OpenLLM-France/wikimedia huggingface dataset:
# Project Management/PMBOK/Cost Management

Project Cost Management is the fourth knowledge area of project management.
It starts with a process of planning that produces a cost management plan. Then there is an iterative process that develops and updates the cost estimates and cost baseline. After these have been produced, a monitoring and controlling process is used to measure project performance and to manage any change requests to the project scope, schedule, and costs.

## Cost Management and Profit

Purpose of every private organization is to maximize the profit and there are two ways to to maximize the profit
* To win the order at maximum highest price (That is not possible in this imperfect competition market)
* To reduce the maximum manufacturing cost by setting different standards, and then management try to achieve these standard.
Second option (controlling manufacturing cost) is in control of the organization that can be apply but it will be necessary that the higher management should accept this mentally, consider this a positive step for growth of the organization, and do not feel it burden / restrictions. Provision of actual data should be as easy as possible so that cost can be calculated near to actual. There should be no hesitation in provision of data because this data will be used especially for internal management and not for external bodies. ( Tax department etc.)

## Project Cost Management Processes

The PMBOK® Guide—5th Edition—describes the following 4 processes as necessary for planning, quantifying and measuring costs, so that the project can be completed within the approved budget:
* Plan Cost Management: used by a project manager to create the cost management plan.
* Estimate Costs: it develops an approximation of costs associated with each of the resources required for all schedule activities of the project.
* Determine Budget: it aggregates the individual activity cost estimates into a total project cost, establishes the costs timing, and measures the performance of the project against the obtained baseline.
* Control Costs: it monitors the project budget and manages variations to the cost baseline included in the project plan.

## References

* A Guide to the Project Management Body of Knowledge (fifth ed.), Project Management Institute, 2013, ISBN 978-1-935589-67-9
* PMP Certification Exam Prep — Project Cost Management
* Earned Value Analysis in Project Management — an Easy Cheat Sheet
# Pinyin/Childhood (4)

See also
Childhood
Childhood is like a painting, looks like simple, but interesting.
Childhood is like running water, although lasting only a short time, but meaningful.
Childhood is like a rainbow after raining, although disappeared, but always remains in my mind.
Tóngnián
Tóngnián hǎoxiàng yī-fú huà, kànsì jiǎndān, què nàirénxúnwèi.
Tóngnián hǎoxiàng liúshuǐ, suīrán duǎnzàn, què fùyǒu yìyì.
Tóngnián hǎoxiàng yǔ-hòu de cǎihóng, suī huì xiāoshī, què yǒngyuǎn liú zài wǒ de nǎohǎi.
# Pinyin/Childhood

See also
# Pinyin/Childbirth

See also
# Pinyin/Chia seed

Search Pinyin chia
Search Pinyin seed
Chia seeds are the edible seeds of Salvia hispanica, a flowering plant in the mint family native to central and southern Mexico,. It is rich in protein, omega-3, and soluble fiber.
chia-zii shih caaobeen zhirwuh zhoongzii, fuhharn dahnbairzhih, omega-3, her keerorng xianweir.
# Pinyin/Chief

See also
# Pinyin/Child

Search child
Search children
A child is a young human who is not yet an adult.
errtorng shih shahngweih cherngniarn d niarnyouh rernleih.
# Pinyin/Childhood (2)

See also
Childhood
Childhood is like a painting, is so colorful.
Childhood have happiness, but also has pain and grief, makes me difficult to forget.
Although have a lot of unhappiness, my childhood is still good.
torngniarn
torngniarn rur 1-fur huah, shih duoziduocaai d.
torngniarn yoou huanleh, yee yoou tohngkuu her beishang, shii woo narnyiiwahnghuair.
suirarn yoou xuuduo bu yurkuaih, woo d torngniarn yirarn meeihaao.
# Pinyin/Childhood (5)

