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Lentis/Golden Rice

Introduction
Rice is a staple food and an important part of society for more than half of the world's population. Small, subsidence farmers are still mostly responsible for rice production, and though 40,000 varieties of rice can be found around the globe, more than 90% of rice worldwide is produced and consumed in Asia.

While rice may be filling and high in calories, it lacks micronutrients such as Vitamin A and its precursor, beta-carotene. The best sources of Vitamin A and beta-carotene are micronutrient-rich foods such as leafy greens, starchy vegetables, and meat-based proteins. For this reason, rice is supposed to be eaten as a side dish, complimentary to a balanced diet. Access to a balanced diet, however, is not common in many areas in Africa and Southeast Asia, specifically India, Bangladesh, and the Philippines. In these places Vitamin A Deficiency (VAD) is a very serious issue, causing childhood blindness and reducing the ability to fight disease. VAD affects 250 million children and 19 million pregnant women globally.

Initial Implementation
In 1999, a group of European scientists led by Dr. Ingo Potrykus genetically engineered rice to contain beta-carotene. They modified the rice by inserting bacteria and daffodil and maize genes into it and named it Golden Rice (GR1) due to the golden color of its grains. Because rice is consumed as a main food source in developing countries that cannot afford a balanced diet, these scientists proposed that Golden Rice could alleviate the problem of Vitamin A Deficiency around the globe.

Time Magazine dubbed Golden Rice “Grains of Hope,” in 2000 and claimed that this rice could save one million kids per year; however, these claims were mere speculations, as the effectiveness of Golden Rice had yet to be evaluated. Though the creation of Golden Rice was met with praise, it was also met with criticism, as it illuminated the global debate surrounding GM crops that first began in the mid-1990’s. On one end of the spectrum, Golden Rice was seen as “a technological leap forward that will bestow incalculable benefits on the world and its people.” On the other end, the GM crop was seen as a “perilous step down a slippery slope that will lead to ecological and agricultural ruin."

Early Failures with Implementation
When Potrykus patented Golden Rice in 2000, he assigned his rights over the technology to Syngenta under the condition that Golden Rice seeds would be given to poor farmers in developing countries free of charge. Syngenta agreed to finance their efforts in exchange for commercial marketing rights in affluent countries. This agreement immediately caused protests due to the fact that Asian farmers would be getting unproven genetically modified rice while Syngenta profited. While Potrykus saw this deal as the best way to get Golden Rice into the hands of poor farmers, corporations felt that the deal was an unethical marketing ploy by Syngenta to sell a new product.

Golden Rice 2 (GR2)
After its initial development, Golden Rice did not contain high enough levels of beta-carotene to be an effective source of Vitamin A. Recognizing the need to improve upon their breakthrough product, Syngenta quickly developed Golden Rice 2 (GR2), containing 23 times higher levels of beta-carotene. Syngenta also stressed that GR2 is meant to compliment, not replace, other efforts to address VAD.

Current Development of Golden Rice
Current Developers of Golden Rice:

 * International Rice Research Institute (IRRI)
 * Philippine Rice Research Institute (PhilRice)
 * Bangladesh Rice Research Institute (BRRI)
 * Indonesian Center for Rice Research (ICRR)

Under the global leadership of the IRRI, these organizations develop varieties of Golden Rice suitable for local farmers. These varieties are field tested to ensure they meet international food standards and retain the qualities of the original grain, such as yield and pest resistance. After field trials, these organizations plan to conduct a nutrition study evaluating the efficacy of Golden Rice in combating vitamin A deficiency. These studies are on hold until Golden Rice has received further regulatory and institutional review board approval.

Regulatory Approval
The most recent version of Golden Rice, GR2E, has met food safety standards for three global regulatory agencies in 2018:

 * May - United States Food and Drug Administration (USFDA)
 * March - Health Canada
 * February - Food Standards Australia New Zealand (FSANZ)

The USFDA stated that "GR2E rice is not materially different in composition, safety, or other relevant parameters" from other existing rice-derived food, "except for the intended β-carotene change. FSANZ and Health Canada also concluded that GR2E is safe for human consumption. While these approvals are significant advancements in the technology's 20 year history, Golden Rice is still not publicly available for farmers.

Beta-Carotene Levels
Despite two decades of development, organizations such as GRAIN and MADGE are still concerned with the beta-carotene levels in Golden Rice. Both reference a 2017 study that found that beta-carotene levels in Golden Rice can degrade to 60% of original levels in 3 weeks in storage and to 13% in 10 weeks of storage. MADGE also points out that based on the study, one would have to eat 4kg of Golden Rice to achieve the same levels of vitamin A in a single carrot. It is important to note that this study was conducted in part by the IRRI.

Distrust of Corporate Intentions
Although the manifest function of Golden Rice is to combat vitamin A deficiency, many believe the latent function for its development is to pave the way for the commercialization of genetically modified (GM) foods. Skeptics from an early stage have believed that Golden Rice was a public relations stunt by agricultural biotechnology firms to normalize the concept of GM foods for future profits.

More recently, a study was conducted to capture the opinions of NGOs with regards to plant genome editing. The NGOs included:

 * Beyond GM
 * Corporate Europe Observatory
 * Econexus
 * FARM
 * Food Ethics Council
 * Friends of the Earth
 * GeneWatch UK
 * GM Freeze
 * GM Watch
 * Greenpeace
 * Logos Environmental
 * Soil Association
 * Sustain
 * Permaculture Association

The study concluded that these organizations were aligned in the belief that genome editing increases corporate power and limits the choice of the consumer. They also believe that corporations have leveraged confusing terminology and crisis framings to advance their agendas.

Greenpeace - Golden Rice's Most Prominent Critic
Greenpeace, a global campaigning organization, claims that Golden Rice has wasted millions of dollars in its 20+ years of development that could have been spent on other practical VAD solutions. They further claim that Golden Rice is "environmentally irresponsible and could compromise food, nutrition, and financial security." Lastly, Greenpeace asserts that "Corporations are overhyping ‘Golden’ rice to pave the way for global approval of other more profitable genetically engineered crops," and that the only true solution to malnutrition is "a diverse healthy diet."

Scientists Advocate for Golden Rice
Despite criticism, the scientific community has shown substantial support for Golden Rice. A 2009 study conducted by The American Journal of Clinical Nutrition found that Golden Rice was as effective as supplements and more effective than natural spinach in providing vitamin A. In 2016 a group of Nobel Laureates signed a letter calling upon Greenpeace to "to cease and desist in its campaign against Golden Rice specifically, and crops and foods improved through biotechnology in general," and advising governments to reject this campaign. The letter has now been signed by 151 Nobel Laureates.

Developers Advocate for Golden Rice
The IRRI still claims that Golden Rice "has the potential to be a suitable complementary intervention to address vitamin A deficiency." They also negate claims from the US FDA about low beta-carotene levels by stating that in target countries, where rice consumption is between 200 to 300 g per day, Golden Rice provides significant amounts vitamin A. The IRRI also believes that Golden Rice is an equitable solution to all farmers and does not pose any serious environmental or health risks.

Anti-GMO Fear Appeals
As the public speculation of the safety of consuming GMO foods grows, corporations have capitalized by creating premium "Non-GMO" food products. Non-GMO verified products exceed $19 Billion in sales annually and are the fastest growing segment of grocery stores in the US. The explosive growth of sales of "Non-GMO" labeled foods has led to extreme advertising against GMO foods. These fear appeal campaigns emphasize the uncertain effects of consuming GMO foods, and portray purchasing premium non-GMO alternatives as a healthy decision. A 2016 poll found that only 37 percent of the public thought that consuming GMO foods is generally safe, compared to 88 percent of scientists. This discrepancy represents the largest gap in trust of all other major scientific controversies, including climate change, childhood vaccination, and evolution.

The Non-GMO Project
The The Non-GMO Project, a nonprofit organization founded in 2007, aims to protect "a Non-GMO food supply through consumer education." Food manufacturers, growers, and distributors partner with the Non-GMO Project to obtain the "Non-GMO Project Verified" label on their products. After obtaining what the Non-GMO Project refers to as "North America's Most Trusted Seal for GMO Avoidance," companies sell non-GMO verified products at a premium price. Opponents of the Non-GMO Project criticize the extension of the verification seal on products where GMOs are never used, meaning all brands of that product are technically "non-GMO." Randy Krotz, CEO of U.S. Farmers and Ranchers Alliance, asserts that the Non-GMO Project has a business model based on "fear and lack of information." He explains that many farmers use GMO crops because of the environmental and sustainability benefits, and he claims that Non-GMO Project attacks "those who dedicate their lives to bring healthy choices to Americans."

Mandatory GMO Food Labeling
Public demands for transparency of food ingredients have sparked controversy regarding mandatory labeling laws for GMO foods. Andrew Kimbrell, executive director at the Center for Food Safety, equates the absence of GMO labels to a betrayal of "public trust by denying Americans the right to know how their food is produced." Conversely, farmers and food companies are among the biggest supporters of relaxed labeling laws. They argue that uniform labeling across all types of GMO products is actually misleading consumers, as different types of GMOs carry very different risks. They cite the complete disappearance of GMO foods in Europe starting in 1999, when mandatory labeling laws worried corporations and caused them to remove nearly all GMO products from shelves. The absence of GMO products in the US would place the biggest burden on the lower class; Northbridge Environmental Management Consultants found that labeling and the subsequent decline in GMOs would increase the average annual food bill of a family of four by $400.

Conclusion
Is Golden Rice a doomed technology developed for the benefit of profit-seeking corporations? Or Is it a case study of how conflicting opinions and unfalsifiable campaigns can destroy a beneficial advancement?

Regardless of the true cause of golden rice's difficult implementation, the reality is that the once dubbed "Grains of Hope" have fallen short of their promising expectations. Nearly two decades after the original Time Magazine article, Golden Rice has not gained regulatory approval in target countries India, Bangladesh, and the Philippines; furthermore it will not be made public until "all necessary permits have been received. Golden Rice sets a clear precedent for the challenges of implementing biotechnology into a skeptical society, and further research could explore how agricultural biotech companies are aiming to gain the trust of the public.

As the world population continues to grow, it is estimated that global food demand will increase by over 70% by 2050. Agricultural biotechnology will play a crucial role in fighting world hunger, malnutrition, and food insecurity in the coming decades. The failure of implementing golden rice and the subsequent spread of anti-GMO campaigns illustrate how different groups' agendas can distract from the original goal of a product and its creators. As President Jimmy Carter said: "Responsible biotechnology is not the enemy. Starvation is."

title: Lentis/Golden Rice


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Visual Basic for Applications/Generate Random Dictionary Words

Summary
This VBA code module works only in MS Word. It generates a pseudo random list of dictionary words, listing them in the immediate window of the VBA editor.

