text:
User:Margav06/sandbox/Click here to continue/Linear Systems/KYP Lemma for Descriptor Systems

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

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

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

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

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

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

 * $$\begin{bmatrix}

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

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

 * $$\begin{bmatrix}

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

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

 * $$\begin{bmatrix}

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

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

 * $$\begin{bmatrix}

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

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

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

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

 * $$\begin{bmatrix}

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

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

 * $$\begin{bmatrix}

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

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

 * $$\begin{bmatrix}

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

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

 * $$\begin{bmatrix}

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

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

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

Related LMIs
KYP Lemma State Space Stability KYP Lemma without Feedthrough


text:
User:Margav06/sandbox/Click here to continue/Linear Systems/Discrete Time Bounded Real Lemma

Discrete-Time Bounded Real Lemma

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

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

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

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

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Discrete-Time Bounded Real Lemma

The LMI formulation

H∞ norm < $$\gamma$$

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

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

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

Related LMIs
- Continuous time KYP_Lemma_(Bounded_Real_Lemma)

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


text:
User:Margav06/sandbox/Click here to continue/Linear Systems/Discrete Time KYP Lemma for QSR Dissipative System

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

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

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

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

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

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

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

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

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

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

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

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

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

 * $$\begin{bmatrix}

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

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

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

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


text:
User:Margav06/sandbox/Click here to continue/Introduction to Linear systems

The Wikipedia page for the following topic is given below:

https://en.wikipedia.org/wiki/Linear_system


text:
Lentis/Shared Space and Woonerven

Introduction
Shared Space is an urban design method that diminishes the physical barriers between modes of road usage. A Woonerf (plural—Woonerven) is an example of residential Shared Space. Both employ tactics to dissuade automobile use, promote nonmotorized travel methods, and redefine streets as social grounds. This casebook chapter summarizes the history of Shared Space and Woonerven, urban design tactics and legislation used to create them, community effects, participants and their agendas, and what these phenomena can teach us.

History
The concept of Shared Space began in the Dutch city of Delft in the late 1960’s, in response to residents’ concerns about traffic in residential areas. Drawing on the work of the British city planner Colin Buchanan, the city developed the concept of a “Woonerf,” or “living street” to provide a space for communities to gather without fear of fast-moving traffic. Generally speaking, a Woonerf is a residential area designed explicitly to favor pedestrians, as opposed to automobiles. The concept spread rapidly throughout continental Europe with over 3500 Woonerven being built in the Netherlands, Belgium and Germany by 1990. Woonerven also proved popular in densely packed countries such as Israel, Japan, and the United Kingdom. Recent estimates suggest that there are now more than 6000 Woonerven in the Netherlands alone, providing housing for over 12% of the Dutch population.

Drawing from the principles of Woonerven, Dutch engineer Hans Monderman developed his principle of “Shared Space” in the early 1980’s. Shared Space seeks to redesign city streets to promote walking and cycling, in contrast to the automobile-centric design philosophy prevalent at the time. As such, Woonerven can be thought of as a subset of Shared Space, where pedestrians are heavily favored. An example of Shared Space in action is in the Dutch city of Amsterdam, which removed all car parking and collaborated with residents to redesign the streets to favor child play areas, gardens, and bike parking. While some parking is still necessary in the form of loading zones, automobiles no longer dominate the city and inhabitants are incentivized to walk and cycle to their destinations. The concept of Shared Space is not unique to the Netherlands, Monderman's work has inspired other European cities such as Oslo and Ljubljana to reorient their cities away from motor vehicle traffic.

Shared Space Elements and Urban Design Tactics
In order to implement Shared Space, Monderman advocates the removal of curbs, road surface markings, signs, and traffic lights in an effort to calm traffic, improve road safety, and increase community cohesion. Some cities, such as Amsterdam, removed vehicle parking spots, while others, such as Nijmegen in the Netherlands, go further and utilize movable barriers, or “Bollards” to limit vehicle traffic inside the city. Taken together, these characteristics can be thought of as a form of “Choice Architecture,” where driving is disincentivized due to the removal of design features that favor automobiles, resulting in more people choosing to walk or cycle. By reducing the barriers between pedestrians, drivers, and cyclists, residents are free to use the mode of transport that they prefer.

Woonerven, themselves being a subset of the Shared Space model, utilize similar urban design techniques to reduce the prevalence of cars in residential areas. A common Woonerf will be well marked and have many design features to force drivers to reduce speed and become more aware. This may consist of road narrowing, changes in elevation, placement of gardens, and strategic alternation of parking between the left- and right-hand side of the road. Since the streets in a Woonerf are open for recreational use, they slowly erode the perception that streets are primarily for cars.

Legislation
Though different nations have slightly different legislation describing Woonerven usage, they share a common theme — pedestrians have more power and protections than automobiles. For example, the Netherlands' Traffic Regulations for Woonerf states that pedestrians may play in and use the full width of the public road, that drivers may not drive faster than walking pace, and that any automobile driver who hits a cyclist is presumed guilty. Belgium's Belgian Traffic Code: Article 22a states that drivers must not endanger or hinder pedestrians and must adhere to a 20 kph speed limit, but it also states that pedestrians must not obstruct traffic unnecessarily. In an attempt to promote Woonerven, United Kingdom's Transport Act 2000: Section 268 allows local authorities to create, mark, and regulate "home zones", which serve the same purpose as Woonerven.

Effects
Implementing Woonerven can have both positive and negative effects on local participants. A major positive effect is an empirical increase in road safety — according to Canin Associates, implementing Woonerven in the Netherlands led to a 40% decrease in traffic accidents. Furthermore, survey respondents have indicated that Woonerven encourage more recreation on public streets and improve communication between neighbors. In addition, Woonerven foster quieter cities with lower automobile usage. Woonerven also have economic benefits; not only do they increase the value of nearby properties, but they also encourage the development and usage of local street markets.

A major obstacle to Woonerf adoption is its impacts on drivers, facilitated by the aforementioned urban design tactics. However, the majority of survey respondents indicated that they would be willing to accept such driving tradeoffs in exchange for Woonerven's benefits. Moreover, Woonerven can be costly to implement; a single Woonerf between two office buildings in Seattle would cost $2.1 million to construct.