See also
Childhood
Childhood is a gust of wind, It gone with the wind. Childhood is a happy memory, It was a happy time. Childhood is a gun salute, It gone with the sound. In my childhood, I had childlike happiness. Childhood is a stamp, Childhood is an ice cream. I want to hold childhood's hand, Travel through the time tunnel, Back to the joy of childhood.
Tóngnián
Shì yī gǔ fēng, yī chuī ér guò. Tóngnián shì jìyì zhōng de kuàilè, shǐ rén huānshēngxiàoyǔ. Tóngnián shì yī zhī lǐpào, yī xiǎng ér guò. Tóngnián zhōng de wǒ, jì yòuxiǎo ér yòu huānlè. Tóngnián jiùshì yī zhāng yóupiào, tóngnián jiùshì yī zhī bīngqílín. Lā qǐ tóngnián de shǒu, chuānyuè shíkōng, huídào huānlè de tóngnián.
# Pinyin/Childhood (3)

See also
Childhood
Still remember? Below the white ceiling, a series of happy laughters.
Still remember? Notes passed around during lessons, paper airplanes played after lessons, games played after school, matters discussed during go to school.
At the blink of the eyes 6 years passed, there is no need for sorrow during departure, let me see the most beautiful you.
torngniarn
hair jihder 'ma? zaih bairseh tianhuabaan xiah, yizhehnzhehn yueh'eer d xiaohsheng.
hair jihder 'ma? shahngkeh shir chuarn d xiaaozhiitiaor, xiahkeh shir warn d zhiifeiji, fahngxuer shir warn d yourxih, shahngxuer shir taaoluhn d huahtir.
yizhaayaan 6 niarn guohquh, lirbier shir bubih shangxin, rahng woo kahnkahn zuih meei d nii.
# LMIs in Control/Click here to continue/Controller synthesis/Apkarian Filter and State Feedback

LMIs in Control/Click here to continue/Controller synthesis/Apkarian Filter and State Feedback

## The System:

The number of LMI constraints needed to check quadratic stability is reduced if all the subsystems in the polytopic model has the same matrix $B$. This can be achieved by adding an Apkarian filter in the input of the system.

## The Optimization Problem:

Apkarian Filter
Let consider our TS-LIA model. This can be re written in linear form as:
       ${\dot {x}}=A(z(t))x+B(z(t))u$
The filter should be such that the equilibrium of the states are the input values and the dynamics should be fast, so we could assume the dynamics of the filter negligible (i.e. the input of the filter is equivalent to the input of the quadrotor). One possible filter is shown , where $A_{F}$ = −100$I_{4}$, $B_{F}$ = 100$I_{4}$ and $I_{4}$ ∈ $R^{4\times 4}$ is the identity matrix.
       ${\dot {x_{F}}}=A_{F}x_{F}+B_{F}u_{F};y_{f}=x_{f}$.
When applying the filter, we are imposing that the output of the filter is the new input of the TS-LIA model (i.e. $u$ = $y_{F}$ ). Then, the extended model is:
       ${\dot {x_{c}}}={\begin{bmatrix}A(z(t))&B(z(t))\\0&A_{F}\\\end{bmatrix}}x_{c}+{\begin{bmatrix}0\\B_{F}\end{bmatrix}}u_{F}=A_{e}(z(t))x_{e}+B_{e}u_{f};x_{e}={\begin{bmatrix}x\\x_{F}\end{bmatrix}}$
This prefiltering does not affect the procedure followed to obtain the TS-LIA model, so the premise variables, membership functions and activations functions remains the same.
State Feedback Controller Design
Let consider the state feedback control law for the extended TS-LIA model:${\dot {x_{e}}}=\sum _{i=1}^{32}h_{i}(z(t))[A_{ei}x_{e}+B_{ei}u_{F}]$, where the state feedback control laws are :$u_{F}=\sum _{i=1}^{32}h_{i}(z(t))K_{i}x(t)$, we get the closed loop system :${\dot {x_{e}}}=\sum _{i=1}^{32}\sum _{j=1}^{32}h_{j}(z(t))[A_{ei}x_{e}+B_{ei}K_{j}]x_{e}$

## The LMI:

The design of the controller is done by solving an LMI problem involving the quadratic stability constraints. In case we want D- stabilization, the following set of LMI constraints are needed:
       $L\otimes P+M\otimes P(A_{ei}+B_{e}K_{i})+MT\otimes (A_{e}i+B_{e}K_{i})^{T}P<0$ ∀i = 1, . . . , 32.