The VBA Code

 * Copy the code listing into a standard VBA module in MS Word and save the file as a docm type. Set the user options in the top section and run the procedure to produce the list.
 * The variable Num sets the number of words as output, and the variable TypoLen should be set to the approximate length of the sought-after words.
 * There is no ready-made collection of English words called a Dictionary in MS Word; the term is used for sets of user-constructed word lists. To access a list of English words, an indirect method needs to be used.  A random word is first generated equal in length to TypoLen.  The spell checker then fails to recognize the word so generates suggestions, this time as a proper collection.   Provided that there is at least one such suggestion in the spell check collection, a random choice is made from among them for the output list.   The loop continues until the chosen number of words has been produced.

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


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LMIs in Control/pages/Unsat Inp Stabilization

LMI For Stabilization Condition for Systems With Unsymmetrical Saturated Control

The LMI in this page gives the feasibility conditions which, if satisfied, imply that the correstponding system can be stabilized.

The System

\begin{align} \dot x(t)&=Ax(t)+BSat(u(t)), \\ x(0)&=x_0, \\ Sat(u)_i = , \end{align}$$

where $$ x \in \|R^n $$ is the state, $$ u \in \|R^m $$ is the control input.

For the system given as above, its symmetrical saturated control form can be derived by following the procedure in the original article. The new system will have the form:

\begin{align} \dot x(t)&=Ax(t)+\tilde{B}sat(z(t))+Ew \end{align}$$

where $$ w_i = u_i - \frac{\alpha_i - \beta_i}{2}, z_i = w_i\frac{2}{\alpha_i + \beta_i} $$

The Data
The system matrices $$ (A,\tilde{B},E) $$, the saturation bounds $$ (\alpha_i, \beta_i) $$ of the control inputs. Positive scalars $$ \rho, \eta $$.

The LMI: The Stabilization Feasibility Condition

\begin{align} &\text{Find} \; X,Y,Z:\\ &\text{subj. to: } X > 0, \\ & \quad \begin{bmatrix} AX + \tilde{B}(D_sY+\hat{D}_s^-Z) \end{bmatrix} + \begin{bmatrix} AX + \tilde{B}(D_sY+\hat{D}_s^-Z) \end{bmatrix}^\top + \frac{\eta}{\rho}X + \frac{1}{\eta}EE^\top < 0, \\ \begin{bmatrix} \mu & Z_i \\ * X \end{bmatrix} > 0, i = 1,...,\bar{m} \end{align}$$

Here $$ D_s $$ is a diagonal matrix with a component either 0 or 1, and $$ D_s+D_s^- = \frac{\Lambda + \Tau}{2} $$ and $$ \hat{D}_s^- = e_{f_m} \times D_s^- $$

Conclusion:
The feasibility of the given LMI implies that the system is stabilizable with control gains $$ K =YX^{-1}, H = ZX^{-1} $$.

Implementation
A link to CodeOcean or other online implementation of the LMI


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LMIs in Control/pages/Hinf Output Optimal Control

$$H_\infty$$ Optimal Output Controllability for Systems With Transients

This LMI provides an $$H_\infty$$ optimal output controllability problem to check if such controllers for systems with unknown exogenous disturbances and initial conditions can exist or not.

The System

\begin{align} \dot x&=Ax+B_1 v + B_2 u, x(0) = x_0, \\ z&=C_1 x+D_{11} v+D_{12} u, \\ y&= C_2x + D_{21}v, \end{align}$$

where $$ x \in \R^n $$ is the state, $$ v \in \R^r $$ is the exogenous input, $$ u\in \R^m $$ is the control input, $$ y \in \R^p $$ is the measured output and $$ z \in \R^s $$ is the regulated output.

The Data
System matrices $$ (A,B_1,B_2,C_1,C_2,D_{11},D_{12},D_{21},D_{22}) $$ need to be known. It is assumed that $$ v \in L_2[0,\infty) $$. $$ N_1, N_2 $$ are matrices with their columns forming the bais of kernels of $$ C_2D_{21} $$ and $$ C_2D_{12} $$ respectively.

The Optimization Problem
For a given $$ \gamma $$, the following $$H_\infty $$ condition needs to be fulfilled:

$$ \gamma_w = sup_{\|v\|^2_\infty+x_0^\top R x_0 \neq 0} \frac{\|z\|_\infty}{(\|v\|_\infty^2+x_0^\top R x_0)^{1/2}} < \gamma_w, $$

The LMI: $$H_\infty$$ Output Feedback Controller for Systems With Transients

\begin{align} &\text{min}_{\gamma, X_{11}, Y_{11}}: \gamma \\ & \text{subj. to: } X_{11} > 0, Y_{11} > 0, \\ & \quad \begin{bmatrix} N_1 & 0 \\ 0 & I \end{bmatrix}^\top \begin{bmatrix} A^\top X_{11} + X_{11}A & X_{11}B_1 & C_1^\top \\ * & -\gamma^2I & D_{11}^\top \\ * & * & -I \end{bmatrix} \begin{bmatrix} N_1 & 0 \\ 0 & I \end{bmatrix} < 0, \\ & \quad \begin{bmatrix} N_2 & 0 \\ 0 & I \end{bmatrix}^\top \begin{bmatrix} A Y_{11} + Y_{11}A^\top & Y_{11}C_1^\top & B_1 \\ * & -I & D_{11} \\ * & * & -\gamma^2I \end{bmatrix} \begin{bmatrix} N_2 & 0 \\ 0 & I \end{bmatrix} < 0, \\ & \quad \begin{bmatrix} X_{11} & I \\ I & Y_{11} \end{bmatrix} \geq 0, X_{11} < \gamma^2 R, \end{align}$$

Conclusion:
Solution of the above LMI gives a check to see if an $$H_\infty$$ optimal output controller for systems with transients can exist or not.

Implementation
A link to CodeOcean or other online implementation of the LMI

Related LMIs
Links to other closely-related LMIs


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LMIs in Control/pages/dt mixed H2 Hinf optimal output feedback control

WIP, Description in progress

This part shows how to design dynamic outpur feedback control in mixed $$\mathcal{H}_2$$ and $$\mathcal{H}_\infty$$ sense for the continuous time.

Problem
Consider the discrete-time generalized LTI plant $$\mathcal{P}$$ with minimal state-space realization

$$ \dot{x}=Ax+\begin{bmatrix}B_{1,1} & B_{1,2} \end{bmatrix}\begin{bmatrix}w_{1} \\ w_{2} \end{bmatrix} +B_{2}u,$$

$$ \begin{bmatrix}z_{1} \\ z_{2} \end{bmatrix}=\begin{bmatrix}C_{1,1} \\ D_{1,2} \end{bmatrix}x_k + \begin{bmatrix}D_{11,11} & D_{11,12} \\ D_{11,21} & D_{11, 22}\end{bmatrix}\begin{bmatrix}w_{1} \\ w_{2} \end{bmatrix}+\begin{bmatrix}D_{12,1} \\ D_{12,2} \end{bmatrix}u,$$

$$y=C_{d2}x + \begin{bmatrix}D_{21,1} & D_{21,2}\end{bmatrix}\begin{bmatrix}w_{1} \\ w_{2} \end{bmatrix}+D_{d22}u$$

Theorem
A continuous-time dynamic output feedback LTI controllerwith state-space realization $$(A_c, B_c,C_c,D_c)$$ is to be designed to minimize the $$\mathcal{H}_2$$ norm of the closed-loop transfer matrix $$T_{11}(s)$$ from the exogenous input $$w_1$$ to the performance output $$z_1$$ while ensuring the H∞ norm of the closed-loop transfer matrix $$T_{22}(s)$$ from the exogenous input $$w_2$$ to the performance output $$z_2$$ is less than $$\gamma_d$$, where

$$T_{11}(s) = C_{CL1,1}(sI-A_{CL})^{-1}B_{CL1,1},$$

$$T_{22}(s) = C_{CL1,2}(sI-A_{CL})^{-1}B_{CL1,2}+D_{CL11,22},$$

$$A_{d_{CL}} = \begin{bmatrix} A+B_2D_c\tilde{D}^{-1}C_{2} & B_{2}(I+D_{c}\tilde{D}^{-1}D_{22})C_{c} \\ B_{c}\tilde{D}^{-1}C_{2} & A_{c}+B_{c}\tilde{D}^{-1}D_{22}C_{c} \end{bmatrix}$$,

$$B_{CL1,1} = \begin{bmatrix}B_{1,1}+B_{2}D_{c}\tilde{D}^{-1}D_{21,1} \\ B_{c}\tilde{D}^{-1}D_{21, 1}\end{bmatrix}$$,

$$B_{CL1,2} = \begin{bmatrix}B_{1,2}+B_{2}D_{c}\tilde{D}^{-1}D_{21,2} \\ B_{c}\tilde{D}^{-1}D_{21, 2}\end{bmatrix}$$,

$$C_{CL1,1} = \begin{bmatrix}C_{1,1}+D_{12,1}D_{c}\tilde{D}^{-1}C_{2,1} & D_{12,1}(I+D_{c}\tilde{D}^{-1}D_{22})C_{c}\end{bmatrix}$$,

$$C_{CL1,2} = \begin{bmatrix}C_{1,2}+D_{12,2}D_{c}\tilde{D}^{-1}C_{2,2} & D_{12,2}(I+D_{c}\tilde{D}^{-1}D_{22})C_{c}\end{bmatrix}$$,

$$D_{CL11,22} = D_{11,22}+D_{12,2}D_{c}\tilde{D}^{-1}D_{21,2}$$,

and $$\tilde{D} = I - D_{22}D_{c}$$.

Synthesis Method
Solve for $$A_{n}\in \mathbb{R}^{n_x\times n_x}, B_{n} \in \mathbb{R}^{n_x \times n_x}, C_{n}\in \mathbb{R}^{n_u \times n_x}, D_{n} \in \mathbb{R}^{n_u\times n_y}, X_1, Y_1\in \mathbb{S}^{n_x}, Z \in \mathbb{S}^{n_{Z_1}},$$ and $$ \mu \in \mathbb{R}_{>0}$$ that minimizes $$\mathcal{J}(\mu) = \mu$$ subjects to $$X_1>0, \ Y_1>0 \ Z>0,$$

$$\begin{bmatrix}N_{11} & A+A_n^T + B_2D_nC_2 & B_{1,1}+B_2D_nD{21,1} \\
 * & X_1A+A^TX_1+B_nC_2+C_2^TB_n^T & X_1B_{1,1} + B_nD_{21,1} \\
 * & * & -I \end{bmatrix} < 0$$,

$$\begin{bmatrix}N_{11} & A+A_n^T + B_2D_nC_2 & B_{1,1}+B_2D_nD{21,1} & Y_1^TC_{1,2}^T+C_n^TD_{12,2}^T\\
 * & X_1A+A^TX_1+B_nC_2+C_2^TB_n^T & X_1B_{1,1} + B_nD_{21,1} & C_{1,2}^T+C_2^TD_n^TD_{12,2}^T\\
 * & * & -\gamma_d I & D_{11,22}^T+D_{21,2}^TD_n^TD_{12,2}^T \\
 * & * & * & -\gamma_d I\end{bmatrix} < 0$$,

$$\begin{bmatrix}Y_1 I Y_1C_{1,1}^T+C_n^TD_{12,1}^T \\
 * & X_1 & C_{1,1}^T+C_2^TD_n^TD_{12,1}^T \\
 * & * & Z \end{bmatrix} > 0$$,

$$D_{11,11}+ D_{12,1}D_{n}D_{21,1}=0,$$

$$\begin{bmatrix}X_1 & I \\
 * & Y_1\end{bmatrix}>0,$$

tr$$Z<\mu,$$

where $$N_{11} = AY_1+Y_1A^T+B_2C_n+C_n^TB_2^T$$.

