$

The Optimization Problem
The system $$\mathcal{G}$$ is QSR disipative if

 * $$\sum_{i \mathop =0}^K (\mathcal{y}_{i}^TQ\mathcal{y}_{i} + 2\mathcal{y}_{i}^TS\mathcal{u}_{i} + \mathcal{u}_{i}^TR\mathcal{u}_{i})\,dt \ge 0, \forall u \in \mathcal{l}_{2e}, \forall k \in \mathcal{Z} \ge 0 $$

where $$\mathcal{u}_{k}$$ is the input to $$\mathcal{G}, \mathcal{y}_{k}$$ is the output of $$\mathcal{G}, Q \in \mathcal{S}^{p}, S \in \mathcal{R}^{p\times m},$$ and $$\mathcal{R} \in \mathcal{S}^{m}$$.

LMI : Discrete-Time KYP Lemma for QSR Dissipative Systems
The system $$\mathcal{G}$$ is also QSR dissipative if and only if there exists $$ P \in \mathcal{S}^{n},$$ where $$ P > 0,$$ such that

 * $$\begin{bmatrix}

A^{T}_{d}PA_{d} - P - C^{T}_{d}QC_{d} & A^{T}_{d}PB_{d} - C^{T}_{d}S - C^{T}_{d}QD_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d}S - C^{T}_{d}QD_{d})^{T} &  B^{T}_{d}PB_{d} - D^{T}_{d}QD_{d} - (D^{T}_{d}S + S^{T}D_{d}) - R \end{bmatrix}\le 0.$$

Conclusion:
If there exist a positive definite $$P$$ for the the selected Q,S and R matrices then the system $$\mathcal{G}$$ is QSR dissipative.

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

Related LMIs
KYP Lemma KYP Lemma for continous Time QSR Dissipative system

title: LMIs in Control/pages/Discrete Time QSR


text:
LMIs in Control/pages/Generalized Lyapunov Theorem

WIP, Description in progress

The theorem can be viewed as a true essential generalization of the well-known continuous- and discrete-time Lyapunov theorems.

Kronecker Product
The Kronecker Product of a pair of matrices $$A\in \mathbb{R}^{m\times n} $$ and $$B \in \mathbb{R}^{p \times q} $$ is defined as follows:

$$A\otimes B=\begin{bmatrix} a_{11}B & a_{12}B & \cdots & a_{1n}B \\ a_{21}B & a_{22}B & \cdots & a_{2n}B \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1}B & a_{m2}B & \cdots & a_{mn}B \end{bmatrix}\in \mathbb{R}^{mp\times nq} $$.

Lemma 1: Manipulation Rules of Kronecker Product
Let $$ A, B, C$$ be matrices with appropriate dimensions. Then, the Kronecker product has the following properties:
 * $$1\otimes A = A$$;
 * $$(A+B)\otimes C = A\otimes C + B\otimes C$$
 * $$(A\otimes B)(C\otimes D)=(AC)\otimes(BD)$$
 * $$(A\otimes B)^T = A^T \otimes B^T$$
 * $$(A\otimes B)^{-1} = A^{-1} \otimes B^{-1}$$
 * $$\lambda(A\otimes B)={\lambda_i(A)\lambda_j{B}}$$

Theorem
In terms of Kronecker products, the following theorem gives the $$\mathbb{D}$$-stability condition for the general LMI region case: Let $$\mathbb{D}=\mathbb{D}_{L,M}$$ be an LMI region, whose characteristic function is

$$F_{\mathbb{D}}=L+sM+\overline{s}M^T$$

Then, a matrix $$A\in \mathbb{R}^{n \times n}$$ is $\mathbb{D}_{L,M}$-stable if and only if there exists symmetric positive definite matrix $$P$$ such that

$$R_{\mathbb{D}}(A, P)=L\otimes P+M\otimes (AP) +M^T\otimes(AP)^T < 0$$,

where $$\otimes$$ represents the Kronecker product.