## Conclusion:

The LMI is feasible.

## Related LMIs

* LMI for Natural Frequency in State Feedback. https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Maximum_Natural_Frequency_in_State_Feedback#The_LMI%3A
* LMI for Minimum Decay Rate in State Feedback. https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Minimum_Decay_Rate_in_State_Feedback#The_LMI%3A

## References

* Control, A. (2016). Gain-scheduling Control of a Quadrotor Using the Takagi-Sugeno Approach.
# LMIs in Control/Click here to continue/Controller synthesis/Controller to achieve the desired Reachable set; Diagonal Norm-bound uncertainty
## Reachable sets with unit-energy inputs; Diagonal Norm-bound uncertainty

A Reachable set is a set of system States reached under the condition $u=Kx$. On this page we will look at the problem of finding an controller $K$, that $E\supseteq RS$ - reachable set.

## The System

    ${\begin{aligned}{\dot {x}}&=Ax+B_{w}w+B_{u}u+B_{q}p\\q&=C_{q}x+D_{qw}w+D_{qu}u+D_{qp}p\\u&=Kx\end{aligned}}$
Where:
    ${\begin{aligned}x&\in R^{n}\\w&\in R^{m}\\u&\in R^{k}\\\end{aligned}}$
In case of Diagonal Norm-bound uncertainty, we have:
    ${\begin{aligned}p_{i}&=\delta _{i}(t)q_{i}\\&|\delta _{i}(t)|\leq 1;\;\;\;\;{\text{for }}i=1,...,N_{q}\end{aligned}}$

## The Data

$N_{p}=N_{q}$
The matrices $A\in R^{n\times n};\;B_{w}\in R^{n\times m};\;B_{u}\in R^{n\times k};B_{p}\in R^{n\times N_{p}};K\in R^{k\times n}$.
$C_{q}\in R^{N_{q}\times n};\;D_{qu}\in R^{N_{q}\times k}D_{qw}\in R^{N_{q}\times m}D_{qp}\in R^{N_{q}\times N_{p}}$.

## Reachable set

The reachable set can be defined:
    ${\begin{aligned}RS&=\{x(T)|u=Kx;\;\;x(0)=0;\;\;T\geq 0;\;\;\int _{0}^{T}w^{T}wdt<1\}\\\end{aligned}}$
The elipsoid $E=\{\varepsilon \in R^{n}|\varepsilon ^{T}Q\varepsilon \leq 1\}\supseteq RS$

## The Optimization Problem

The following optimization problem should be solved:
    ${\begin{aligned}{\text{Find}}\;&M>0:Y\\&{\begin{bmatrix}QA^{T}+AQ+B_{u}Y+Y^{T}B_{u}^{T}+B_{w}B_{w}^{T}+B_{p}MB_{p}^{T}&(C_{q}Q+D_{qu}Y)^{T}\\C_{q}Q+D_{qu}Y&-MI\end{bmatrix}}<0\\&K=YQ^{-1}\end{aligned}}$

## Conclusion:

This LMI allows us to investigate stability for the robust control problem in the case of polytopic uncertainty and gives on the controller for this case

## Implementation:

* - Matlab implementation using the YALMIP framework and Mosek solver

## Related LMIs:

* - Reachable sets with unit-energy inputs; Polytopic uncertainty
* - Reachable sets with unit-energy inputs; norm bound uncertainty
* Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes. (3.20.2 page 64)
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

## Return to Main Page:

* Main Page.
# LMIs in Control/Click here to continue/Controller synthesis/Controller to achieve the desired Reachable set; Norm bound uncertainty
## Reachable sets with unit-energy inputs; norm bound uncertainty

A Reachable set is a set of system States reached under the condition $u=Kx$. On this page we will look at the problem of finding an controller $K$, that $E\supseteq RS$ - reachable set.