Adoption of Woonerven depends heavily on local culture. In the U.S., for example, cars are considered more essential and are used more often than in most European nations. This can lead to backlash from U.S. drivers and increase the frequency that drivers are unaware of or disregard the characteristics and road rules of Woonerven. Similarly, the U.S. has multiple road regulations that contradict those of Woonerven, such as the requirement that roads must have an unobstructed width of 6 meters (and as few turns as possible) for fire truck accessibility.

Participants and Agendas
Several groups advocate for and against Shared Space ideals. Residents of Woonerven typically support the movement and claim that design elements "make it nice for people to live here instead of nice for the cars" (Albrecht, 2019). Dutch people seeking to convert their own street organize with urban design professionals on an online network called WoonERFgoed. Government transportation agencies play a role in defining Shared Space areas by law, enforcing the traffic rules, and working closely with urban designers and communities to construct the road projects. In areas where Shared Space is normalized, drivers tend to give way to pedestrians. However, in areas where Shared Space is just emerging, drivers are mostly stubborn to the new changes. Some instances have found that overall behavioral changes are slow to come about because "if as a driver you stop to let pedestrians across you are often abused by other drivers". (Driver qtd. in UCU, 2016)

Shared Space can also displace certain groups who rely on familiar obstacles to travel, like the visually impaired and disabled population. Some blind individuals who have tried to walk in Shared Space "wouldn’t really know where the road was" and "don’t feel safe at all" (Participant 3 qtd. in Brown & Norgate, 2019) (Participant 5 qtd. in Brown & Norgate, 2019). Also, the "Say No to Shared Streets" campaign has the backing of over 20 disability organizations. Shared Space and the economy are also interwoven, as shared street design in commercial districts tends to drastically change the way people use the road. However, it is difficult to draw a single conclusion on how business owners view Shared Space. Some enjoy increased pedestrian traffic that Shared Space can bring, but some fear Shared Space will damage business from non-local customers due to reduced parking.

Conclusion
Shared Space and Woonerven are a specific example of how people are taking back car dominated streets, or "street reclaiming", a concept studied in a book of the same name by David Engwicht. They exist to fight the notion that cars are a solution to travel rather than a tool. Some mistake a tool and a solution as synonyms, however their distinction is important. A tool is used to aid in solving a problem, similar to how a carpenter uses several tools to build a house. A solution is an all-encompassing method of solving a problem. This same story of mistaking a "tool" for a "solution" can be observed in many other cases throughout history (the use of environmentally dangerous insecticides as a solution to pest control) and should be learned from in the future endeavors of our civilization (autonomous vehicles being the sole of travel). In order to expand this chapter, future contributors may want to study Shared Space efforts outside of Europe, investigate other participants with stake in Shared Space ideals, and identify parallels to the Shared Space case study where similar challenges were faced.

title: Lentis/Shared Space and Woonerven


text: The Azure Guide/Printable version

title: The Azure Guide/Printable version


text:
User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory

LMIs in system and stability Theory

 * 1) Lyapunov Inequalities
 * 2) /Continuous- time Lyapunov Stability//
 * 3) /Discrete-time Lyapunov Stability//
 * 4) /Generalized Lyapunov Stability//
 * 5) Bounded Real Lemma and Hinf Norm
 * 6) /Continuous - time Bounded Real Lemma//
 * 7) /Discrete - time Bounded Real Lemma//
 * 8) H2 Norm
 * 9) /Continuous- time H2 Norm//
 * 10) /Discrete - time H2 Norm//
 * 11) /Generalized H2 Norm//
 * 12) /Peak-to-Peak Norm//
 * 13) Kalman-Yakubovich-Popov(KYP) Lemma
 * 14) /KYP Lemma for QSR Dissipative Systems/
 * 15) /Discrete-Time KYP Lemma for QSR Dissipative Systems/
 * 16) /KYP Lemma Without Feedthrough/
 * 17) /KYP Lemma with Feedthrough/
 * 18) /Discrete-Time KYP Lemma With Feedthrough/
 * 19) /KYP Lemma for Descriptor systems/
 * 20) Conic Sectors
 * 21) /Conic Sector Lemma/
 * 22) /Exterior Conic Sector Lemma/
 * 23) /Modified Exterior Conic Sector Lemma/
 * 24) /Generalized KYP Lemma for Conic Sectors/
 * 25) Minimum Gain
 * 26) /Minimum Gain Lemma/
 * 27) /Modified Minimum Gain Lemma/
 * 28) /Discrete-Time Minimum Gain Lemma/
 * 29) /Discrete-Time Modified Minimum Gain Lemma/
 * 30) Negative Imaginary Systems
 * 31) /Negative Imaginary Lemma/
 * 32) /Discrete-time Negative Imaginary Lemma/
 * 33) /Generalized Negative Imaginary Lemma/
 * 34) /Algebraic Riccati Inequalities/
 * 35) /Discrete-time Algebraic Riccati Inequalities/
 * 36) Stabilizability
 * 37) /Continuous-time Stabilizability/
 * 38) /Discrete-time Stabilizability/
 * 39) Detectability
 * 40) /Continuous-time Detectability/
 * 41) /Discrete-time Detectability/
 * 42) Static Output Feedback Stabilizability
 * 43) /Continuous-time Static Output Feedback Stabilizability/
 * 44) /Discrete-time Static Output Feedback Stabilizability/
 * 45) Strong Stabilizability
 * 46) /Continuous-time strong stabilizability/
 * 47) /Discrete-time strong stabilizability/
 * 48) System Zeros
 * 49) /System Zeros without Feedthrough/
 * 50) /System Zeros with Feedthrough/
 * 51) D-Stability
 * 52) /Conic Sector Region Stability via the Dilation Lemma/
 * 53) /alpha-Region via the Dilation Lemma/
 * 54) /Circular Region Stability via the Dilation Lemma/
 * 55) /DC Gain of a Transfer Matrix/
 * 56) /Kharitonov-Bernstein-Haddad(KBH) Theorem/
 * 57) Quadratic Stability
 * 58) /Continuous-time Quadratic Stability/
 * 59) /Discrete-time Quadratic Stability/
 * 60) Stability of Time Delay Systems
 * 61) /Delay-Independent Condition/
 * 62) /Delay-Dependent Condition/
 * 63) /mu-Analysis/
 * 64) /Static Output Feedback Algebraic Loop/
 * 65) Continuous Time
 * 66) Time-Delay Systems
 * 67) /Delay Dependent Time-Delay Stabilization/
 * 68) /Delay Independent Time-Delay Stabilization/
 * 69) /Parametric, Norm-Bounded Uncertain System Quadratic Stability/
 * 70) /Stability of Structured, Norm-Bounded Uncertainty/
 * 71) /Stability under Arbitrary Switching/
 * 72) /Quadratic Stability Margins/
 * 73) /Stability of Linear Delayed Differential Equations/
 * 74) /H infinity Norm for Affine Parametric Varying Systems/
 * 75) /Entropy Bond for Affine Parametric Varying Systems/
 * 76) /Dissipativity of Affine Parametric Varying Systems/
 * 77) /Hankel Norm of Affine Parameter Varying Systems/
 * 78) /Positive Orthant Stabilizability/
 * 79) /LMI For Stabilization Condition for Systems With Unsymmetrical Saturated Control/
 * 80) /LMI Condition For Exponential Stability of Linear Systems With Interval Time-Varying Delays/
 * 81) /Polytopic Quadratic Stability/
 * 82) /Optimization Over Affine Family of Linear Systems/
 * 83) /Hurwitz Stabilizability/
 * 84) /Quadratic Hurwitz Stabilization for Polytopic Systems/
 * 85) /Output Energy Bound for Autonomous LTI Systems/
 * 86) /Transient State Bound for Non-Autonomous LTI Systems/
 * 87) Discrete-time
 * 88) D-stability
 * 89) /Controller D-Stability/
 * 90) /Observer D-stability/
 * 91) /Discrete-Time Lyapunov Stability/
 * 92) /LMI for Schur Stabilization/
 * 93) /L2-Gain of Systems with Multiplicative noise/
 * 94) /Discrete-Time Quadratic Stability/
 * 95) /Output Energy Bound for Discrete-Time Autonomous LTI Systems/
 * 96) /Transient Bound for Discrete-Time Non-Autonomous LTI Systems/
 * 97) Lure's System
 * 98) /Stability Of Lure's System/
 * 99) /L2 Gain of Lure's Systems/
 * 100) /Output Energy Bound for Lure's Systems/
 * 101) /Stability of Quadratic Constrained Systems/