Lemma 2
Given two LMI regions $$\mathbb{D}_1$$ and $$\mathbb{D}_2$$, a matrix $$A$$ is both $$\mathbb{D}_1$$-stable and $$\mathbb{D}_2$$-stable if there exists a positive definite matrix $$P$$, such that $$R_{\mathbb{D}_1}(A, P)<0$$ and $$R_{\mathbb{D}_2}(A, P)<0$$.

$$$$ WIP, additional references to be added


text:
LMIs in Control/pages/Discrete Time KYP Lemma without Feedthrough

The Concept
It is assumed in the Lemma that the state-space representation (A, B, C, D) is minimal. Then Positive Realness (PR) of the transfer function C(SI − A)-1B + D  is equivalent to the solvability of the set of LMIs given in this page. Consider now the following scalar example, where (A, B, C, D)=(−α, 0, 0, 1), with α > 0. The transfer function is H(s) = 0 that is PR

The System
Consider a contiuous-time LTI system, $$\mathcal{G} : \mathcal{L}_{2e} \rightarrow \mathcal{L}_{2e}$$, with minimal state-space relization (A, B, C, 0), where $$\mathcal{A} \in \mathcal{R}^{n\times n}, \mathcal{B} \in \mathcal{R}^{n\times m}, $$ and $$\mathcal{C} \in \mathcal{R}^{m\times n}, $$.

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

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

LMI : KYP Lemma without Feedthrough
The system $$\mathcal{G}$$ is positive real (PR) under either of the following equivalet necessary and sufficient conditions.

 * 1. There exists $$ P \in \mathcal{S}^{n}, $$ where $$ p > 0 $$ such that

 * $$\begin{align}

PA + A^{T}P \ge 0 \\ PB = C^{T} \end{align}$$

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

 * $$\begin{align}

AQ + QA^{T} \ge 0 \\ B = QC^{T} \end{align}$$

This is a special case of the KYP Lemma for QSR dissipative systems with Q = 0, Q = 0.5 and R = 0.

The system $$\mathcal{G}$$ is strictly positive real (SPR) under either of the following equivalet necessary and sufficient conditions.

 * 1. There exists $$ P \in \mathcal{S}^{n}, $$ where $$ p > 0 $$ such that

 * $$\begin{align}

PA + A^{T}P < 0 \\ PB = C^{T} \end{align}$$

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

 * $$\begin{align}

AQ + QA^{T} < 0 \\ B = QC^{T} \end{align}$$

This is a special case of the KYP Lemma for QSR dissipative systems with Q = ε1, Q = 0.5 and R = 0. where ε $$ \in \mathcal{R}_{>0}.$$

Conclusion:
If there exist a positive definite $$P$$ for the the selected Q,S and R matrices then the system $$\mathcal{G}$$ is Positive Real.

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

Related LMIs
KYP Lemma State Space Stability


text:
LMIs in Control/pages/KYP Lemma without Feedthrough

The Concept
It is assumed in the Lemma that the state-space representation (A, B, C, D) is minimal. Then Positive Realness (PR) of the transfer function C(SI − A)-1B + D  is equivalent to the solvability of the set of LMIs given in this page. Consider now the following scalar example, where (A, B, C, D)=(−α, 0, 0, 1), with α > 0. The transfer function is H(s) = 0 that is PR

The System
Consider a contiuous-time LTI system, $$\mathcal{G} : \mathcal{L}_{2e} \rightarrow \mathcal{L}_{2e}$$, with minimal state-space relization (A, B, C, 0), where $$\mathcal{A} \in \mathcal{R}^{n\times n}, \mathcal{B} \in \mathcal{R}^{n\times m}, $$ and $$\mathcal{C} \in \mathcal{R}^{m\times n}, $$.

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

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

LMI : KYP Lemma without Feedthrough
The system $$\mathcal{G}$$ is positive real (PR) under either of the following equivalet necessary and sufficient conditions.

 * 1. There exists $$ P \in \mathcal{S}^{n}, $$ where $$ p > 0 $$ such that

 * $$\begin{align}

PA + A^{T}P \ge 0 \\ PB = C^{T} \end{align}$$

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

 * $$\begin{align}

AQ + QA^{T} \ge 0 \\ B = QC^{T} \end{align}$$

This is a special case of the KYP Lemma for QSR dissipative systems with Q = 0, Q = 0.5 and R = 0.

The system $$\mathcal{G}$$ is strictly positive real (SPR) under either of the following equivalet necessary and sufficient conditions.