The controller is recovered by

$$A_{c} = A_{K}-B{c}(I-D_{22}D_{c})^{-1}D_{22}C_{c}, $$

$$B_{c} = B_{K}(I-D_{c}D_{22}),$$

$$C_{c} = (I-D_{c}D_{22})C_{K},$$

D_{c} = (I+D_{K}D{22})^{-1}D_{K},

where

$$\begin{bmatrix} A_{K} & B_{K} \\ C_{K} & D_{K} \end{bmatrix} = \begin{bmatrix} X_2 & X_1B_{2} \\ 0 & I \end{bmatrix}^{-1}(\begin{bmatrix} A_{n} & B_{n} \\ C_{n} & D_{n} \end{bmatrix}-\begin{bmatrix} X_1A_dY_1 & 0 \\ 0 & 0 \end{bmatrix})\begin{bmatrix} Y_2^T & 0 \\ C_{2}Y_1 & I \end{bmatrix}^{-1}$$, and the matrices $$X_2$$ and $$Y_2$$ satisfy $$X_2Y_2^T=I-X_1Y_1$$. If $$D_{22}=0$$, then $$A_{c}=A_{K}, B_{c}=B{K}, C_{c}=C_{K}$$ and $$D_{c}=D_{K}$$.

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

If $$D_{11,11} = 0, D_{12,1} \neq 0, \text{ and } D_{21,1} \neq 0,$$ then it is often simplest to choose $$D_{n} = 0$$ in order to satisfy the equality constraint $$D_{11,11}+ D_{12,1}D_{n}D_{21,1}=0,$$.

WIP, additional references to be added


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LMIs in Control/pages/dt mixed H2 Hinf optimal dynamic output feedback control

WIP, Description in progress

This part shows how to design dynamic outpur feedback control in mixed $$\mathcal{H}_2$$ and $$\mathcal{H}_\infty$$ sense for the discrete time.

Problem
Consider the discrete-time generalized LTI plant $$\mathcal{P}$$ with minimal state-space realization

$$ x_{k+1}=A_dx_k+\begin{bmatrix}B_{d1,1} & B_{d1,2} \end{bmatrix}\begin{bmatrix}w_{1,k} \\ w_{2,k} \end{bmatrix} +B_{d,2}u_k,$$

$$ \begin{bmatrix}z_{1,k} \\ z_{2,k} \end{bmatrix}=\begin{bmatrix}C_{d1,1} \\ D_{d1,2} \end{bmatrix}x_k + \begin{bmatrix}D_{d11,11} & D_{d11,12} \\ D_{d11,21} & D_{d11, 22}\end{bmatrix}\begin{bmatrix}w_{1,k} \\ w_{2,k} \end{bmatrix}+\begin{bmatrix}D_{12,1} \\ D_{12,2} \end{bmatrix}u_k,$$

$$y_k=C_{d2}x_k + \begin{bmatrix}D_{21,1} & D_{21,2}\end{bmatrix}\begin{bmatrix}w_{1,k} \\ w_{2,k} \end{bmatrix}+D_{d22}u_k$$

Theorem
A discrete-time dynamic output feedback LTI controller with state-space realization $$(A_{dc}, B_{dc},C_{dc},D_{dc})$$ is to be designed to minimize the $$\mathcal{H}_2$$ norm of the closed loop transfer matrix $$T_{11}(z)$$ from the exogenous input $$w_1,k$$ to the performance output $$z_1,k$$ while ensuring the $$\mathcal{H}_\infty$$ norm of the closed-loop transfer matrix $$T_{22}(z)$$ from the exogenous input $$w_2,k$$ to the performance output $$z_2,k$$ is less than $$\gamma_d$$, where

$$T_{11}(z) = C_{d_{CL}1,1}(zI-A_{d_{CL}})^{-1}B_{d_{CL}1,1},$$

$$T_{22}(z) = C_{d_{CL}1,2}(zI-A_{d_{CL}})^{-1}B_{d_{CL}1,2}+D_{d_{CL}11,22},$$

$$A_{d_{CL}} = \begin{bmatrix} A_d+B_{d2}D_{dc}\tilde{D}_d^{-1}C_{d2} & B_{d2}(I+D_{dc}\tilde{D}_d^{-1}D_{d22})C_{dc} \\ B_{dc}\tilde{D}^{-1}_dC_{d2} & A_{dc}+B_{dc}\tilde{D}^{-1}_dD_{d22}C_{dc} \end{bmatrix}$$,

$$B_{d_{CL}1,1} = \begin{bmatrix}B_{d1,1}+B_{d2}D_{dc}\tilde{D}_d^{-1}D_{d21,1} \\ B_{dc}\tilde{D}_d^{-1}D_{d21, 1}\end{bmatrix}$$,

$$B_{d_{CL}1,2} = \begin{bmatrix}B_{d1,2}+B_{d2}D_{dc}\tilde{D}_d^{-1}D_{d21,2} \\ B_{dc}\tilde{D}_d^{-1}D_{d21, 2}\end{bmatrix}$$,

$$C_{d_{CL}1,1} = \begin{bmatrix}C_{d1,1}+D_{d12,1}D_{dc}\tilde{D}_d^{-1}C_{d2,1} & D_{d12,1}(I+D_{dc}\tilde{D}_d^{-1}D_{d22})C_{dc}\end{bmatrix}$$,

$$C_{d_{CL}1,2} = \begin{bmatrix}C_{d1,2}+D_{d12,2}D_{dc}\tilde{D}_d^{-1}C_{d2,2} & D_{d12,2}(I+D_{dc}\tilde{D}_d^{-1}D_{d22})C_{dc}\end{bmatrix}$$,

$$D_{d_{CL}11,22} = D_{d11,22}+D_{d12,2}D_{dc}\tilde{D}^{-1}_dD_{d21,2}$$,

and $$\tilde{D}_d = I - D_{d22}D_{dc}$$.

Synthesis Method
Solve for $$A_{dn}\in \mathbb{R}^{n_x\times n_x}, B_{dn} \in \mathbb{R}^{n_x \times n_x}, C_{dn}\in \mathbb{R}^{n_u \times n_x}, D_{dn} \in \mathbb{R}^{n_u\times n_y}, X_1, Y_1\in \mathbb{S}^{n_x}, Z \in \mathbb{S}^{n_{Z_1}},$$ and $$ \mu \in \mathbb{R}_{>0}$$ that minimizes $$\mathcal{J}(\mu) = \mu$$ subjects to $$X_1>0, \ Y_1>0 \ Z>0,$$

$$\begin{bmatrix}X_1 & I & X_1A_d + B_{d_n}C_{d_2} & A_{d_n} & X_1B_{d1,1}+B{d_n}D_{d21,1} \\ \end{bmatrix}>0,$$
 * & Y_1 & A_d+B_{d2}C_{dn}D_{d2} & A_dY_1+B_{d2}C_{dn} & B_{d1, 1}+B_{d2}D_{dn}D_{d21, 1} \\
 * & * & X_1 & I & 0 \\
 * & * & * & Y_1 & 0 \\
 * & * & * & * & I

$$\begin{bmatrix}X_1 & I & X_1A_d + B_{d_n}C_{d_2} & A_{d_n} & X_1B_{d1,2}+B{d_n}D_{d21,2} & 0\\ \end{bmatrix}>0$$
 * & Y_1 & A_d+B_{d2}C_{dn}D_{d2} & A_dY_1+B_{d2}C_{dn} & B_{d1, 2}+B_{d2}D_{dn}D_{d21, 2} & 0 \\
 * & * & X_1 & I & 0 & C_{d1,2}^T + C_{d2}^TD_{dn}^TD_{d12,2}^T \\
 * & * & * & Y_1 & 0 & Y_1 C_{d1,2}^T+C_{dn}^TD_{d12,2}^T\\
 * & * & * & * & -\gamma_d I & D^T_{d11,22} + D_{d21,2}^TD_{dn}^TD{d12,2}^T \\
 * & * & * & * & * & -\gamma_d I

$$\begin{bmatrix}Z & C_{d1,1}+D_{d12,1}D_{dn}C_{d2} & C_{d1,1,}Y_1^T+D_{d12,1}C_{dn} \\
 * & X_1 & I \\
 * & * & Y_1\end{bmatrix}>0,$$

$$D_{d11,11}+ D_{d12,1}D_{dn}D_{d21,1}=0,$$

$$\begin{bmatrix}X_1 & I \\
 * & Y_1\end{bmatrix}>0,$$

tr$$Z<\mu.$$

The controller is recovered by

$$A_{dc} = A_{dK}-B{dc}(I-D_{d22}D_{dc})^{-1}D_{d22}C_{dc}, $$

$$B_{dc} = B_{dK}(I-D_{dc}D_{d22}),$$

$$C_{dc} = (I-D_{dc}D_{d22})C_{dK},$$

D_{dc} = (I+D_{dK}D{d22})^{-1}D_{dK},

where

$$\begin{bmatrix} A_{dK} & B_{dK} \\ C_{dK} & D_{dK} \end{bmatrix} = \begin{bmatrix} X_2 & X_1B_{d2} \\ 0 & I \end{bmatrix}^{-1}(\begin{bmatrix} A_{dn} & B_{dn} \\ C_{dn} & D_{dn} \end{bmatrix}-\begin{bmatrix} X_1A_dY_1 & 0 \\ 0 & 0 \end{bmatrix})\begin{bmatrix} Y_2^T & 0 \\ C_{d2}Y_1 & I \end{bmatrix}^{-1}$$, and the matrices $$X_2$$ and $$Y_2$$ satisfy $$X_2Y_2^T=I-X_1Y_1$$. If $$D_{22}=0$$, then $$A_{dc}=A_{dK}, B_{dc}=B{dK}, C_{dc}=C_{dK}$$ and $$D_{dc}=D_{dK}$$.

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

If $$D_{d11,11} = 0, D_{d12,1} \neq 0, \text{ and } D{d21,1} \neq = 0,$$ then it is often simplest to choose $$D_{dn} = 0$$ in order to satisfy the equality constraint $$D_{d11,11}+ D_{d12,1}D_{dn}D_{d21,1}=0,$$.