text:
User:Margav06/sandbox/Click here to continue/Optimal control systems

Optimal control systems

 * 1) H2-optimal control
 * 2) /H2-optimal Full-state Feedback control/
 * 3) /Discrete-time H2-Optimal Full-state Feedback control/
 * 4) /H2-optimal Dynamic output Feedback control/
 * 5) /Discrete-time H2-optimal Dynamic output Feedback control/
 * 6) Hinf -Optimal Control
 * 7) /Hinf-optimal Full-state Feedback Control/
 * 8) /Discrete-time Hinf-Optimal Full State Feedback Control/
 * 9) /Hinf-optimal Dynamic output Feedback control/
 * 10) /Discrete-time Hinf-optimal Dynamic output Feedback control/
 * 11) Mixed H2-Hinf Optimal Control
 * 12) /Mixed H2-Hinf-optimal Full-state Feedback Control/
 * 13) /Discrete-time H2-Hinf-optimal Full state Feedback control/
 * 14) /Mixed H2-Hinf-optimal Dynamic output Feedback control/
 * 15) /Discrete-time H2-Hinf-optimal Dynamic output Feedback control/
 * 16) H2-Optimal State Estimation
 * 17) /H2-Optimal Observer/
 * 18) /Discrete-Time H2-Optimal observer/
 * 19) Hinf-Optimal State Estimation
 * 20) /Hinf-Optimal Observer/
 * 21) /Discrete-time Hinf-Optimal Observer/
 * 22) Mixed H2-Hinf Optimal State Estimation
 * 23) /Mixed H2-Hinf-Optimal Observer/
 * 24) /Discrete-Time Mixed H2-Hinf Optimal Observer/
 * 25) Optimal Filtering
 * 26) /H2-Optimal Filtering/
 * 27) /Hinf-Optimal Filtering/


text: Category:Book:Dad ‘n Me Walkthrough

title: Category:Book:Dad ‘n Me Walkthrough


text: IB Music/Printable version

title: IB Music/Printable version


text: Category:Book:LLVM Compiler

title: Category:Book:LLVM Compiler


text: Sardinian/Printable version

title: Sardinian/Printable version


text: LLVM Compiler/Printable version

title: LLVM Compiler/Printable version


text: A-level French/Printable version

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text: A-level Biology/Printable version

title: A-level Biology/Printable version


text: Issues in Interdisciplinarity 2018-19/Printable version


text: How to Write a Program/Printable version


text:
Hacking/Attack/Post exploitation

Goals

 * Manipulation. For example, ,.
 * Altering or destroying any kind of information. For example, ,.
 * Stealing private information. For example :, ,.
 * Finance. For example,.
 * Access for physical resources. For example,.
 * report
 * report
 * report

Maintaining control
The ultimate gratification for a network intruder always is to obtain administrator privileges for a network. When an intruder is inside, one of his or her first undertakings is often to install a so-called  on a target computer. This is a collection of programs to facilitate durable influence on a system. Some of these programs are used to compromise new or new computers on the network. Other programs are to obscure the presence of the intruder. These obscuring programs may include false versions of standard network utilities such as, or programs that can remove all data from the log files of a computer that relate to the intruder. Yet other programs of a rootkit may be used to survey the network or to overhear more passwords that are travelling over it. Rootkits may also give the means to change the very operating system of the computer it is installed on.

The network intruder then proceeds with creating one or more so called . These are access provisions that are hard to find for system administrators, and they serve to prevent the logging and monitoring that results from normal use of the network. A back door may be a concealed account or an account of which the privileges have been escalated. Or it may be a utility for remote access, such as, that has been configured to operate with a that is not customary.

The network intruder then proceeds with stealing files, or stealing credit card information, or preparing a computer to send spam emails at will. Another goal is to prepare for the next intrusion. A cautious intruder is protective against discovery of his or her location. The method of choice is to use a computer that already has been attacked as an intermediary. Some intruders use a series of intermediate computers, making it impracticable to locate them.