 * 1. There exists $$ P \in \mathcal{S}^{n}, $$ where $$ p > 0 $$ such that

 * $$\begin{align}

PA + A^{T}P < 0 \\ PB = C^{T} \end{align}$$

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

 * $$\begin{align}

AQ + QA^{T} < 0 \\ B = QC^{T} \end{align}$$

This is a special case of the KYP Lemma for QSR dissipative systems with Q = ε1, Q = 0.5 and R = 0. where ε $$ \in \mathcal{R}_{>0}.$$

Conclusion:
If there exist a positive definite $$P$$ for the the selected Q,S and R matrices then the system $$\mathcal{G}$$ is Positive Real.

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

Related LMIs
KYP Lemma State Space Stability Discrete Time KYP Lemma with Feedthrough


text:
LMIs in Control/pages/Discrete Time KYP Lemma with Feedthrough

The Concept
It is assumed in the Lemma that the state-space representation (Ad, Bd, Cd, Dd) is minimal. Then Positive Realness (PR) of the transfer function Cd(SI − Ad)-1Bd + Dd  is equivalent to the solvability of the set of LMIs given in this page. Consider now the following scalar example, where (Ad, Bd, Cd, Dd)=(−α, 0, 0, 1), with α > 0. The transfer function is H(s) = 0 that is PR

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

 * $$x(k+1)=\mathcal{A}_{d}x(k)+\mathcal{B}_{d}u(k)$$
 * $$y(k)=\mathcal{C}_{d}x(k)+\mathcal{D}_{d}u(k), k=0,1... $$

The Data
The matrices $$\mathcal{A}_{d},\mathcal{B}_{d},\mathcal{C}_{d} $$ and $$\mathcal{D}_{d}$$

LMI : Discrete-Time KYP Lemma with Feedthrough
The system $$\mathcal{G}$$ is positive real (PR) under either of the following equivalet necessary and sufficient conditions.

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

 * $$\begin{bmatrix}

A^{T}_{d}PA_{d} - P &  A^{T}_{d}PB_{d} - C^{T}_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d})^{T} &  B^{T}_{d}PB_{d} - (D^{T}_{d}+ D_{d}) \end{bmatrix}\le 0.$$

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

 * $$\begin{bmatrix}

A_{d}QA^{T}_{d} - Q &  A_{d}QC^{T}_{d} - B_{d} \\ (A_{d}QC^{T}_{d} - B_{d})^{T} &  C_{d}PC^{T}_{d} - (D^{T}_{d}+ D_{d}) \end{bmatrix}\le 0.$$

 * 3. There exists $$ P \in \mathcal{S}^{n}, $$ where $$ Q > 0 $$ such that

 * $$\begin{bmatrix}

P &  PA_{d} & PB_{d} \\ (PA_{d})^{T} &  P & C^{T}_{d}\\ (PB_{d})^{T} & C_{d} & D^{T}_{d}+ D_{d} \end{bmatrix}\ge 0.$$

 * 4. There exists $$ Q \in \mathcal{S}^{n}, $$ where $$ Q > 0 $$ such that

 * $$\begin{bmatrix}

Q &  A_{d}Q & B_{d} \\ (A_{d}Q)^{T} &  Q & QC^{T}_{d}\\ (B_{d})^{T} & (QC^{T}_{d})^{T} & D^{T}_{d}+ D_{d} \end{bmatrix}\ge 0.$$

This is a special case of the KYP Lemma for QSR dissipative systems with Q = 0, Q = 0.5 and R = 0.

The system $$\mathcal{G}$$ is strictly positive real (SPR) under either of the following equivalet necessary and sufficient conditions.

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

 * $$\begin{bmatrix}

A^{T}_{d}PA_{d} - P &  A^{T}_{d}PB_{d} - C^{T}_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d})^{T} &  B^{T}_{d}PB_{d} - (D^{T}_{d}+ D_{d}) \end{bmatrix} < 0.$$

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

 * $$\begin{bmatrix}

A_{d}QA^{T}_{d} - Q &  A_{d}QC^{T}_{d} - B_{d} \\ (A_{d}QC^{T}_{d} - B_{d})^{T} &  C_{d}PC^{T}_{d} - (D^{T}_{d}+ D_{d}) \end{bmatrix} < 0.$$