## The System

    ${\begin{aligned}{\dot {x}}&=Ax+B_{w}w+B_{u}u+B_{q}p\\q&=C_{q}x+D_{qw}w+D_{qu}u+D_{qp}p\\u&=Kx\end{aligned}}$
Where:
    ${\begin{aligned}x&\in R^{n}\\w&\in R^{m}\\u&\in R^{k}\\\end{aligned}}$
In case of norm-bound uncertainty, we have:
    ${\begin{aligned}p&=\Delta (t)q\\&\|\Delta \|\leq 1\end{aligned}}$

## The Data

The matrices $A\in R^{n\times n};\;B_{w}\in R^{n\times m};\;B_{u}\in R^{n\times k};B_{p}\in R^{n\times N_{p}};K\in R^{k\times n}$.
$C_{q}\in R^{N_{q}\times n};\;D_{qu}\in R^{N_{q}\times k}D_{qw}\in R^{N_{q}\times m}D_{qp}\in R^{N_{q}\times N_{p}}$.

## Reachable set

The reachable set can be defined:
    ${\begin{aligned}RS&=\{x(T)|u=Kx;\;\;x(0)=0;\;\;T\geq 0;\;\;\int _{0}^{T}w^{T}wdt<1\}\\\end{aligned}}$
The elipsoid $E=\{\varepsilon \in R^{n}|\varepsilon ^{T}Q\varepsilon \leq 1\}\supseteq RS$

## The Optimization Problem

The following optimization problem should be solved:
    ${\begin{aligned}{\text{Find}}\;&\mu >0:Y\\&{\begin{bmatrix}QA^{T}+AQ+B_{u}Y+Y^{T}B_{u}^{T}+B_{w}B_{w}^{T}+\mu B_{p}B_{p}^{T}&(C_{q}Q+D_{qu}Y)^{T}\\C_{q}Q+D_{qu}Y&-\mu I\end{bmatrix}}<0\\&K=YQ^{-1}\end{aligned}}$
Or
    ${\begin{aligned}{\text{Find}}\;&\mu >0:\sigma \\&{\begin{bmatrix}QA^{T}+AQ+\sigma B_{u}B_{u}^{T}+B_{w}B_{w}^{T}+\mu B_{p}B_{p}^{T}&(C_{q}Q-\sigma D_{qu}B_{u}^{T}+\mu D_{qp}B_{p}^{T})^{T}\\C_{q}Q-\sigma D_{qu}B_{u}^{T}+\mu D_{qp}B_{p}^{T}&-\mu (I-D_{qp}D_{qp}^{T})\end{bmatrix}}<0\\&K=-{\frac {\sigma }{2}}B_{u}^{T}Q^{-1}\end{aligned}}$

## Conclusion:

This LMI allows us to investigate stability for the robust control problem in the case of polytopic uncertainty and gives on the controller for this case

## Implementation:

* - Matlab implementation using the YALMIP framework and Mosek solver

## Related LMIs:

* Reachable sets with unit-energy inputs; Polytopic uncertainty
* Reachable sets with unit-energy inputs; Diagonal Norm-bound uncertainty
* H-inf Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes. (3.20.2 page 64)
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

## Return to Main Page

* Main Page.
# LMIs in Control/Click here to continue/Controller synthesis/Design for Insensitive Strip Region

Insensitive Strip Region Design with Minimum $H_{2}$ Gain
When designing controllers with insensitive region conditions, the aim is to place the closed-loop poles of the system in a particular region defined by its inner boundary. These regions are specified based on their insensitivity to perturbations to the system parameter matrices.
One type of such design is the Insensitive Strip Region Design. In this section, building upon that, optimization problems will be provided that ensure that the conditions for insensitive strip region design are satisfied with some bounds on the $H_{2}$ gain of the closed-loop system.

## The System

A state-space representation of a linear system as given below:
    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)\end{aligned}}$
where $x(t)\in \mathbb {R} ^{n}$, $y(t)\in \mathbb {R} ^{m}$ and $u(t)\in \mathbb {R} ^{r}$ are the system state, output, and the input vector respectively.

## The Data

To solve the design optimization problem, the linear system matrices A,B,C are required. Furthermore, to define the strip region on the eigenvalue-space, two parameters $\gamma _{1}$ and $\gamma _{2}$ are required.