Back doors
The purpose of a back door is to maintain a communication channel and having methods to control a host that has been gained entry to. These methods include those for file transfer and the execution of programs. It is often important to make sure that the access or communication remains secret. And access control is desirable in order to prevent others from using the back door.

was designed as a back door. The server runs on Windows, and there are for Windows, Linux and other operating systems. The server is configured easily with a utility. After configuration, the server needs to be uploaded to the target and then started. Back Orifice 2000 supports file transfer, file execution,, and control of connections. There is also an AES for traffic encryption and an STCPIO plug-in for further obfuscation of the traffic. The first plug-in adds security and the combination of these plug-ins makes it much harder for an IDS to relate the traffic to a back door. More information can be found at http://www.bo2k.com.

Rootkits
Rootkits specialize in hiding themselves and other programs.

Hacker Defender (hxdef) is an open source rootkit for Windows. It can hide its files, its process, its entries, and its port in multiple. Although it has a simple command-line interface as a back door, it is often better to use its ability to hide a more appropriate tool.

Tools

 * https://docs.rapid7.com/metasploit/about-post-exploitation
 * https://www.offensive-security.com/metasploit-unleashed/msf-post-exploitation/
 * https://www.offensive-security.com/metasploit-unleashed/maintaining-access/

title: Hacking/Attack/Post exploitation


text: American Football in the National Football League/Printable version


text: University of Alberta Guide/Printable version


text:
User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Continuous- time Lyapunov Stability

The System:

\begin{align} \dot x(t)&=Ax(t)\\ x(0)&=x_0 \end{align}$$

The Optimization Problem:

The LMI: The Lyapunov Inequality

\begin{align} \text{Find} \; X:&\\ \begin{bmatrix} X\end{bmatrix}&>0\\ \begin{bmatrix} A^T X+XA\end{bmatrix}&<0 \end{align}$$

Conclusion:

If feasible, System is internally exponentially Stable for any $$ x_0 \in \R^n $$.

Related LMIs:


text:
User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Discrete-time Lyapunov Stability

Discrete-Time Lyapunov Stability

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

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

The System
Discrete-Time System

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

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

The Optimization Problem
The following feasibility problem should be optimized:

Find P obeying the LMI constraints.

The LMI:
Discrete-Time Bounded Real Lemma

The LMI formulation

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

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

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

Related LMIs
Continuous_Time_Lyapunov_Inequality - Lyapunov_Inequality


text:
User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Generalized Lyapunov Stability

WIP, Description in progress

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

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

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

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

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

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

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

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

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

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

$$$$ WIP, additional references to be added


text:
User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Discrete - time Bounded Real Lemma

Discrete-Time Bounded Real Lemma

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

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

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

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

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Discrete-Time Bounded Real Lemma

The LMI formulation

H∞ norm < $$\gamma$$

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

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

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

Related LMIs
- Continuous time KYP_Lemma_(Bounded_Real_Lemma)


text: Written ASL/Printable version

title: Written ASL/Printable version


text: Optimal Classification/Printable version


text: Diagnostic Radiology/Printable version

title: Diagnostic Radiology/Printable version


text: ActionScript Programming/Printable version


text:
Programming in LOLGraphics 3.4

Table of Contents

 * 1) /Preface/
 * 2) /Interface, hello world program, subprograms, & labels/
 * 3) /Variables & Input/
 * 4) /Conditions & Loops/
 * 5) /Graphics in LOLGraphics/
 * 6) /Making a simple game in LOLGraphics/
 * 7) /LOLGraphics Cheat Sheet/
 * 8) /External links/
 * 9) /Authors & Contributors/

title: Programming in LOLGraphics 3.4


text:
Programming in LOLGraphics 3.4/Interface, hello world program, subprograms, & labels

Editor Interface

When opening LOLGraphics, you will be greeted to the LOLGraphics Editor. Let’s go over what each button does:

The save button saves the current code to example.lulz in the same folder that the jar file of LOLGraphics is located in. The load button reads example.lulz and displays its contents on the editor. This is assuming you wrote in the file field the word “example”. Otherwise, it will read and write to whatever file name you wrote there with a .lulz extension.

The run button takes what is written in the editor, removes all blank lines and comments (a line that begins with BTW, or a line that begins with OBTW all the way till the line TLDR) and trims all the lines (removes whitespace characters from the start and end of it). Then it will send what’s left to the interpreter to run it line by line.

Interpreter interface
Here’s how the interface of the interpreter works:

The screen of the interpreter consists of 3 main parts: a console for text output, a field for input, and a graphics area. Don’t worry if you don’t understand how to use them just yet as all of it will covered later on in the book.

You can check out the example codes available in the editor. Just take into account that not all of them use the latest version of LOLGraphics. This book will only cover version 3.4 which is the latest.

Code Structure
When you open the editor, you will see the default code which looks like this:

HAI 3.4 0 100 IM IN UR CODE EXECUTIN UR KOMANDZ BTW UR CODE GOEZ HER IM OUTTA UR CODE

Let’s go over what each line does. The first line might seem useless for the untrained eye but it actually has multiple purposes. The first is to specify the version of LOLGraphics you are using. Believe it or not but if you download LOLGraphics Editor x.y, you are actually downloading all the versions up to this version. That means that a code written for LOLGraphics 0.1 will work fine on all versions of the editor, and will load using LOLGraphics Interpreter 0.1. The two numbers represent the delays between two commands. In this example 0 milliseconds since the interpreter I launched till the first command starts, and 100 milliseconds (0.1 second) in between every 2 commands. If you write just one number it will be the delay between every 2 commands and the initial delay will be 0. If you don’t write any numbers it will run as if you wrote 0 and 100.

This is also why throughout this book, the efficiency of the code will be measured purely by the number of lines executed by the interpreter.

Also it’s possible to change the delay in the middle of the code using the command.

The line  does NOTHING🎉🎉🎉 but is necessary for the code to run. Otherwise you will get a message saying that you forgot to specify where the code starts. Remember, this is a meme programming language.

Finally the line  ends the code. There’s no limitation on where you can put, or how many times you can put it. It’s just a command that ends the code and can be placed anywhere. You can theoretically omit the line and it will run until it has executed the last line. However, if you want to have subprograms it’s a must to have this line otherwise the interpreter will run on all the subprograms.