 * 3. There exists $$ P \in \mathcal{S}^{n}, $$ where $$ Q > 0 $$ such that

 * $$\begin{bmatrix}

P &  PA_{d} & PB_{d} \\ (PA_{d})^{T} &  P & C^{T}_{d}\\ (PB_{d})^{T} & C_{d} & D^{T}_{d}+ D_{d} \end{bmatrix} > 0.$$

 * 4. There exists $$ Q \in \mathcal{S}^{n}, $$ where $$ Q > 0 $$ such that

 * $$\begin{bmatrix}

Q &  A_{d}Q & B_{d} \\ (A_{d}Q)^{T} &  Q & QC^{T}_{d}\\ (B_{d})^{T} & (QC^{T}_{d})^{T} & D^{T}_{d}+ D_{d} \end{bmatrix} > 0.$$ This is a special case of the KYP Lemma for QSR dissipative systems with Q = ε1, Q = 0.5 and R = 0. where ε $$ \in \mathcal{R}_{>0}.$$

Conclusion:
If there exist a positive definite $$P$$ for the the selected Q,S and R matrices then the system $$\mathcal{G}$$ is Positive Real.

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

Related LMIs
KYP Lemma State Space Stability KYP Lemma without Feedthrough


text:
LMIs in Control/pages/Stability of Quadratic Constrained Systems

The System

\begin{align} \dot x(t)&=Ax(t)+B_pp(t)+B_u u(t)+B_ww(t),\\ q(t)&=C_qx(t)+D_{qp}p(t)+D_{qu}u(t)+D_{qw}w(t),\\ z(t)&=C_zx(t)+D_{zp}p(t)+D_{zu}u(t)+D_{zw}w(t)\\ \int_0^t &p^\top(\tau)p(\tau) \ d\tau \leq \int_{0}^t q^\top(\tau)q(\tau) \ d\tau. \end{align}

$$

The Data
The matrices $$ A,B_p,B_w,C_q,C_z,D_{qp},D_{zw} $$.

The LMI:
The following feasibility problem should be solved.

\begin{align} \text{Find} \; &\{P\succ0,\lambda \geq 0\}:\\ &  s.t. \quad \begin{bmatrix} A^\top P+PA+\lambda C^\top_q C_q & PB_p+ \lambda C^\top_q D_{qp}   \\ (PB_p+ \lambda C^\top_q D_{qp})^\top &  \lambda (I-D^\top_{qp}D_{qp})  \end{bmatrix} \prec 0. \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/38f3b55ca7060a1260384a96e9dc31142af07a9a

Conclusion
The integral quadratic constrained system is stable if the provided LMI is feasible

Remark
The key point of the proof is to satisfy $$ \dot{V}<0 $$ whenever $$ p^\top p \leq q^\top q $$, using $$ \mathcal{S} $$-procedure.


text:
LMIs in Control/pages/KYP Lemma for Descriptor Systems

The Concept
Descriptor system descriptions frequently appear when solving computational problems in the analysis and design of standard linear systems. The numerically reliable solution of many standard control problems like the solution of Riccati equations, computation of system zeros, design of fault detection and isolation filters (FDI), etc. relies on using descriptor system techniques.

Many algorithm for standard systems as for example stabilization techniques, factorization methods, minimal realization, model reduction, etc. have been extended to the more general descriptor system descriptions. An important application of these algorithms is the numerically reliable computation with rational and polynomial matrices via equivalent descriptor representations. Recall that each rational matrix R(s) can be seen as the transfer-function matrix of a continuous- or discrete-time descriptor system. Thus, each R(s) can be equivalently realized by a descriptor system quadruple (A-sE, B, C, D) satisfying R(S)= C(SE-A)-1B+D

Many operations on standard matrices (e.g., finding the rank, determinant, inverse or generalized inverses), or the solution of linear matrix equations can be performed for rational matrices as well using descriptor system techniques. Other important applications of descriptor techniques are the computation of inner-outer and spectral factorisations, or minimum degree and normalized coprime factorisations of polynomial and rational matrices. More explanation can be found in the website of Institute of System Dynamics and control