WIP, additional references to be added


text:
LMIs in Control/pages/reduced order state estimation

WIP, Description in progress

In this page, we investigate an LMI approach for the design of the problem of reduced-order observer design for the linear system.

System Setting
$$ \dot{x} = Ax + Bu $$

$$ y = Cx$$

where $$x \in \mathbb{R}^n, \ u \in \mathbb{R}^r, \text{ and } y \in \mathbb{R}^m$$ are the state vector, the input vector, and the output vector, respectively. Without loss of generality, it is assumed that rank$$(C) = m \leq n$$.

In the design of reduced-order state observers for linear systems, the following lemma performs a fundamental role.

Lemma
Given the linear system, and let $$R \in \mathbb{R}^{(n-m)\times n}$$ be an arbitrarily chosen matrix which makes the matrix

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

nonsingular, then

$$CT^{-1}=\begin{bmatrix}I_m & 0 \end{bmatrix}$$.

Furthermore, let

$$TAT^{-1}=\begin{bmatrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}$$, $$A_{11} \in \mathbb{R}^{m\times m}$$,

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

Let

$$Tx=\begin{bmatrix}x_1 \\ x_2 \end{bmatrix} $$, $$TB=\begin{bmatrix}B_1 \\ B_2 \end{bmatrix} $$,

then it follows from the relations in previous 3 equations that system is equivalent to

$$\begin{bmatrix}\dot{x}_1 \\ \dot{x}_2 \end{bmatrix}=\begin{bmatrix}A_{11} & A_{12} \\ A_{21} & A_{22}\end{bmatrix}\begin{bmatrix}x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix}B_1 \\ B_2 \end{bmatrix}u$$,

$$y=x_1$$

In the equivalent system, the substate vector $$x_1$$ is directly equal to the output $$y$$ of the original system. Thus to reconstruct the state of the original system, we suffice to get an estimate of the substate vector $$x_2$$, namely, $$\hat{x}_2$$, from the earlier equivalent system. Once an estimate $$\hat{x}_2$$ is obtained, an estimate of $$x(t)$$, that is, the state vector of original system, can be obtained as

$$\hat{x}(t)=T^{-1}\begin{bmatrix}y(t) \\ \hat{x}_2(t) \end{bmatrix} $$.

Problem Formulation
For the equivalent continuous-time linear system, design a reduced-order state observer in the form of

$$\dot{z} = Fz+Gy+Hu$$

$$\hat{x}_2=Mz+Ny$$

such that for arbitrary control input $$u(t)$$, and arbitrary initial system values $$x1(0),\ x2(0), \text{ and } z(0)$$, there holds

$$\text{lim}_{t\rightarrow \infty} (x_2(t) - \hat{x}_2(t) = 0$$.

Solution/Theorem
Problem has a solution if and only if one of the following two conditions holds:

1. There exist a symmetric positive definite matrix P and a matrix W satisfying

$$PA_{22}+A_{22}^TP+WA_{12}+A_{12}^TW^T < 0$$

2. There exists a symmetric positive definite matrix P satisfying

$$PA_{22}+A_{22}^TP-A_{12}A_{12}^T < 0$$

In this case, a reduced-order state observer can be obtained as in problem with

$$F=A_{22}+LA_{12}, \ G=(A_{21}+LA_{11})-(A_{22}+LA_{12})L,$$

$$H=B_{2}+LB_{1}, \ M=I, \ N=-L,$$

where

$$ L = P^{-1}W $$ with W and $$P >0$$ being a pair of feasible solutions to the first inequality condition or $$ L = -\frac{1}{2}P^{-1}A_{12}^T $$ with $$P>0$$ being a solution to the second inequality condition.

WIP, additional references to be added


text:
LMIs in Control/pages/full order Hinf H2 state observers

WIP, Description in progress

In this section, we treat the problem of designing a full-order state observer for system such that the effect of the disturbance $$w(t)$$ to the estimate error is prohibited to a desired level.

System Setting
The system is following

$$\dot{x}(t) = Ax(t) + B_1u(t) + B_2w(t), x(0) = x_0, $$

$$y(t) = C_1x(t) + D_1u(t) + D_2w(t), $$

$$z(t) = C_2x(t), $$

where $$x\in \mathbb{R}^n, \ y\in \mathbb{R}^l, \ z\in \mathbb{R}^m$$ are respectively the state vector, the measured output vector, and the output vector of interests.

$$ w \in \mathbb{R}^p$$ are the disturbance vector and control vector, respectively.

$$ A, \ B_1, \ B_2, \ C_1, \ C_2, \ D_1, \text{ and } D_2$$ are the system coefficient matrices of appropriate dimensions.

Problem Formulation
For the system, we introduce a full-order state observer in the following form:

$$\dot{\hat{x}}=(A+LC_1)\hat{x}-Ly+(B_1+LD_1)u$$

where $$\hat{x}$$ is the state observation vector and $$L\in \mathbb{R}^{n\times m}$$ is the observer gain. Obviously, the estimate of the interested output is given by

$$\hat{z}(t) = C_2\hat{x}(t)$$

which is desired to have as little affection as possible from the disturbance $$w(t)$$.

Using system dynamics,

$$\begin{align}\dot{x}(t) &= Ax(t) + B_1u(t) + B_2w(t) \\ &=Ax(t)+Ly-Ly+B_1u(t)+B_2w(t) \\ &=(A+LC_1)x(t)-Ly(t)+(B_1+LD_1)u(t)+(B_2+LD_2)w(t)\end{align}$$

Denoting

$$\dot{e} = (A+LC_1)e+(B_2+LD_2)w$$

$$\tilde{z}(t)=C_2e$$.

The transfer function of the system is clearly given by

$$ G_{\tilde{z}w}(s) = C_2(sI-A-LC_1)^{-1}(B_2+LD_2)$$.

With the aforementioned preparation, the problems of $$\mathcal{H}_\infty \text{ and } \mathcal{H}_2$$ state observer designs can be stated as follows.

Problem 1
($$\mathcal{H}_\infty$$ state observers) Given system (9.22) and a positive scalar $$\gamma$$ , find a matrix $$L$$ such that

$$||G_{\tilde{z}w}(s)||_\infty < \gamma$$.

Problem 2
($$\mathcal{H}_2$$ state observers) Given system (9.22) and a positive scalar $$\gamma$$ , find a matrix $$L$$ such that

$$||G_{\tilde{z}w}(s)||_2 < \gamma$$

As a consequence of the requirements in the previous problems, the error system is asymptotically stable, and hence we have

$$e(t) = x(t) - \hat{x}(t) \rightarrow 0, \text{ as } t\rightarrow \ \infty$$

This states that $$\hat{x}(t)$$ is an asymptotic estimate of $$x (t)$$.

Solution/Theorem
Regarding the solution to the problem of H∞ state observers design, we have the following theorem.

Theorem 1
The $$\mathcal{H}_\infty$$ state observers problem 1 has a solution if and only if there exist a matrix $$W$$ and a symmetric positive definite matrix $$P$$ such that

$$ \begin{bmatrix} A^TP+C_1^TW^T+PA+WC_1 & PB_2+WD_2 & C_2^T \\ (PB_2+WD_2)^T & -\gamma I & 0 \\ C_2 & 0 & -\gamma I \end{bmatrix}<0$$

When such a pair of matrices W and P are found, a solution to the problem is given as

$$L=P^{-1}W$$

With a prescribed attenuation level, the problem of H∞ state observers design is turned into an LMI feasibility problem in the form problem stated before. The problem with a minimal attenuation level $$\gamma$$ can be sought via the following optimization problem:

min $$\gamma$$

s.t. $$ \begin{align}P&>0 \\ \begin{bmatrix} A^TP+C_1^TW^T+PA+WC_1 & PB_2+WD_2 & C_2^T \\ (PB_2+WD_2)^T & -\gamma I & 0 \\ C_2 & 0 & -\gamma I \end{bmatrix}&<0 \end{align}$$

Theorem 2
The $$\mathcal{H}_2$$ state observers problem 2 has a solution the following 2 conclusions hold.

1.It has a solution if and only if there exists a matrix W, a symmetric matrix Q, and a symmetric matrix X such that

$$ \begin{bmatrix} XA+WC_1+(XA+WC_1)^T & XB_2+WD_2 \\ (XB_2+WD_2)^T & -I \end{bmatrix}<0$$,

$$\begin{bmatrix}-Q & C_2 \\ C_2^T & -X\end{bmatrix}<0$$,

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

When such a triple of matrices are obtained, a solution to the problem is given as

$$L=X^{-1}W$$.

2. It has a solution if and only if there exists a matrix V, a symmetric matrix Z, and a symmetric matrix Y such that

$$A^TY+C_1^TV^T+YA+VC_1+C_2^TC_2<0$$,

$$\begin{bmatrix}-Z & (YB_2+VD_2)^T \\ YB_2+VD_2 & -Y\end{bmatrix}<0$$,

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

When such a triple of matrices are obtained, a solution to the problem is given as

$$L=Y^{-1}V$$.

In applications, we are often concerned with the problem of finding the minimal attenuation level $$\gamma$$. This problem can be solved via the optimization

min $$\rho$$

s.t. $$ \begin{bmatrix} XA+WC_1+(XA+WC_1)^T & XB_2+WD_2 \\ (XB_2+WD_2)^T & -I \end{bmatrix}<0$$,

$$\begin{bmatrix}-Q & C_2 \\ C_2^T & -X\end{bmatrix}<0$$,

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

or

min $$\rho$$

$$A^TY+C_1^TV^T+YA+VC_1+C_2^TC_2<0$$

$$\begin{bmatrix}-Z & (YB_2+VD_2)^T \\ YB_2+VD_2 & -Y\end{bmatrix}<0$$

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

When a minimal ρ is obtained, the minimal attenuation level is $$\gamma=\sqrt{\rho}$$.

WIP, additional references to be added


text:
LMIs in Control/pages/H inf Aircraft Optimization

Robust $$H_\infty $$ Aircraft Dynamics

The Optimization Problem
This Optimization problem involves the use of optimizing aircraft dynamics using the regulator framework, and optimizing the given aircraft parameters using the following set of aircraft dynamics. these aircraft dynamics are given non-dimensional characteristics defined by various parameters of the aircraft being tested. By making these characteristics non-dimensional, it allows for the problem to be scaled to larger porportions. For instance the longitudinal dynamics for an aircraft system are defined as suchː

\begin{align} A_p = \begin{bmatrix} X_u & X_w & 0 & - g_0 \cos(\vartheta_0) \\ \frac{Z_u}{1-Z_{\dot{w}}} & \frac{Z_w}{1-Z_{\dot{w}}} & \frac{u_0 + Z_q}{1-Z_{\dot{w}}} & \frac{-g_0 \sin(\vartheta_0)}{1 - Z_{dot{w}}}\\ M_u + \frac{M_{\dot{w}} Z_u}{1 - Z_{\dot{w}}} & M_w + \frac{M_{\dot{w}} Z_w}{1 - Z_{\dot{w}}} & M_w + \frac{(u_0 + Z_q) M_{\dot{w}}}{1 - Z_{dot{w}}} & \frac{-M_{\dot{w}} g_0 \sin(\vartheta_0)}{1 - Z_{dot{w}}}\\ 0 & 0 & 1 & 0\\	\end{bmatrix}\\ B_p = \begin{bmatrix} X_{\delta_e} & X_{\delta_T} \\ \frac{Z_{\delta_e}}{1-Z_{\dot{w}}} & \frac{Z_{\delta_T}}{1-Z_{\dot{w}}} \\ M_{\delta_e} + \frac{M_{\dot{w}} Z_{\delta_e}}{1 - Z_{\dot{w}}} & M_{\delta_T} + \frac{M_{\dot{w}} Z_{\delta_T}}{1 - Z_{\dot{w}}} \\ 0 & 0\\	\end{bmatrix}\\ C_p = I \\ D_p = 0 \\ \end{align}$$

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 a robust $$H_\infty $$ optimization.