Error handling
LOLGraphics forces you to be polite and most (not all) commands begin with PLZ. If you type a command that doesn’t exist it will check if it begins with PLZ. If yes it will tell that it doesn’t know what to do and ask to give a command that actually exists. Otherwise it will yell at you that you forgot the magic word. Don’t worry about the wording to much and definitely don’t add PLZ to commands that don’t have it otherwise it won’t work.

When you get yelled at, the interpreter will not tell you what line to fix so it’s recommended that when writing your code, you write it in small parts and test each one before you continue to the next.

Printing strings
There are two ways to print strings in LOLGraphics:

PLZ TYPE TEXT HELLO WORLD PLZ PRINT TEXT HELLO WORLD

The first one prints Hello World to the console, and the second one prints hello world and then goes to the next line. If you want to print variable x (variables will be taught in a later chapter), as x: 5, adding a space at the end of the type command won’t help since the LOLGraphics Editor deleted spaces at the beginning and end of the line. To compensate there is a command  that types that character to the screen.

Hello World
This is how a hello world program in LOLGraphics looks like:

HAI 3.4 0 100 IM IN UR CODE EXECUTIN UR KOMANDZ PLZ PRINT TEXT HAI WORLD! IM OUTTA UR CODE

Clearing the console & changing text color
You can clear the console using.

You can also change the text color using the command. There are 3 ways to specify the color:
 * Type its name for the following colors: black, blue, cyan, dark gray, gray, green, light gray, magenta, orange, pink, red, white, and yellow.

PLZ CHANGE TEXT COLOR BLUE
 * Type its RGB values.

PLZ CHANGE TEXT COLOR 255 0 0
 * Type the word random for a random color.

PLZ CHANGE TEXT COLOR RANDOM

Subprograms
A subprogram begins with. In order to end the subprogram, you type  which like the command to end the code, can be placed anywhere in the subprogram. After the subprogram ends it will continue running starting from the line after where the subprogram was called. In later chapters, you will learn how to use subprograms in loops and conditions. You can also call a subprogram inside a subprogram and create recursion!🎊 Heres a code that prints  indefinitely using recursion.

HAI 3.4 0 100 IM IN UR CODE EXECUTIN UR KOMANDZ PLZ RUN SUBPROGRAM PRINT IM OUTTA UR CODE IM IN UR SUBPROGRAM DAT IZ KALLED PRINT PLZ PRINT TEXT I CAN HAS A CHEEZBURGER? PLZ RUN SUBPROGRAM PRINT

In this code there’s no point writing  since it will never get there anyways. That’s another thing about LOLGraphics, you can add commands between the “main area” and the first subprogram or in between 2 subprograms but that probably isn’t very smart since the interpreter will never run them. Also they can be used as comments since they won’t be run anyways but it’s probably smarter to write regular comments with BTW or OBTW/TLDR since they will be wasted before the code is sent to the interpreter.

Labels
When you start a subprogram, the program remembers which line it was that called the subprogram so that it can go back and continue running from that line. Usually that’s useful, but you might not always want that. Take for example the recursive code above. There’s not any reason for the code to remember where this subprogram was called. That’s why labels exist. To define a label, you write  (and yes, the exclamation mark is mandatory) and to jump to a label, write. Now let’s rewrite the infinite recursion above only with labels.

HAI 3.4 0 100 IM IN UR CODE EXECUTIN UR KOMANDZ DIS IZ MY LABEL! IT IZ KALLED L PLZ PRINT TEXT I CAN HAS A CHEEZBURGER? PLZ GOTO LABEL L IM OUTTA UR CODE

This code does exactly the same. The only differences are that it’s shorter and more efficient.


text:
Lentis/The War on Drugs

Introduction
The war on drugs is a global campaign led by the U.S. federal government aimed at reducing the illegal drug trade through the use of drug prohibition and military intervention. The term was first used in 1969 by Richard Nixon but was popularized by him in 1971 during a speech to Congress about drug abuse. At this time, the Vietnam War led to high rates of heroin addiction among servicemen. As a result of increasing drug usage in America, Nixon claimed that drug abuse is “public enemy number one” and created the Drug Enforcement Administration (DEA) in 1973 to combat drug crimes.

In 1980, Ronald Reagan used “the War on Drugs” as a campaigning method, and once elected, implemented stricter regulations on drug trafficking. This caused the price of drugs to increase (specifically cocaine), leading to the invention of “crack”, which is a cheaper, smokable form of cocaine. During his second term, President Reagan signed the Anti-Drug Abuse Act, which defined mandatory minimum sentences for drug offenses and allocated $1.7 billion to the War on Drugs. A mandatory minimum is “a sentence, created by Congress or a state legislature, which the court must give to a person convicted of a crime, no matter what the unique circumstances of the offender or the offense are.” These mandatory minimums have created disparities in sentencing for different socioeconomic and racial groups (see Racial Disparity and Disproportionate Incarceration Rates section). Collateral consequences were also faced by those convicted of drug crimes. These are additional state-level civil penalties attached to a criminal conviction such as loss of ability to purchase a firearm, loss of eligibility for food stamps, loss of eligibility for Federal Student Aid, loss of ability to live in federal housing, loss of ability to vote, and deportation.

Under President George H.W. Bush, the Office of National Drug Control Policy (ONDCP) was formed in 1989 and the position of “Drug Czar” was created, which was originally a term coined by Nixon. The Drug Czar was promoted to a cabinet-level position by President Clinton in 1993. In 1995, the U.S. Sentencing Commission acknowledged disparities between crack and powder cocaine sentencing, but Congress overrules recommendations that would make sentencing more equitable. As a result, by the 21st century, the U.S. accounts for 25% of the world’s prison population, with approximately 20% of the prisoners are in for drug offenses. The War on Drugs has been coined “the new Jim Crow” and because of the mass incarceration of people of color in the United States.

Controlled Substance Act of 1970
The Controlled Substance Act is the start of the Nixon administration attempting to control the possession and distribution of certain drugs and substances. All substances that are regulated by the government were categorized into one of five schedules. Of the five schedules, schedule one denotes drugs that were deemed highly addictive and destructive. Examples of substances classified under schedule one include Marijuana, LSD, Heroin, MDMA (Ecstasy).

Comprehensive Crime Control Act of 1984
Comprehensive Crime Control Act of 1984 was passed by the Reagan administration in an attempt to take the War on Drugs a step further. In this act, penalties for the cultivation, possession, and transfer of marijuana increased. Also, mandatory minimum sentencing was established as well as the elimination of federal parole.