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

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

LMI : KYP Lemma for Descriptor Systems
The system $$\mathcal{G}$$ is extended strictly positive real (ESPR) if and only if there exists $$ X \in \mathcal{R}^{n\times n} $$ and $$ W \in \mathcal{R}^{n\times m}$$ such that
 * $$ E^{T}X = X^{T}E \ge 0$$
 * $$ E^{T}W = 0 $$
 * $$\begin{bmatrix}

X^{T}A + A^{T}X &  A^{T}W + X^{T}B - C^{T}  \\ (A^{T}W + X^{T}B - C^{T})^{T} &  W^{T}B + B^{T}W - (D^{T} + D) \end{bmatrix} < 0.$$

The system is also ESPR if there exists $$ \mathcal{X} \in \mathcal{R}^{n\times n} $$ such that
 * $$ E^{T}X = X^{T}E \ge 0$$
 * $$\begin{bmatrix}

X^{T}A + A^{T}X &  X^{T}B - C^{T}  \\ (X^{T}B - C^{T})^{T} &  -(D^{T} + D) \end{bmatrix} < 0.$$

Conclusion:
If there exist a X and W matrix satisfying above LMIs then the system $$\mathcal{G}$$ is Extended Strictly Positive Real.

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

Related LMIs
KYP Lemma State Space Stability Discrete Time KYP Lemma with Feedthrough


text:
LMIs in Control/pages/KBH theorem

WIP, Description in progress

Kharitonov-Bernstein-Haddad (KBH) Theorem
Consider the set of matrices

$$\mathcal{A}=\{\textbf{A}=\begin{bmatrix}\textbf{0}_{(n-1)\times 1} & & \textbf{1}_{(n-1)\times(n-1)} \\ -a_0 & \cdots & -a_{n-1} \end{bmatrix}| \underline{a_j}\leq a_j \leq \overline{a_j}, j=0,1,2,\cdots,n-1\}$$

Every matrix in the set $$\mathcal{A}$$ is Hurwitz if and only if there exist $$\textbf{P}_i \in mathbb{S}^n, i = 1, 2, 3, 4$$ where $$\textbf{P}_i>0, i = 1, 2, 3 ,4,$$ such that

$$\textbf{P}_iA_i + \textbf{A}_i^T\textbf{P}_i<0, \quad i = 1, 2, 3, 4, $$

where

$$\textbf{A}_i =\begin{bmatrix} \begin{bmatrix}\textbf{0}_{(n-1)\times 1} & \textbf{1}_{(n-1)\times(n-1)} \end{bmatrix} \\ \textbf{a}_i \end{bmatrix}, \quad i = 1, 2, 3, 4,$$

$$ \textbf{a}_1 = -[\underline{a}_0 \quad \underline{a}_1 \quad \overline{a}_2 \quad \overline{a}_3 \quad \cdots \quad \underline{a}_{n-4} \quad \underline{a}_{n-3} \quad \overline{a}_{n-2} \quad \overline{a}_{n-1} ], $$

$$ \textbf{a}_2 = -[\underline{a}_0 \quad \overline{a}_1 \quad \overline{a}_2 \quad \underline{a}_3 \quad \cdots \quad \underline{a}_{n-4} \quad \overline{a}_{n-3} \quad \overline{a}_{n-2} \quad \underline{a}_{n-1} ], $$

$$ \textbf{a}_3 = -[\overline{a}_0 \quad \underline{a}_1 \quad \underline{a}_2 \quad \overline{a}_3 \quad \cdots \quad \overline{a}_{n-4} \quad \underline{a}_{n-3} \quad \underline{a}_{n-2} \quad \overline{a}_{n-1} ], $$

$$ \textbf{a}_3 = -[\overline{a}_0 \quad \underline{a}_1 \quad \underline{a}_2 \quad \overline{a}_3 \quad \cdots \quad \overline{a}_{n-4} \quad \underline{a}_{n-3} \quad \underline{a}_{n-2} \quad \overline{a}_{n-1} ], $$

$$ \textbf{a}_4 = -[\overline{a}_0 \quad \overline{a}_1 \quad \underline{a}_2 \quad \underline{a}_3 \quad \cdots \quad \overline{a}_{n-4} \quad \overline{a}_{n-3} \quad \underline{a}_{n-2} \quad \underline{a}_{n-1} ]. $$

WIP, additional references to be added

title: LMIs in Control/pages/KBH theorem


text:
LMIs in Control/pages/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 : 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 equivalent necessary and sufficient conditions.
 * 1. There exists P $$ \in \mathcal{S}^{n}$$, where P $$\ge 0$$, such that

 * $$\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 0.$$

 * 2. There exists P $$ \in \mathcal{S}^{n}$$, where P $$\ge 0$$, such that

 * $$\begin{bmatrix}

PA + A^{T}P - C^{T}C &  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


text:
LMIs in Control/pages/Discrete Time Lyapunov Stability

Discrete-Time Lyapunov Stability

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.