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: Robust $$H_\infty$$ Optimal Aircraft Dynamics Control
There exists the scalar, $$ \gamma $$, along with the matrices $$ S > 0 $$, and $$ R $$ where:

\begin{align} \begin{bmatrix} N_R & 0 \\ 0 & I \\ \end{bmatrix}^{T} \begin{bmatrix} A R+R A^{T} & R C^T_1 & B_1 \\ C_1 R & -\gamma I & D_{11} \\ B^{T}_{1} & D^{T}_{11} & -\gamma I \\ \end{bmatrix} \begin{bmatrix} N_R & 0 \\ 0 & I \\ \end{bmatrix} < 0 \\

\begin{bmatrix} N_S & 0 \\ 0 & I \\ \end{bmatrix}^{T} \begin{bmatrix} A^{T} S+S A & S B_1 & C^{T}_1 \\ B^{T}_1 S & -\gamma I & D^{T}_{11} \\ C_1 & D_{11} & -\gamma I \\ \end{bmatrix} \begin{bmatrix} N_S & 0 \\ 0 & I \\ \end{bmatrix} < 0 \\

\begin{bmatrix} R & I \\ I & S \\ \end{bmatrix} \geq 0 \end{align}$$

Where the optimized controller matrices are defined as followsː

\begin{align} A_K = -N[A^T + S(A + B_2 F+L C_2)R+1/\gamma S(B_1 + L D_{21})B^{T}_1 + 1/\gamma C^{T}_1 (C_1 + D_{12}F)R]M^{T} \\ \\	B_K = N^{-1} S L \\ \\	C_K = F R M^{-T} \\ \\	D_K = 0; \end{align} $$

Where:

\begin{align} N = S \\ \\	M = S^{-1} - R \\ \\	F = -(D^{T}_{12} D_{12})^{-1} (\gamma B^{T}_2 R^{-1} + D^{T}_{12} C_1) \\ \\       L = -(\gamma S^{-1} C^{T}_2 + B_1 D^{T}_{21}) (D_{21} D^{T}_{21})^{-1} \\ \\ \end{align} $$

Conclusion:
The results from this LMI give a controller that is a robust $$ H_\infty $$, optimization which would be capable of stabilizing various aircraft Dynamics.

Implementation

 * Robust H_inf optimization example


text:
Lentis/Abortion in America as a Sociotechnical Controversy

Introduction
Abortion will be explored as an American socio-technical controversy from the 19th century up until present day. Key influential figures leading up to the abortion ban in 1873 will be analyzed; abortion technologies in the 20th century and current controversies will be addressed.

Menstruation, Fertilization, and Pregnancy
Pregnancy begins with the menstrual cycle, a 28 day process that is divided into the menstrual, follicular, ovulation, and luteal phases. The lining of the uterus is shed during the menstrual stage. During the follicular phase, the uterine lining thickens and the egg matures. Two weeks into the cycle, the egg is released from the ovary during ovulation to be fertilized, surviving for 24 hours; this is the only time when a woman can conceive. Pregnancy requires the buildup of a uterine lining, so “doctors date pregnancy from the first day of a woman's last menstrual period, which is typically 2 weeks before fertilization".

Trimester I (Week 1 to Week 13)
Fertilization marks the beginning of Week 3 of a pregnancy. The fertilized egg travels towards the uterus, undergoing cell division and implantation. The embryo is formed from the inner layer of the cells. At Week 11, the embryo is considered a fetus and measures 2 inches in length.

Trimester II (Week 13 to Week 27) & III (Week 27 to Week 40)
By 24 weeks of pregnancy, a fetus can survive outside of pregnancy with the help of a neonatal unit.

The fetus’ skeleton, neural network, and senses continue developing through the last trimester. A baby is considered full term by the end of Week 36 but most women give birth between 38 and 42 weeks.

Abortion in the Modern Era
As of 2015, 65.4% of abortions were performed at or before Week 8 of pregnancy, 91.1% were performed at or before 13 weeks of pregnancy and 98.7% of abortions were performed before Week 21.

Early Abortion
Up until the late 19th century, abortion prior to quickening was considered legal. Quickening is when pregnant women are able to feel and/or perceive fetal movements, usually around the 13 to 25 week mark during pregnancy. It was largely believed that the fetus was not alive until quickening was felt, thus making it acceptable to terminate a pregnancy up until this point. Leading up to and around the 19th century, abortions were commonly performed with abortifacients, or drugs which cause an abortion. Advertisements in newspapers would advertise these drugs, oftentimes by the name "female monthly pills." Madame Restell was a leading figure whose advertisements to women were often found in newspapers. One such advertisement read: "Female monthly pills prepared by Madame Costello...the very best medicine that ladies laboring under a suppression of their natural illness can take, and they very seldom fail to relieve when taken according to the directions".

Dr. Horatio Storer

Dr. Horatio Storer was a Boston Surgeon who started practicing medicine in 1855. He was a strong leader in the 'physicians crusade against abortion,' and an individual with prominent anti-feminist sentiments. In his book ''Why Not? A Book for Every Woman'', published in 1868, he declared abortions to be "proof of actual insanity" and that women's influence to do so came from the extravagance of living, fashion, or lust. It was commonly believed in the 19th century that physical and psychological problems were rooted in a woman's womb. Storer stated: "Woman's mind is prone to depression, and, indeed, to temporary actual derangement, under the stimulus of uterine excitation". Storer was influential among the physician profession, and declared: "Physicians have now arrived at the unanimous opinion, that the [fetus] is alive from the very moment of conception". Despite many women believing that a child was not alive prior to quickening, Storer stated quickening was merely a sensation, and one that some women never felt. Thus, he concluded, it could not be an indication of the existence of a child during pregnancy.

American Medical Association (AMA)
Established in 1847, the American Medical Association (AMA) was an organization of medical professions that in its early years largely condemned abortion. In May 1857, a Committee was appointed by the American Medical Association in order to investigate criminal abortion, of which Dr. Horatio Storer was made Chairman. In the Report on Criminal Abortion submitted by the Committee, it stated that guilt of criminal abortion was acknowledged by medical men everywhere and proceeded to conclude: "As a profession we are unanimous in our condemnation of the crime". It was advised by the Committee that all physicians express abhorrence to the crime of abortion. Later in 1860, in the Address of Henry Miller, M.D., President of the Association, Miller stated that the resolutions from the Report on Criminal Abortion would be brought before Congress in order for laws punishing abortion to be revised and further legal action be provided.

Hippocratic Oath
The Hippocratic Oath, taken by all new physicians entering the medical field, establishes the professional conduct for medical personnel to uphold when practicing medicine. In particular, the oath requires physicians to swear to do no harm and thus can be attributed to the beliefs by many physicians at the time believing abortion to be ethically immoral and going directly against this sworn oath.

Comstock Act of 1873
A Statute passed by the Forty Second Congress, Session III on March 3, 1873, titled as An Act for the Suppression of Trade in, and Circulation of, obscene Literature and Articles of immoral Use, banned the possession of articles which could be used to either prevent conception or cause an abortion. The consequence of doing so included imprisonment for hard labor and a fine of up to two thousand dollars. This Statute later became known as the Comstock Act of 1873.

Vacuum Aspiration
Vacuum aspiration abortions were first investigated in 1957 in China and were developed throughout the 1970s. By the late 1970s, vacuum aspiration had become the most common first-trimester method and it is still used today. During vacuum aspiration, the cervical canal is dilated and a small tube is placed inside to suck the tissue out of the uterus. Vacuum aspiration has an extremely low complication rate and a lower risk of death than carrying a pregnancy to term, with the risk decreasing the earlier an abortion is performed. This proved to be a key factor in the Roe v. Wade (1973) decision: “Because early induced abortion was much safer than a continued pregnancy, the Court recognized the right of a woman and her physician to choose to terminate a pregnancy during the first trimester”. Increasing safety of abortions resulting from new technologies led to an increase in abortions between 1973 and the 1980s. This pattern is consistent with Jevon’s Paradox and was one of the supporting points of abortion opponents who were concerned that safer, faster, and cheaper procedures would result in abortions becoming routine and frequent. Note that abortions did level out in the 1980s around 1.5 million per year.

Medication Abortion
In 1982, the French company Roussel Uclaf announced that mifepristone (RU-486) could abort a pregnancy. Mifepristone causes the zygote to detach from the uterine lining. This medication is followed up with misoprostol, which induces contractions to get rid of the tissues. Once it was approved for use in France, American feminist groups like the Feminist Majority Foundation tried and failed to bring the drug to the U.S. Political pressure from oppositional groups was strong; in 1989, the FDA banned the importation of mifepristone. Despite the ban, the AMA announced their support of mifepristone on the grounds that it was less invasive but still as effective than surgical abortions—an interesting change from the AMA's stance on abortion in the 19th century. The tide turned with the Clinton administration, which succeeded in overturning the ban. The FDA issued a tentative approval for the drug in 1996; however, it was not approved for public use until 2000. More recently, oppositional groups like Americans United for Life have tried to push for stricter medical supervision during medication abortions, arguing that they are dangerous.

Fetal Pain
Fetal pain has been a social issue since 1984 when the National Right to Life Committee (NRLC) produced The Silent Scream, a film depicting a first-trimester abortion where the fetus is seen opening its mouth as though it were in pain. There is significant debate whether a fetus can feel pain associated with abortion, particularly regarding late-term abortion methods. Determining if a fetus can feel pain is challenging since pain involves physical, psychological, and emotional responses. Therefore, researchers must take developmental and neurological factors from the fetus into consideration. The American Medical Association has denied a fetus can feel pain at 20 weeks while many conflicting studies have been published on the issue. Various members of the United States legislature have proposed the Unborn Child Safety Awareness Act since 2004 and as recently as 2013. This bill mandates abortion providers to give information to abortion-seeking women about the possibility of their fetus to feel pain. The state of Utah also passed the “Administration of Anesthetic or Analgesic to an Unborn Child” law which mandates that a fetus be anesthetized prior to abortion with few exceptions. These laws have primarily been presented by Republican congressmen and women as the Republican party has opposed abortions since Roe v. Wade.