Anti-Drug Abuse Act of 1986
Signed into law by President Ronald Reagan, the Anti-Drug Abuse Act of 1986 established the disparate treatment of powdered and crack cocaine. Creating a 100:1 disparity, this act put in place a five-year minimum sentence for the possession of both 5 grams of crack cocaine and 500 grams of powdered cocaine. In 2010, the Obama administration passed the Fair Sentencing Act, which reduced this disparity from 100:1 to 18:1.

Racial Disparity and Disproportionate Incarceration Rates
Many different racial minorities were negatively impacted by discriminatory policies enacted during this time. Specifically, those from the Black and Latinx communities have been primary targets of anti-drug policies and rhetoric. Although statistics prove that they use drugs at approximately the same rates, People of Color are 2.5 times more likely to be arrested for possession. In turn, mandatory minimum sentencing and escalating laws led to mass incarceration rates for the Black and Latinx communities. Overall, almost 60% of people incarcerated in state prisons, and nearly 80% of people incarcerated in federal prisons for drug offenses are Black and Latino.

Black Community
Black Americans only comprise 13% of the U.S. population, however, they comprise nearly 40% of those incarcerated in state or federal prisons for drug law violations. Due to the media frenzy on the unsubstantiated crack epidemic in the black community, the government was prompted to pass laws against crack cocaine. The 100:1 disparity between crack and powdered cocaine makes the false distinction between the two clear given the statistics. Powdered cocaine is more commonly used by white people while crack cocaine, often known as the cheaper alternative, is dealt primarily by Black people. As a result of the establishment of the difference, U.S. prisons were filling up and incarceration rates were skyrocketing. By 2001, over 80% of federal crack defendants were black.

The manner in which drug laws were being enforced heavily impacted the Black community. Due to the U.S Supreme Court Case Terry v. Ohio, police officers were given the power to stop an individual for a pat-down search if the officer had reasonable suspicion that said individual is engaged, or about to be engaged, in criminal activity. This has resulted in a massive racial profiling issue as People of Color are more likely to be stopped, searched, and arrested by the police. For example, in 2019 in New York, out of all of the reported stop-and-frisks, 66% of those individuals were innocent and 59% were Black. The 2013 court case Floyd v. City of New York ruled that New York City violated the Fourth Amendment, which protects citizens from unreasonable searches and seizures.

Latinx Community
Similarly, the Latinx community comprises only 17% of the U.S. population, but they represent almost half of all cases in federal courts for drug offenses, specifically marijuana. Cannabis has been in the process of being decriminalized since 1996. Colorado laws legalized marijuana for recreational use and sales in 2012. However, In 2013, simple marijuana possession was the fourth most common cause for deportation. Today, the U.S. has created the largest immigrant exclusion, detention, and deportation structure in the world.

Private Prison Companies
Since the early stages of the War on Drugs incarceration rates in the U.S. have nearly tripled to 800 prisoners per 100,00 people. This has led to overcrowding and funding issues for government owned prisons. In response local and state governments have contracted out to private companies to manage and run prisons. CoreCivic, formerly known as Corrections Corporation of America, is one of the largest private prison corporations in the world. It currently operates over 65 correctional facilities with yearly revenues of over $1.7 billion.

Private prison companies, like CoreCivic, have faced a wide variety of legal issues, ranging from inadequate staffing to treatment of inmates. However, CoreCivic has done little to change their practices. Instead, these lawsuits have resulted in state and local governments attempting to terminate their contracts and take back control of the prisons.

The Communities Being Affected
Community members feel the policing and enforcement policies used during the War on Drugs were excessive and targeted, particularly towards black Americans. A member of the community stated that "they're spending so much money on these prisons to keep kids locked up, they don't even spend a fraction of that money sending them to college or some kind of school." Despite this disapproval of federal spending, the national drug control budget is expected to hit a historic high of $41 billion in 2022. This had led community members to believe that the War on Drugs, and its disastrous effects, will not be over anytime soon.

Advocacy Groups
Advocacy groups range from broad international organizations, like the Global Commission on Drug Policy, to more specific groups like the National Harm Reduction Coalition. These groups oppose the tactics of the War on Drugs and propose a reassessment of current policy to favor a more rehabilitative and healthcare related approach to the drug problem. The Global Commission on Drug Policy states “putting health and community safety first requires a fundamental reorientation of policy priorities and resources, from failed punitive enforcement to proven health and social interventions,” demonstrating their recommended reevaluation of drug policy. To accomplish their agendas both advocacy groups have supplied grants to local drug treatment facilities and have taken on lobbying roles to push Congress to reevaluate its current drug policy.

Conclusion
The War on Drugs remains a continuous issue as many of the laws enacted during the 1980’s and 1990’s are still in effect today. The United States still has the highest incarceration rate in the world and Black Americans are still arrested at higher rates. In the future, it may be helpful to extend this research to include the use of the term “war” and the effect the use of this metaphor has on the participants. Additionally, as the philosophy on how to deal with drug abuse is ever changing it may be helpful to include updates on new legislation that is passed and developments in the areas of rehabilitation and treatment.

title: Lentis/The War on Drugs


text: Template:Editnotices/Group/Lentis

title: Template:Editnotices/Group/Lentis


text:
User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Continuous- time H2 Norm

$${H_2}$$-norm of System

The $${H_2}$$-norm is conceptually identical to the Frobenius (aka Euclidean) norm on a matrix. It can be used to determine whether the system representation can be reduced to its simplest form, thereby allowing its use in performing effective block-diagram algebra.

The System
Suppose we define the state-space system$$G: {L_2}\rightarrow{L_2}\text{ by }y=Gu$$ if:

\begin{align} \dot x(t)&=Ax(t)+Bu(t)\\ y(t)&=Cx(t)+Du(t)\\ \end{align}$$ where $$A\in\mathbb{R}^{mxm}$$, $$B\in\mathbb{R}^{mxn}$$, $$C\in\mathbb{R}^{pxm}$$, and $$D\in\mathbb{R}^{qxn}$$ for any $$t\in\mathbb{R}$$. Then the $${H_2}$$-norm of the system can be determined as described below.