Stability of DT LTI systems can be determined by solving Lyapunov Inequality.

The System
Discrete-Time System

\begin{align} x(t)_{k+1}&=A_dx(t)_k,&A_d \in \bf{R^{n*n}} \;\\ \end{align}$$

The Data
The matrices: System $$ A_d, P $$.

The Optimization Problem
The following feasibility problem should be optimized:

Find P obeying the LMI constraints.

The LMI:
Discrete-Time Bounded Real Lemma

The LMI formulation

\begin{align} P \in \bf{S^{n}} \;\\ \text{Find}\; &P>0,\\ \begin{bmatrix} A_d^T P A_d - P\end{bmatrix}&<0 \end{align}$$

Conclusion:
If there exists a $$P \in \bf{S^{n}}$$ satisfying the LMI then, $$|\lambda_i(A_d)|\leq 1, \forall i = 1,2,...,n;$$ and the equilibrium point $$\bar{x}=0$$  of the system is Lyapunov stable.

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

Related LMIs
Continuous_Time_Lyapunov_Inequality - Lyapunov_Inequality


text:
LMIs in Control/pages/H2 Deduced Condition

WIP, Description in progress

This gives LMI of the deduced condition for $$H_2$$-norm of a system.

Deduced Condition
Given a positive $$\gamma$$, the transfer function matrix

$$G(s)=C(sI-A)^{-1}B$$

satisfies

$$||G(s)||_2<\gamma $$

if and only if there exists symmetric matrices $$Z, P$$ and a matrix $$V$$ such that,

$$\begin{cases} \text{trace}(Z)\leq \gamma^2 \\ \begin{bmatrix} -Z & C \\ C^T & P \end{bmatrix} < 0 \\ \begin{bmatrix} -(V+V^T) & V^TA+P & V^TB & V^T \\ A^TV+P & -P & 0 & 0\\ B^TV & 0 & -I & 0 \\ V & 0 & 0 & -P\end{bmatrix} < 0 \end{cases}$$

$$\begin{cases} \text{trace}(Z)\leq \gamma^2 \\ \begin{bmatrix} -Z & B^T \\ B & P \end{bmatrix} < 0 \\ \begin{bmatrix} -(V+V^T) & V^TA^T+P & V^TC^T & V^T \\ AV+P & -P & 0 & 0\\ CV & 0 & -I & 0 \\ V & 0 & 0 & -P\end{bmatrix} < 0 \end{cases}$$

WIP, additional references to be added


text:
LMIs in Control/pages/Discrete Time Bounded Real Lemma

Discrete-Time Bounded Real Lemma

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 Bounded Real Lemma or the H∞ norm can be found by solving a LMI.

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 $$ (A_d,B_d,C_d,D_d), P $$.

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Discrete-Time Bounded Real Lemma

The LMI formulation

H∞ norm < $$\gamma$$

$$ \begin{align} P \in {S^{n}}; \gamma \in {R_{>0}} \;\\ &P>0,\\ \begin{bmatrix}A_d^TPA_d-P & A_d^TPB_d & C_d^T\\ \end{align}$$
 * & B_d^TPB_d-\gamma I & D_d^T\\
 * & * & -\gamma I\end{bmatrix}&<0,

Conclusion:
The H∞ norm is the minimum value of $$ \gamma \in {R_{>0}} $$that satisfies the LMI condition. If $$(A_d,B_d,C_d,D_d)$$ is the minimal realization then the inequalities can be non-strict.

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

Related LMIs
- Continuous time KYP_Lemma_(Bounded_Real_Lemma)


text:
LMIs in Control/pages/Positive Orthant Stability

Positive Orthant Stabilizability

The positive orthant stability of a linear system refers to the property of the system states being real and positive for all $$ t \geq 0 $$ and decaying down to zero over time. In this section, the feasibility problem for systems to be positive orthant stable, and the stabilizability conditions to make the system positive orthant stable will be covered.