Late-Term Abortions
The term "late-term abortion" is often used politically; however, it's important to note that it "has no medical definition and is not used in a clinical setting"  due to its ambiguity. Colloquially, it is often understood as referring to abortions occurring after the point of viability. A fetus is considered viable between 24 and 28 weeks as defined in the opinion of the court in Roe v. Wade. As of 2019, 43 states have bans on abortion at some point during pregnancy, including 20 at viability and one at the beginning of the third trimester (25 weeks). Reasons for late-term abortions are most commonly due to the risk of death of the fetus or the mother due to abnormalities only detectable at later stages of pregnancy. The American College of Obstetricians and Gynecologists (ACOG) places importance on a physician-patient relationship and believes political interference disrupts this.

Fetal Heartbeats
In 2019, six states passed “heartbeat bills,” those being Georgia, Kentucky, Louisiana, Missouri, Mississippi and Ohio. A heartbeat bill prevents abortions after six to eight weeks of pregnancy, the typical timepoint at which an embryonic heartbeat can first be detected. The heartbeat at six weeks is "a little flutter in the area that will become the future heart of the baby," said Dr. Saima Aftab, medical director of the Fetal Care Center at Nicklaus Children's Hospital. These states based their bills after a model bill drafted by Faith2Action founder and president Janet Porter. Faith2Action is a religious non-profit whose mission is to “win the cultural war for life, liberty, and the family”. Many groups and individuals have vocalized their opposition to these bills. The American Civil Liberties Union (ACLU) and Planned Parenthood, the largest provider of reproductive health services in the U.S., have filed lawsuits in each state to block the laws from being enacted. They have been successful in each case. Their defense rests on the claim that “women have a categorical right to a pre-viability abortion” as established in Roe v. Wade. Planned Parenthood also implemented an Abortion Care Finder tool on the homepage of their website to alleviate the confusion surrounding the effects of the bills and their blockage.

Conclusion
Throughout the past three centuries, each technological advancement in abortion has inflamed American abortion opponents, giving rise to new controversies. Safer abortion methods have pushed the debate away from medical hazards and further into the realm of religion and philosophy. Accordingly, the role of religion in abortion controversies remains open to investigation as well as other abortion technologies, a more detailed analysis of Roe v. Wade, and related court decisions.


text:
Talk:Lentis/Dakota Access Pipeline

Azar - offline addition of the "allusion" and "environmental impacts" sections. Hermans - offline addition of the entire "Pipeline Opponents" section (section 2.3 including 2.3.1, 2.3.1.1, 2.3.1.2, and 2.3.2). Also supported writing part of the conclusion section. Zaman - Wrote the background, "History and Future Prospects", "Federal Agencies", "Pipeline Supporters", and part of the "Conclusions" sections.

title: Talk:Lentis/Dakota Access Pipeline


text:
LMIs in Control/pages/Discrete Time Minimum Gain Lemma

The Concept
The output of the system y(t) is fed back through a sensor measurement F to a comparison with the reference value r(t). The controller C then takes the error e (difference) between the reference and the output to change the inputs u to the system under control P. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.

This is called a single-input-single-output (SISO) control system; MIMO (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through vectors instead of simple scalar values. For some distributed parameter systems the vectors may be infinite-dimensional (typically functions).

If we assume the controller C, the plant P, and the sensor F are linear and time-invariant (i.e., elements of their transfer function C(s), P(s), and F(s) do not depend on time), the systems above can be analysed using the Laplace transform on the variables. This gives the following relations:

 * $$Y(s) = P(s) U(s)$$
 * $$U(s) = C(s) E(s)$$
 * $$E(s) = R(s) - F(s)Y(s).$$

Solving for Y(s) in terms of R(s) gives

 * $$Y(s) = \left( \frac{P(s)C(s)}{1 + P(s)C(s)F(s)} \right) R(s) = H(s)R(s).$$

The expression $$H(s) = \frac{P(s)C(s)}{1 + F(s)P(s)C(s)}$$ is referred to as the closed-loop transfer function of the system. The numerator is the forward (open-loop) gain from r to y, and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If $$|P(s)C(s)| \gg 1$$, i.e., it has a large norm with each value of s, and if $$|F(s)| \approx 1$$, then Y(s) is approximately equal to R(s) and the output closely tracks the reference input. This page gives an LMI to reduce the gain so that the ouput closely tracks the reference input.

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 Minimum Gain Lemma
The system $$\mathcal{G}$$ has minimium gain γ under any of the following equivalent sufficient conditions.

 * 1. There exists $$ P \in \mathcal{S}^{n}, $$ and γ $$\in \mathcal{R}_{\ge 0}$$ where $$ P \ge 0 $$ such that

 * $$\begin{bmatrix}

A^{T}_{d}PA_{d} - P - C_{D}^{T}C_{D} & A^{T}_{d}PB_{d} - C^{T}_{d}D_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d}D_{d})^{T} &  B^{T}_{d}PB_{d} + \gamma^{2}I - (D^{T}_{d}+ D_{d}) \end{bmatrix}\le 0.$$

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

 * $$\begin{bmatrix}

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

$$proof$$ : Applying the Schur complement lemma to the γ2 term in equation 1 gives equation 2.

Conclusion:
If there exist a positive definite $$P$$ for the system $$\mathcal{G}$$, then the minimum gain of the system is γ can be obtaied from above defined LMIs.

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/Discrete Time Modified Minimum Gain Lemma

The Concept
The output of the system y(t) is fed back through a sensor measurement F to a comparison with the reference value r(t). The controller C then takes the error e (difference) between the reference and the output to change the inputs u to the system under control P. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.

This is called a single-input-single-output (SISO) control system; MIMO (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through vectors instead of simple scalar values. For some distributed parameter systems the vectors may be infinite-dimensional (typically functions).

If we assume the controller C, the plant P, and the sensor F are linear and time-invariant (i.e., elements of their transfer function C(s), P(s), and F(s) do not depend on time), the systems above can be analysed using the Laplace transform on the variables. This gives the following relations:

 * $$Y(s) = P(s) U(s)$$
 * $$U(s) = C(s) E(s)$$
 * $$E(s) = R(s) - F(s)Y(s).$$

Solving for Y(s) in terms of R(s) gives

 * $$Y(s) = \left( \frac{P(s)C(s)}{1 + P(s)C(s)F(s)} \right) R(s) = H(s)R(s).$$

The expression $$H(s) = \frac{P(s)C(s)}{1 + F(s)P(s)C(s)}$$ is referred to as the closed-loop transfer function of the system. The numerator is the forward (open-loop) gain from r to y, and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If $$|P(s)C(s)| \gg 1$$, i.e., it has a large norm with each value of s, and if $$|F(s)| \approx 1$$, then Y(s) is approximately equal to R(s) and the output closely tracks the reference input. This page gives an LMI to reduce the gain so that the ouput closely tracks the reference input.

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 Modified Minimum Gain Lemma
The system $$\mathcal{G}$$ has minimium gain γ under any of the following equivalent sufficient conditions.

 * 1. There exists $$ P \in \mathcal{S}^{n}, $$ and γ $$\in \mathcal{R}_{\ge 0}$$ where $$ P \ge 0 $$ such that

 * $$\begin{bmatrix}

A^{T}_{d}PA_{d} - P & A^{T}_{d}PB_{d} - C^{T}_{d}D_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d}D_{d})^{T} &  B^{T}_{d}PB_{d} + \gamma^{2}I - (D^{T}_{d}+ D_{d}) \end{bmatrix}$$

$$Proof$$. The term $$ -C_{d}^{T}C_{d} $$ in Discrete Time Minimum Gain Lemma makes the matrix inequality more negative definite. Therefore,
 * $$\begin{bmatrix}

A^{T}_{d}PA_{d} - P - C_{D}^{T}C_{D} & A^{T}_{d}PB_{d} - C^{T}_{d}D_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d}D_{d})^{T} &  B^{T}_{d}PB_{d} + \gamma^{2}I - (D^{T}_{d}+ D_{d})\end{bmatrix} \le \begin{bmatrix} A^{T}_{d}PA_{d} - P & A^{T}_{d}PB_{d} - C^{T}_{d}D_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d}D_{d})^{T} &  B^{T}_{d}PB_{d} + \gamma^{2}I - (D^{T}_{d}+ D_{d}) \end{bmatrix}$$

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

 * $$\begin{bmatrix}

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

$$proof$$ : Applying the Schur complement lemma to the γ2 term in equation 1 gives equation 2.

Conclusion:
If there exist a positive definite $$P$$ for the system $$\mathcal{G}$$, then the minimum gain of the system γ can be obtaied from above defined LMIs.

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 Discrete Time Minimum Gain Lemma


text:
User:Tanbirzx

I am Tanbir Islam.And this is Wiki Books.Wikibooks is a section of Wikimedia.Anyone can search or find a topic in Wiki books.This is a free open site for the book reader's.

title: User:Tanbirzx


text:
User talk:Ainz Ooal Gown

Welcome to Wikibooks, Masumrezarock100!

 First steps tutorial Wikibooks is for collaborative development of free textbooks.

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title: User talk:Ainz Ooal Gown


text:
LMIs in Control/Controller Synthesis/Takagi Sugeno Models for Polytopic Controller Design

 * Quadratic Stability
 * Apkarian Filter and State Feedback
 * Minimum Decay Rate in State Feedback
 * Maximum Natural Frequency in State Feedback


text:
LMIs in Control/pages/Quadratic Stability

The System:
A TS fuzzy model allows the representation of a non-linear model as a set of local LTI (Linear Time Invariant) models $$[1, p. 10]$$, each one called subsystem. A subsystem is the local representation of the system in the space of premise variables $$z(t)$$ = $$[z1(t) z2(t). . . zp(t)]$$ which are known and could depend on the state variables and input variables.

The Optimization Problem:
Let consider an autonomous system $$x$$=$$Ax$$ with $$A$$ being a constant matrix. If we define the Lyapunov function $$V(x)$$=$$x^T P x$$, then the system is stable if there exist $$P > 0$$ such that condition is satisfied.

$$A^T P + P A < 0 $$

If we have a family of matrices $$A(\delta(t))$$ (where $$\delta(t)$$ is a parameter that is bounded by a polytope ∆) instead of a single matrix A, then the system equation becomes $$x$$=$$A(\delta(t))x$$ and condition should be satisfied for all possible values of $$\delta(t)$$. If exists $$P>0$$ such that following condition is satisfied then the system is quadratically stable.

$$A(\delta(t))^T P + P A(\delta(t)) < 0$$ ∀$$\delta(t)$$ ∈ ∆.