The Data
In order to determine the $${H_2}$$-norm of the system, we need the matrices $${A}$$, $${B}$$, and $${C}$$.

The Optimization Problem
Suppose we wanted to to infer properties of the system behaviour (which is represented in the form (A,B,C,D)). Then it becomes necessary to ensure that the overall system forms an algebra, as the standard use of block-diagram algebra would otherwise be invalid. The only way this is possible is by calculating $${H_2}$$ and/or $${H_\infty}$$-norms - both of which are signal norms that (in a certain sense) measure the size of the transfer function.

The LMI: The $$H_2$$ Norm
Assuming that $$\hat{P}(s)=C(sI-A)^{-1}B$$, this means that the following are equivalent:

 * $$ 1)\quad A\text{ is Hurwitz and }||\hat{P}||^2_{H_2}<\gamma$$

2) \begin{align} \begin{cases} trace(CXC^T )&<\gamma\\   AX+XA^T +BB^T &<0\\    X&>0 \end{cases} \end{align}$$

Conclusion:
The LMI can be used to minimize the $${H_2}$$-norm of the system. It is worth noting that a finite $${H_2}$$-norm does not guarantee finite $${H_\infty}$$-norm, and that in order for the block diagram algebra to be valid, $${H_\infty}$$-norm must be finite.

Implementation

 * Example Code - A GitHub link that contains code (titled "H2Norm.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIs

 * $H_2$-Filtering - LMI for $$H_2$$-Filtering
 * Discrete-Time $H_2$ Norm - LMI for $$H_2$$-norm in the Discrete-Time case.


text:
User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Discrete - time H2 Norm

Discrete-Time H2 Norm

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

Discrete-Time LTI systems' H2 norm can be found by solving a LMI.

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

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

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Discrete-Time Bounded Real Lemma

The LMI formulation

H2 norm < $$\mu$$

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

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

trZ<\mu ^2

\end{align}$$

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

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

Related LMIs
- Continuous time H2 norm.


text:
User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Generalized H2 Norm

Generalized $$H_{2}$$ Norm
The $$ H_2 $$ norm characterizes the average frequency response of a system. To find the H2 norm, the system must be strictly proper, meaning the state space represented $$ D $$ matrix must equal zero. The H2 norm is frequently used in optimal control to design a stabilizing controller which minimizes the average value of the transfer function, $$ G $$ as much as possible. This optimal control problem is also called the Linear Quadratic Gaussian.

The System
Consider a continuous-time, linear, time-invariant system $$ G : L_{2e} \to L_{2e} $$ with state space realization $$ (A,B,C,0) $$ where $$ A\in\mathbb{R}^{n\times n} $$, $$B\in\mathbb{R}^{n\times m}$$, $$C\in\mathbb{R}^{p\times n}$$, amd $$ A $$ is Hurwitz. The generalized $$ H_{2} $$ norm of $$ G $$ is:

$$ \left\| G \right\|_{2,{\infty}} = \sup_{u\in L_{2}, {u}\neq\text{zero}} $$ $$ \frac{\left\| Gu \right\|_{\infty}}{\ \left\| u \right\|_{2}}$$

The Data
The transfer function $$G$$, and system matrices $$A$$, $$B$$, $$C$$ are known and $$ A $$ is Hurwitz.

The LMI: Generalized $$H_{2}$$ Norm LMIs
The inequality $$ \left\| G \right\|_{2,{\infty}} < \mu $$ holds under the following conditions:

1. There exists $$ P\in\mathbb{S}^{n}$$ and $$ \mu\in\mathbb{R}_{>0}$$ where $$ P > 0 $$ such that:

 * $$\begin{bmatrix} A^{T}P + PA & PB \\ * & -\mu 1 \end{bmatrix}<0 $$.
 * $$\begin{bmatrix} P & C^{T} \\ * & \mu 1\end{bmatrix}>0 $$.

2. There exists $$ Q\in\mathbb{S}^{n}$$ and $$ \mu\in\mathbb{R}_{>0}$$ where $$ Q > 0 $$ such that:
 * $$\begin{bmatrix} QA^{T} +AQ & B \\ * & -\mu 1 \end{bmatrix}<0 $$.
 * $$\begin{bmatrix} Q & QC^{T} \\ * & \mu 1 \end{bmatrix}>0 $$.

3. There exists $$ P\in\mathbb{S}^{n}, V\in\mathbb{R}^{n\times n}$$ and $$ \mu\in\mathbb{R}_{>0}$$ where $$ P > 0 $$ such that:
 * $$\begin{bmatrix} -(V+V^{T}) & V^{T}A+P & V^{T}B & V^{T} \\ * & -P & 0 & 0 \\ * & * * -\mu 1 & 0 \\ * & * & * & -P \end{bmatrix}<0 $$.
 * $$\begin{bmatrix} P & C^{T} \\ * & \mu 1 \end{bmatrix}>0 $$.

Conclusion:
The generalized $$ H_{2} $$ norm of $$ G $$ is the minimum value of $$ \mu\in\mathbb{R}_{>0} $$ that satisfies the LMIs presented in this page.

Implementation
This implementation requires Yalmip and Sedumi.

Generalized $ H_{2} $ Norm - MATLAB code for Generalized $$ H_{2} $$ Norm.

Related LMIs
LMI for System H_{2} Norm

Return to Main Page:
LMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Peak-to-Peak Norm

Peak-to-peak norm performance of a system

The System
Considering the following system:

\begin{align} \dot{x} = A x + B u\\ z     = C x + D u\\ \end{align}$$

Where $$ x(t) \in \mathbb{R}^{n} $$ is the state signal, $$ u(t) \in \mathbb{R}^{m} $$ is the input signal, and $$ z(t) \in \mathbb{R}^{p} $$ is the output. When given an initial condition $$ x(0) = 0 $$, the system can be defined to map the output and input signals for the peak-to-peak performance.

\begin{align} ||T||_{\infty,\infty} := \sup\limits_{0<||u||_{\infty}<\infty} \frac{||z||_{\infty}}{||u||_{\infty}}\\ \end{align}$$

The Data
The matrices $$ A $$, $$ B $$, $$ C $$, and $$ D $$ are the only data sets required for this optimization problem.