## The Optimization Problem

The problem of designing an $H_{2}$ optimal controller that results in the closed loop system insensitive to a certain strip region involves two sub-problems:
* Finding a control gain $K$ such that: $\left\|K\right\|_{2}<\gamma$.
* The conditions for insensitive strip region design for the closed-loop system, as provided in the section Insensitive Strip Region Design are fulfilled.
* The optimization goal is to minimize $\gamma$ such that above two hold.

## The LMI: $H_{2}$ Optimal Control Design for Insensitive Strip Region

The problem above has a solution if and only if the following optimization problem has a solution $(K,\gamma )$:
${\begin{aligned}{\text{min }}&\gamma \\{\text{s.t. }}&{\begin{bmatrix}-\gamma I&K\\K^{\top }&-\gamma I\end{bmatrix}}<0\\&2\gamma _{1}I<(A+BKC)^{\top }+(A+BKC)<2\gamma _{2}I\end{aligned}}$

## Conclusion:

By using the design problem provided here, an optimal $H_{2}$ controller is designed to make the closed-loop system robust to perturbations in the system matrices.

## Implementation

To solve the optimization problem with LMI presented here, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:
https://github.com/smhassaan/LMI-Examples/blob/master/H2_Strip_example.m

## Related LMIs

Insensitive Strip Region Design

## External Links

A list of references documenting and validating the LMI.
* LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 10.1.1 pp. 322–323.
# LMIs in Control/Click here to continue/Controller synthesis/Controller to achieve the desired Reachable set; Polytopic uncertainty
## Reachable sets with unit-energy inputs; Polytopic uncertainty

A Reachable set is a set of system States reached under the condition $u=Kx$. On this page we will look at the problem of finding an controller $K$, that $E$ contains $RS$ - reachable set.

## The System

    ${\begin{aligned}{\dot {x}}&=Ax+B_{w}w+B_{u}u\\u&=Kx\end{aligned}}$
Where:
    ${\begin{aligned}x&\in R^{n}\\w&\in R^{m}\\u&\in R^{k}\\\end{aligned}}$
In case of polytopic uncertainty, we have:
    ${\begin{aligned}A(t)\;\;B_{w}(t)\;\;B_{u}(t)&\in {\textbf {Co}}\{[A_{1}\;\;B_{w,1}\;\;B_{u;1}],...,[A_{L}\;\;B_{w;L}\;\;B_{u;L}]\}\\\end{aligned}}$

## Reachable set

The reachable set can be defined:
    ${\begin{aligned}RS&=\{x(T)|u=Kx;\;\;x(0)=0;\;\;T\geq 0;\;\;\int _{0}^{T}w^{T}wdt<1\}\\\end{aligned}}$
The elipsoid $E=\{\varepsilon \in R^{n}|\varepsilon ^{T}Q\varepsilon \leq 1\}$ contains $RS$

## The Data

The matrices $A,A_{i}\in R^{n\times n};\;B_{w},B_{w;i}\in R^{n\times m};\;B_{u},B_{u;i}\in R^{n\times k};Q\in R^{n\times n}$. And $$

## The Optimization Problem

The following optimization problem should be solved:
    ${\begin{aligned}{\text{Find}}\;&Y>0:\\&QA_{i}^{T}+A_{i}Q+B_{u;i}Y+Y^{T}B_{u;i}^{T}+B_{w;i}B_{w;i}^{T}<0{\text{ for all }}i=1,...n\\&K=YQ^{-1}\end{aligned}}$

## Conclusion:

This LMI allows us to investigate stability for the robust control problem in the case of polytopic uncertainty and gives on the controller for this case

## Implementation:

* - Matlab implementation using the YALMIP framework and Mosek solver

## Related LMIs:

* Reachable sets with unit-energy inputs; Norm-bound uncertainty
* Reachable sets with unit-energy inputs; Diagonal Norm-bound uncertainty
* Quadratic polytopic stabilization

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes. (3.20.2 page 64)
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

## Return to Main Page:

* Main Page.
# LMIs in Control/Click here to continue/Controller synthesis/Design for Insensitive Disk Region

Insensitive Disk Region Design with Minimum $H_{2}$ Gain
Apart from the design for the insensitive strip region with minimum $H_{2}$ gain, another type of such design is the Insensitive Disk Region Design. In this section, optimization problems will be provided that ensure that the conditions for insensitive disk region design are satisfied with some bounds on the $H_{2}$ gain of the closed-loop system.