Since there are an infinite number of matrices A(δ(t)) there is also an infinite number of constraints like that for quadratic stability mentioned previously that should be fulfilled. From a practical point of view this makes the problem impossible to be solved. Let consider now that the system $$x = A(\delta(t))x$$ can be written in a polytopic form as a Takagi-Sugeno (TS) polytopic system with premise variables $$z(t)$$ and a set of r subsystems $$Ai$$ for $$i = {1,. . ., r}$$.

$$x(t)=\sum_{i=1}^{r}(h_i(z(t)) A_i x(t)$$.

It can be proven that a polytopic autonomous system is quadratically stable if previous condition  is satisfied in the vertices (subsystems) of the polytope. Therefore there is no need to check stability in an infinite number of matrices, but only in subsystems matrices $$A_i$$.

$$A^T_i P + P Ai < 0$$ ∀i = 1,. . ., r.

Stability conditions can be applied to the closed-loop system  and the following set of conditions are obtained.

$$G^T_{ii} P + P G_{ii} < 0$$ ∀i = 1,. . ., r.

$$((G_{ij} + G_{ji})/2)^T P + P((G_{ij} + G_{ji})/2)<= 0$$ ∀i, j ∈ {1,. . ., r}, i < j.

where $$G_{ij} = A_i + B_i K_j $$and $$ hi(z(t))hj (z(t)) \neq 0$$.

In the special case where matrices Bi are constant (i.e. $$B_i = B$$), the first set of inequalities are enough to prove stability. Therefore, assuming constant B for all the subsystems, if there exist P > 0 such that conditions are fulfilled, then the polytopic TS model (2.2) with state feedback control is quadratically stable inside the polytope.

$$(A_i + B K_i)^T P + P (A_i + B K_i) < 0$$ ∀i = 1,. . ., r.

The assumption of constant B can be achieve using a prefiltering of the input. This change is not restrictive and the main consequence is the addition of some new state variables (the ones from the filter) to the TS model.

The LMI:
The design of the controller that stabilizes the closed-loop system boils down to solve the Linear Matrix Inequality (LMI) problem of finding a positive definite matrix P and a set of matrices $$K_i$$ such that conditions are fulfilled. However, since the constraints should be linear combinations of the unknown variable, the following change of variables is applied: $$W_i = K_i Q$$ where $$ Q = P^-1$$. The solution of the LMI problem is the set of matrices $$W_i$$ such that conditions are fulfilled.

$$Q > 0$$

$$ A_i Q + Q A^T_i + B W_i + W^T_i B^T < 0$$. ∀i = 1,. . ., r.

The i-th controller is computed from the solution as $$K_i$$ = $$W_i$$ $$Q^{-1}$$

Conclusion:
The LMI is feasible.


text:
LMIs in Control/pages/Mu Analysis

Mu Synthesis. The technique of $$ \mu $$ synthesis extends the methods of $$ H\infin $$ synthesis to design a robust controller for an uncertain plant. You can perform $$ \mu $$ synthesis on plants with parameter uncertainty, dynamic uncertainty, or both using the "musyn" command in MATLAB. $$ \mu $$ analysis is an extremely powerful multivariable technique which has been applied to many problems in the almost every industry including Aerospace, process industry etc.

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

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

where it is assumed that $$ D $$ is Invertible.

The Data
The matrices needed as inputs are only, $$ A $$ and $$ D $$.

The LMI: $$ \mu $$- Analysis
The inequality $$ \overline{\sigma}(DAD^{-1})< \gamma $$ holds if and only if there exist $$ X \in \mathbb{S}^n $$ and $$ \gamma \in \mathbb{R}_{>0}$$, where $$ X>0 $$, satisfying:

\begin{align}

A^TXA - \gamma^2X <0 \end{align}$$

Conclusion:
The inequality $$ \overline{\sigma}(DAD^{-1}) < \gamma $$ holds for $$ D= X^{1/2}, $$ where X satisfies the above Inequality.

Implementation

 * https://github.com/Ricky-10/coding107/blob/master/Mu%20Analysis - $$ \mu $$-Analysis

title: LMIs in Control/pages/Mu Analysis


text:
LMIs in Control/pages/H2 index

H2 Index Deduced LMI

Although there are ways to evaluate an upper bound on the H2, the verification of the bound on the H2-gain of the system can be done via the deduced condition.

The System
We consider the generalized Continuous-Time LTI system with the state space realization of $$(A,B,C,D)$$

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

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

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

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

The Optimization Problem
For an arbitrary $$ \gamma >0 $$(a given scalar), the transfer function satisfies

 * $$ \left\| C(sI-A)^{-1}B+D \right\|_{2} < \gamma $$

The H2-norm condition on Transfer function holds only when the matrix A is stable. And this can be conveniently converted to an LMI problem

if and only if 1. There exists a symmetric matrix $$X > 0 $$ such that: $$ AX+XA^T+BB^T < 0 $$, $$ trace(CXC^T)< \gamma^2 $$

2. There exists a symmetric matrix $$Y > 0 $$ such that: $$ AY+YA^T+C^TC < 0 $$, $$ trace(B^TYB)< \gamma^2 $$

The LMI - Deduced Conditions for H2-norm
These deduced condition can be derived from the above equations. According to this

For an arbitrary $$ \gamma >0 $$(a given scalar), the transfer function satisfies

 * $$ \left\| G(s) \right\|_{2} < \gamma $$

if and only if there exists symmetric matrices $$ Z$$ and $$ P$$; and a matrix $$ V$$ such that $$ trace(Z) < \gamma^2$$ $$ \begin{bmatrix} -Z & B^T \\ B & -P \end{bmatrix}$$$$\begin{align}< 0\end{align}$$ $$ \begin{bmatrix} -(V+V^T) & V^TA^T+P & V^TC^T & V^T \\ AV+P & -P & 0 & 0 \\ CV & 0 & -I & 0 \\ V & 0 & 0 & -P \end{bmatrix}$$$$\begin{align}< 0\end{align}$$

The above LMI can be combined with the bisection method to find minimum $$ \gamma $$ to find the minimum upper bound on the H2 gain of $$ G(s) $$.

Conclusion:
If there is a feasible solution to the aforementioned LMI, then the $$ \gamma $$ upper bounds the norm of the system G(s).

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

https://github.com/yashgvd/ygovada

Related LMIs
Bounded Real Lemma Deduced LMIs for H-infinity index

title: LMIs in Control/pages/H2 index


text:
LMIs in Control/pages/Hurwitz detectability

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

The System

\begin{align} \dot x(t)&=Ax(t)+Bu(t),\\ y(t)&=Cx(t)+Du(t)\\ x(0)&=x_0\\ \end{align}$$

where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$, $$u(t)\in \R^q$$, at any $$t\in \R$$.

The Data

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

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

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

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

Implementation
Find the MATLAB implementation at this link below Hurwitz detectability

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

Return to Main Page:

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


text:
LMIs in Control/pages/SSFP

We are attempting to stabilizing The Static State-Feedback Problem

The System
Consider a continuous time Linear Time invariant system

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

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

The Optimization Problem
The Problem's main aim is to find a feedback matrix such that the system

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

\begin{align} u(t) = Kx(t)\\ \end{align} $$ is stable Initially we find the $$K$$ matrix such that $$(A+BK)$$ is Hurwitz.

The LMI: Static State Feedback Problem
This problem can now be formulated into an LMI as Problem 1:
 * $$ X(A+BK)+(A+BK)^TX<0 $$

From the above equation $$ X>0 $$ and we have to find K

The problem as we can see is bilinear in $$ K,X$$ Problem 2:
 * The bilinear in X and K is a common paradigm
 * Bilinear optimization is not Convex. To Convexify the problem, we use a change of variables.
 * $$ AP+BZ+PA^T+Z^TB^T < 0 $$

where $$ P > 0$$ and we find $$Z$$ $$ K = ZP^{-1} $$

The Problem 1 is equivalent to Problem 2

Conclusion
If the (A,B) are controllable, We can obtain a controller matrix that stabilizes the system.

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
Hurwitz Stability

title: LMIs in Control/pages/SSFP


text:
LMIs in Control/pages/Schur Detectability

Schur Detectability

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

The System
We consider the following system:

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

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

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

The state feedback control law is defined as follows:

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

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

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

The Data

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

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

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

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

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

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

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

Return to Main Page
LMIs in Control/Tools


text:
LMIs in Control/pages/Apkarian Filter

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 $$I_4$$ ∈ $$R^{4\times4}$$ 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:

 * $$L \otimes P + M \otimes P (A_{ei} + B_e K_i) + MT \otimes (A_ei + B_e K_i)^T P < 0$$ ∀i = 1, . . ., 32.

Conclusion:
The LMI is feasible.

title: LMIs in Control/pages/Apkarian Filter


text:
LMIs in Control/pages/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 $$I_4$$ ∈ $$R^{4\times4}$$ 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:

 * $$L \otimes P + M \otimes P (A_{ei} + B_e K_i) + MT \otimes (A_ei + B_e K_i)^T P < 0$$ ∀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


text:
LMIs in Control/pages/Minimum Decay Rate in 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 $$I_4$$ ∈ $$R^{4\times4}$$ 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:

 * $$L \otimes P + M \otimes P (A_{ei} + B_e K_i) + MT \otimes (A_ei + B_e K_i)^T P < 0$$ ∀i = 1, . . ., 32.

A pair of conjugate complex poles s of the closed loop system can be written as $$s$$ = − $${\xi} \omega_n \pm j\omega_d $$where $$\xi$$ is the damping ratio,$$\omega_n$$ is the undamped natural frequency and $$\omega_d $$ is the frequency response defined as $$ \omega_d = \omega_n$$ $$ \sqrt{1-\xi^2}$$.Three different LMI regions have been considered, each one related with a performance specification regarding $$ \alpha = \xi \omega_n,\omega_n$$ and $$\xi$$:

Minimum Decay Rate:

If we want to set a minimum decay rate α in the closed loop system response, the poles should be inside the LMI region defined in : $$S_{\alpha}$$ = [s = x + j y | x < − $$ \alpha $$].where $$ \alpha $$ > 0. In this case L$$_\alpha$$ = $$ 2 \alpha $$ and M$$_\alpha$$ = 1.

Applying condition to the closed-loop system, the LMI condition associated to this LMI region is:

 * $$ 2 \alpha P + (A_{ei} + B_e K_i)^T P + P (A_{ei} + B_e K_i) < 0$$ ∀i = 1, . . ., 32.

Conclusion:
The LMI is feasible.

Related LMIs

 * Apkarian Filter and State Feedback. https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Apkarian_Filter-and_State_Feedback
 * Maximum Natural Frequency in State Feedback https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Maximum_Natural_Frequency_in_State_Feedback#The_LMI%3A


text:
LMIs in Control/pages/mixedhinfh2desiredpole

LMI for Mixed $$H_{2}/H_{\infty}$$ with desired pole location Controller

The mixed $$ H_{2}/H_{\infty}$$ output feedback control has been known as an example of a multi-objective optimal control problem. In this problem, the control feedback should respond properly to several specifications. In the $$ H_{2}/H_{\infty}$$ controller, the $$H_{\infty}$$ channel is used to improve the robustness of the design while the $$H_{2}$$ channel guarantees good performance of the system and additional constraint is used to place poles at desired location.