The System
Consider a linear state-space representation of a system as:

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

where $$x(t)\in \R^n$$ and $$u(t)\in \R^r$$ are the system state and the input vector respectively. A and B are system coefficient matrices of appropriate dimensions.

The Data
Number of states n and number of control inputs r need to be known. Moreover, the system matrices A,B are also required to be known.

The Feasibility LMI
An LTI system is positive orthant stable if $$ x(0) \in \R^n_+ $$ implies that $$ \forall t \geq 0, x(t) \in \R^n_+ $$. Moreover, as $$t \rightarrow \infty$$, $$ x(t) \rightarrow 0 $$. This is possible if and only if the following conditions hold:

\begin{align} A_{ij} & \geq 0, \forall i \neq j, \\ \exists P & > 0 \text{ s.t. } PA^\top + AP < 0, \end{align} $$

The above LMI feasibility is the positive orthant stability criteria. To convert it into a positive orthant stabilizability check, the problem can be modified so that we check if $$ \dot{x}=(A+BK)x $$ is positive orthant stable. As $$ K $$ is also a design variable here, the second inequality in the above LMI will result in bilinearity. A simple change of variables can overcome that to result in the following LMI feasibility problem for checking positive orthant stabilizability of the LTI system:

\begin{align} \text{Find } Q,Y & \text{ subj. to:} \\ & Q > 0, (AQ+BY)_{ij} \geq 0, \forall i \neq j, \\ & \text{ s.t. } QA^\top + AQ + BY + Y^\top B^\top < 0, \end{align} $$

If the above LMI is feasible, the LTI system is stabilizable with controller $$ K = YQ^{-1} $$.

Conclusion:
The feasibility of the above LMIs guarantees that the system is positive orthant stable if the first LMI is feasible or stabilizable with a controller if the second LMI holds.

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/Positive_Orthant_LMI.m


text:
LMIs in Control/pages/Discrete-Time H2 Norm

Discrete-Time H2 Norm

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' H2 norm can be found by solving a LMI.

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 $$ (A_d,B_d,C_d,D_d), P, Z $$.

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Discrete-Time Bounded Real Lemma

The LMI formulation

H2 norm < $$\mu$$

$$ \begin{align} P \in {S^{n}}; Z \in {S^{p}}; \mu \in {R_{>0}} \;\\ &P>0, &Z>0\\ \begin{bmatrix}P & A_dP & B_d\\
 * & P & 0\\
 * & * & I\end{bmatrix}&>0,\\

\begin{bmatrix}Z & C_dP\\
 * & P\end{bmatrix}&>0,\\

trZ<\mu ^2

\end{align}$$

Conclusion:
The H2 norm is the minimum value of $$ \mu \in {R_{>0}} $$that satisfies the LMI condition.

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

Related LMIs
- Continuous time H2 norm.


text:
LMIs in Control/pages/Discrete Time Stabilizability

Discrete-Time Stabilizability

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 stable using controller gain K, which can be found using LMI optimization, such that the close loop system is stable.

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 K is found.

The LMI:
Discrete-Time Stabilizability

The LMI formulation

$$ \begin{align} P \in {S^{n}}; W \in {R^{m*n}}\;\\ &P>0\\ \begin{bmatrix}P & A_dP+B_dW\\
 * & P\end{bmatrix}&>0,\\

K_d = WP^{-1}

\end{align}$$

Conclusion:
The system is stabilizable iff there exits a $$P$$, such that $$P>0$$. The matrix $$A_d+B_dK_d$$ is Schur with $$K_d = WP^{-1}$$

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

Related LMIs
- Continuous Time Stabilizability


text:
Talk:How to Enthrone the Sacred Heart

I had material about Enthroning the Sacred Heart that I wanted to post on Wikipedia. I was told some of the information was too much about how to do an Enthronement and was more appropriate for Wikibooks. So, I created the article here. I still would like to include the manuals for doing a 3-day and 7-day Enthronement. That is left to be done.Shenaw2016 (discuss • contribs) 11:02, 7 December 2019 (UTC)

title: Talk:How to Enthrone the Sacred Heart