## The System

A state-space representation of a linear system as given below:
    ${\begin{aligned}\rho x&=Ax+Bu\\y&=Cx\end{aligned}}$
where $x\in \mathbb {R} ^{n}$, $y\in \mathbb {R} ^{m}$ and $u\in \mathbb {R} ^{r}$ are the system state, output, and the input vector respectively. $\rho$ represents the differential operation for continuous time systems, or the one-step shift forward operator for discrete time case.

## The Data

To solve the design optimization problem, the linear system matrices A,B,C are required. Furthermore, to define the disk region on the eigenvalue-space, its radius $\gamma _{0}$ is required.

## The Optimization Problem

The problem of designing an $H_{2}$ optimal controller that results in the closed loop system insensitive to a certain disk region involves two sub-problems:
* Finding a control gain $K$ such that: $\left\|K\right\|_{2}<\gamma$.
* The conditions for insensitive disk region design for the closed-loop system, as provided in the section Insensitive Disk Region Design are fulfilled.
* The optimization goal is to minimize $\gamma$ such that above two hold.

## The LMI: $H_{2}$ Optimal Control Design for Insensitive Disk Region

The problem above has a solution if and only if the following optimization problem has a solution $(K,\gamma )$:
${\begin{aligned}{\text{min }}&\gamma \\{\text{s.t. }}&{\begin{bmatrix}-\gamma I&K\\K^{\top }&-\gamma I\end{bmatrix}}<0\\&{\begin{bmatrix}-\gamma _{0}I&A+BKC+qI\\(A+BKC+qI)^{\top }&-\gamma _{0}I\end{bmatrix}}<0\end{aligned}}$

## Conclusion:

By using the design problem provided here, an optimal $H_{2}$ controller is designed to make the closed-loop system robust to perturbations in the system matrices.

## Implementation

To solve the optimization problem with LMI presented here, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:
https://github.com/smhassaan/LMI-Examples/blob/master/H2_Disk_example.m

## Related LMIs

Insensitive Disk Region Design

## External Links

A list of references documenting and validating the LMI.
* LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 10.1.2 pp. 323–325.
# LMIs in Control/Click here to continue/Controller synthesis/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 $(A_{d},B_{d},C_{d},D_{d})$ ${\begin{aligned}&A_{d}\in {\bf {{R^{n*n}},}}&B_{d}\in {\bf {{R^{n*m}},}}&C_{d}\in {\bf {{R^{p*n}},}}&D_{d}\in {\bf {{R^{p*m}}\;}}\\\end{aligned}}$

## 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
${\begin{aligned}P\in {S^{n}};W\in {R^{m*n}}\;\\&P>0\\{\begin{bmatrix}P&A_{d}P+B_{d}W\\*&P\end{bmatrix}}&>0,\\K_{d}=WP^{-1}\end{aligned}}$

## Conclusion:

The system is stabilizable iff there exits a $P$, such that $P>0$. The matrix $A_{d}+B_{d}K_{d}$ is Schur with $K_{d}=WP^{-1}$

## Implementation

A link to CodeOcean or other online implementation of the LMI MATLAB Code

## Related LMIs

- Continuous Time Stabilizability

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
# Pinyin/Mining region

See also
# Pinyin/Minute

A minute is a period of 60 seconds.
fenzhong shih 60 miaao d qijian.
# Pinyin/Ministry

See also
# Pinyin/Minority

See also
# Pinyin/Miner

Search Pinyin miner
# Pinyin/Mineral

Search
A mineral is a natural substance in the Earth, for example coal, salt, gold, or diamonds.
kuahngwuh shih Dihqiur d tianrarn wuhzhih, rur meir, yarn, jin, huoh zuahnshir.
# Pinyin/Mine

Search Pinyin mine
# Pinyin/Missile

Search Pinyin missile