The System
We consider the following state-space representation for a linear system:

$$

\begin{align} \dot{x} &= Ax + B_1u + B_2w \\ z_{\infty} &= C_{\infty} + D_{\infty1}u + D_{\infty2}w\\ z_2 &= C_2x+D_{21}u \end{align} $$

where

 * $$x\in \R^n$$, $$z_2,z_{\infty} \in \R^m$$are the state vector and the output vectors, respectively

 * $$w\in \R^p$$, $$u\in \R^r$$ are the disturbance vector and the control vector

 * $$ A $$, $$ B_1$$,$$ B_2$$, $$ C_{\infty}$$,$$ C_2$$,$$ D_{\infty1}$$,$$ D_{\infty2}$$, and $$D_{21} $$ are the system coefficient matrices of appropriate dimensions

The Data
We assume that all the four matrices of the plant,$$ A $$, $$ B_1$$,$$ B_2$$, $$ C_{\infty}$$,$$ C_2$$,$$ D_{\infty1}$$,$$ D_{\infty2}$$, and $$D_{21} $$ are given.

The Optimization Problem
For the system with the following feedback law: $$ u = Kx $$ The closed loop system can be obtained as: $$

\begin{align} \dot{x} &= (A + B_1K)x + B_2w \\ z_{\infty} &= (C_{\infty} + D_{\infty1}K)x + D_{\infty2}w\\ z_2 &= (C_2+D_{21}K)u \end{align} $$

the transfer function matrices are $$ G_{z\infty w}(s) $$ and $$ G_{z2w}(s) $$ Thus the $$ H_{\infty} $$ performance and the $$ H_{2} $$ performance requirements for the system are, respectiverly $$ ||G_{z\infty w}(s)||_{\infty} < \gamma_{\infty} $$ and $$ ||G_{z2 w}(s)||_{2} < \gamma_2 $$. For the performance of the system response, we introduce the closed-loop eigenvalue location requirement. Let $$ D =, $$ It is a region on the complex plane, which can be used to restrain the closed-loop eigenvalue locations. Hence a state feedback control law is designed such that, $$ \, \, \, \, \, $$ $$ \lambda (A+B_1K) \subset D $$.
 * The $$ H_{\infty} $$ performance and the $$ H_{2} $$ performance are satisfied.
 * The closed-loop eigenvalues are all located in $$ D $$, that is,

The LMI: LMI for mixed $$H_{2}$$/$$H_{\infty}$$ with desired Pole locations
The optimization problem discussed above has a solution if there exist two symmetric matrices $$ X, Z $$ and a matrix $$ W $$, satisfying

min $$ c_2 \gamma_2^2 + c_{\infty} \gamma_{\infty} $$ s.t $$ \begin{align}

&\begin{bmatrix} (AX+B_1W)^T+AX+B_1W & B_2 & (C_{\infty}X+D_{\infty 1}W)^T \\ B_2^T & -\gamma_{\infty}I & D_{\infty 2}^T \\ C_{\infty}X+D_{\infty 1}W & D_{\infty 2} &-\gamma_{\infty}I\\ \end{bmatrix} < 0 \\

&AX + B_1W+(AX+B_1W)^T+B_2B_2^T < 0\\

&\begin{bmatrix} -Z  & C_2X+D_{21}W \\ (C_2X+D_{21}W)^T & -X\\ \end{bmatrix} > 0 \\ & \text{trace}(Z) < \gamma_{2}^{2} \\ & L \otimes + M \otimes (AX+B_1W) + M^T \otimes (AX+B_1W)^T < 0 \\ \end{align} $$

where $$ c_2 > 0 $$ and $$ c_{\infty} > 0 $$ are the weighting factors.

Conclusion:
The calculated scalars $$ \gamma_{\infty} $$ and $$ \gamma_{2} $$ are the $$ H_{2}$$ and $$ H_{\infty}$$ norms of the system, respectively. The controller is extracted as $$ K = WX^{-1} $$

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

Related LMIs
Mixed H2 Hinf with desired pole location for perturbed system

Return to Main Page
LMIs in Control/Tools


text:
LMIs in Control/pages/H2SO

We treat the problem of designing a full-order state observer for the system mentioned below. The aim of it is to mitigate the effect of disturbance $$ w(t) $$ to the estimate error is prohibited to a desired level.

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

\begin{align} \dot x(t)&=Ax(t)+B_1u(t)+B_2w(t) & x(0)=x_0\\ y(t)&=C_1x(t)+D_{1}u(t)+D_2w(t)\\ &z(t) = C_2x(t) \end{align}$$

where it is assumed that $$(A,C_2)$$ is detectable. An observer of the form

The Data
output vector, and the output vector of interests
 * $$x$$ ∈ $$\begin{align}\mathbb{R}\end{align}$$n, $$y$$ ∈ $$\begin{align}\mathbb{R}\end{align}$$l , $$z$$ ∈ $$\begin{align}\mathbb{R}\end{align}$$m are respectively the state vector, the measured

respectively
 * $$w$$ ∈ $$\begin{align}\mathbb{R}\end{align}$$p and $$u$$ ∈ $$\begin{align}\mathbb{R}\end{align}$$r are the disturbance vector and the control vector,

appropriate dimensions
 * A, B1, B2, C1, C2, D1, and D2 are the system coefficient matrices of

The Optimization Problem
For the system we introduce a full state observer in the following form: $$ \dot \hat x = (A+LC_1)\hat x - Ly + (B_1+LD_1)u$$ $$ \hat x, L$$ are the observation vector and the observer gain. The transfer function for this case is $$ G_{zw}(s) = C_2(sI-A-LC_1)^{-1}(B_2+LD_2)$$ and thus the problem of $$H_2$$ state observer design is to find L such that ||$$G_{zs}(s)||_2 < \gamma $$  The error :$$e = x - \hat x$$

The LMI: LMI for H2 State Observer Design
The H2 state observer problem has a solution if and only if there exists a matrix $$W$$, a symmetric matrix $$Q$$ and a symmetric matrix $$X$$ such that

\begin{bmatrix} XA + WC_1 + (XA+WC_1)^T & XB_2 +WD_2 \\ (XB_2 +WD_2)^T & -I \end{bmatrix}$$$$\begin{align}< 0\end{align}$$

\begin{bmatrix} -Q & C_2 \\ C_2^T & -X \end{bmatrix}$$$$\begin{align}< 0\end{align}$$

\begin{align} trace(Q)< \gamma^2 \end{align} $$

and from the solution of the above LMIs we can obtain the observer matrix as

L = X^{-1}W $$

Conclusion
Thus by formulation, we have converted the problem of H2 state observer design into an LMI feasibility problem by optimizing the above LMIs. In application we are often concerned with the problem of finding the minimal attenuation level  $$\gamma$$  On implementation and optimization of the above LMI using YALMIP and MOSEK (or) SeDuMi we get 3 matrices as output, $$X,W and Q$$ and also $$\rho$$ which is used to calculate $$\gamma$$ which is the H2 norm of the system.

There exists another set of LMIs which holds true for the same optimization problem as above.

\begin{align} A^TY+C_1^TV^T+YA+VC_1+C_2^TC_2 < 0\\ \end{align}$$

\begin{bmatrix} -Z & YB_2 +VD_2 \\ (YB_2 +VD_2)^T & -Y \end{bmatrix}$$$$\begin{align}< 0\end{align}$$ '''$$ \begin{align} trace(Z)< \gamma^2 \end{align} $$'''

When a minimal $$\rho$$ is obtained, the minimal attenuation level is $$\gamma = \sqrt\rho$$

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
H$\infty$ State Observer Design

Discrete time H2 State Observer Design

title: LMIs in Control/pages/H2SO


text:
LMIs in Control/pages/Maximum Natural Frequency in 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 $$I_4$$ ∈ $$R^{4\times4}$$ 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:

 * $$L \otimes P + M \otimes P (A_{ei} + B_e K_i) + MT \otimes (A_ei + B_e K_i)^T P < 0$$ ∀i = 1, . . ., 32.

A pair of conjugate complex poles s of the closed loop system can be written as $$s$$ = − $${\xi} \omega_n \pm j\omega_d $$where $$\xi$$ is the damping ratio,$$\omega_n$$ is the undamped natural frequency and $$\omega_d $$ is the frequency response defined as $$ \omega_d = \omega_n$$ $$ \sqrt{1-\xi^2}$$.Three different LMI regions have been considered, each one related with a performance specification regarding $$ \alpha = \xi \omega_n,\omega_n$$ and $$\xi$$:

Maximizing Natural Frequency:

Natural frequency is related with the maximum frequency response in the undamped case ($$ \xi $$ = 0). If we want to set a maximum $$ \omega_n $$ condition, the LMI region associated is $$S_\rho$$ = [s = x + jy | |x + jy| < $$\rho$$], ::$$ L_\rho = \begin{bmatrix}-{\rho}&0\\0&\rho \end{bmatrix}, M_\rho = \begin{bmatrix}0&1\\0&0 \end{bmatrix} $$. Resulting LMI condition is:

\begin{bmatrix} - \rho P&P(A_{ei}+B_{ei} K_i)\\ (A_{ei}+B_{ei} K_i)^T P& - \rho P \end{bmatrix} < 0 $$ ∀i = 1,. . ., 32.

Conclusion:
The LMI is feasible.

Related LMIs

 * Apkarian Filter and State Feedback. https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Apkarian_Filter-and_State_Feedback
 * Minimum Decay Rate in State Feedback. https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Minimum_Decay_Rate_in_State_Feedback#The_LMI%3A


text:
LMIs in Control/pages/TDSIC

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

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

where

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

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

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

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

$$P>0$$

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

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

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

Implementation
The implementation of the above LMI can be seen here

https://github.com/yashgvd/LMI_wikibooks

Related LMIs
Time Delay systems (Delay Dependent Condition)

title: LMIs in Control/pages/TDSIC


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

 * LMI for Time-Delay system on delay Independent Condition
 * LMI for Time-Delay system on delay Dependent Condition
 * LMI for Stability of Retarded Differential Equation with Slowly-Varying Delay
 * LMI for Robust Stability of Retarded Differential Equation with Norm-Bounded Uncertainty
 * Bounded Real Lemma under Slowly-Varying Delay
 * LMI for L2-Optimal State-Feedback Control under Time-Varying Input Delay


text:
LMIs in Control/pages/TDSDC

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

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

where

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

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

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

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

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

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

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

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

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

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

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

Implementation
The implementation of the above LMI can be seen here

https://github.com/yashgvd/LMI_wikibooks

Related LMIs
Time Delay systems (Delay Independent Condition)

title: LMIs in Control/pages/TDSDC