The Optimization Problem
Consider a continuous-time LTI system, $$ G: L_{2e} \to L_{2e} $$, given that: $$ A $$, $$ B $$, $$ C $$, and $$ D $$ $$ A \in \mathbb{R}^{n \times n} $$, $$ B \in \mathbb{R}^{n \times m} $$, $$ C \in \mathbb{R}^{p \times n} $$, and $$ D \in \mathbb{R}^{p \times m} $$. Given that the matrix $$ A $$ is Hurwitz,The peak-to-peak norm of $$ G $$ is given as:

\begin{align} ||T||_{\infty,\infty} := \sup\limits_{0<||w||_{\infty}<\infty} \frac{||z||_{\infty}}{||u||_{\infty}}\\ \end{align}$$

The LMI: Peak-to-Peak norm
There exists a matrix $$ P \in \mathbb{S}^{n} $$ and $$ \gamma $$, $$ \epsilon $$, $$ \mu \in \mathbb{R}_{>0} $$, where the following constraints are used: $$ \min \mu $$

\begin{align} P < 0 \\ \begin{bmatrix} A^T P + P A + \gamma P & P B \\ B' P & \epsilon I\end{bmatrix}&<0\\ \begin{bmatrix} \gamma P & 0 & C^T \\ 0 & (\mu - \epsilon) I & D^T \\ C & D & \mu I\end{bmatrix}&>0\\ \end{align}$$

Since this optimization has $$ \gamma P $$ in the constraints, this does make this optimization bi-linear. attempting to solve this LMI is not feasible unless some type of substitute is implemented to the variables $$ \gamma P $$.

Conclusion:
The results from this LMI will give the peak to peak norm of the system:

\begin{align} \sup\limits_{0<||u||_{\infty}<\infty} \frac{||z||_{\infty}}{||u||_{\infty}} < \mu \end{align}$$

Implementation

 * Peak-to-Peak Norm Example


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/KYP Lemma for QSR Dissipative Systems

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

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

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

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

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

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

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

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

The Data
The matrices $$ A,B,C $$ and $$D $$ which defines the state space of the system

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

 * $$\int_0^T (y^T(t)Qy(t) + 2y^TSu(t) + u^T(t)Ru(t))\,dt \ge 0, \forall u \in \mathcal{L}_{2e}, \forall T \ge 0 $$

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

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

 * $$\begin{bmatrix}

PA + A^{T}P - C^{T}QC & PB - C^{T}S - C^{T}QD \\ (PB - C^{T}S - C^{T}QD)^{T} &  -D^{T}QD - (D^{T}S + S^{T}D) - R \end{bmatrix}\le 0.$$

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

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

Related LMIs
KYP Lemma


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Discrete-Time KYP Lemma for QSR Dissipative Systems

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

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

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

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

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

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

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

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

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

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

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

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

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

 * $$\begin{bmatrix}

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

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

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

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


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/KYP Lemma Without Feedthrough

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

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

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

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

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

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

 * $$\begin{align}

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

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

 * $$\begin{align}

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

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

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

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

 * $$\begin{align}

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

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

 * $$\begin{align}

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

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

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

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

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


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/KYP Lemma with Feedthrough

Definition
Consider a square, continuous-time LTI system, $$\mathcal{G}:\mathcal{L}_{2e}\longrightarrow \mathcal{L}_{2e}$$, with minimal state-space realization ($$A$$, $$B$$, $$C$$, $$D$$), where $$A\in\mathbb{R}^{n \times n}$$, $$B\in\mathbb{R}^{n \times m}$$, $$C\in\mathbb{R}^{m \times n}$$, and $$D\in\mathbb{R}^{m \times m}$$.

LMI 1.1
The system $$\mathcal{G}$$ is positive real (PR) under either of the following equivalent necessary and sufficient conditions.

 * 1) There exists $$P\in\mathbb{S}^{n}$$, where $$P>0$$, such that $$\begin{bmatrix} PA+A^{T}P & PB-C^{T} \\ * & -(D+D^{T}) \end{bmatrix} \leq0$$.
 * 2) There exists $$Q\in\mathbb{S}^{n}$$, where $$Q>0$$, such that $$\begin{bmatrix} AQ+QA^{T} & B-QC^{T} \\ * & -(D+D^{T}) \end{bmatrix} \leq0$$

This is a special case of the KYP Lemma for QSR dissipative systems with $$Q=0$$, $$S=\frac{1}{2}\cdot1$$, and $$R=0$$.

LMIs 1.2
The system $$\mathcal{G}$$ is strictly positive real (SPR) under either of the following equivalent necessary and sufficient conditions.

 * 1) There exists $$P\in\mathbb{S}^{n}$$, where $$P>0$$, such that $$\begin{bmatrix} PA+A^{T}P & PB-C^{T} \\ * & -(D+D^{T}) \end{bmatrix} \leq0$$.
 * 2) There exists $$Q\in\mathbb{S}^{n}$$, where $$Q>0$$, such that $$\begin{bmatrix} AQ+QA^{T} & B-QC^{T} \\ * & -(D+D^{T}) \end{bmatrix} \leq0$$

This is a special case of the KYP Lemma for QSR dissipative systems with $$Q=\epsilon1$$, $$S=\frac{1}{2}\cdot1$$, and $$R=0$$, where $$\epsilon\in\mathbb{R}_{>0}$$.


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Discrete-Time KYP Lemma With Feedthrough

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

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

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

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

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

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

 * $$\begin{bmatrix}

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

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

 * $$\begin{bmatrix}

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

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

 * $$\begin{bmatrix}

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

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

 * $$\begin{bmatrix}

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

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

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

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

 * $$\begin{bmatrix}

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

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

 * $$\begin{bmatrix}

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

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

 * $$\begin{bmatrix}

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

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

 * $$\begin{bmatrix}

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

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

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

Related LMIs
KYP Lemma State Space Stability KYP Lemma without Feedthrough


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/KYP Lemma for Descriptor systems

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

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

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

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

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

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

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

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

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

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

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

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

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


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Conic Sector Lemma

Conic Sector Lemma

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Related LMIs
Exterior Conic Sector Lemma.

KYP Lemma


text:
User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Exterior Conic Sector Lemma

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

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

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

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

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

 * $$\begin{bmatrix}

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

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

 * $$\begin{bmatrix}

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

Proof, Applying the Schur complement lemma to the $$ r^{2}I $$ terms in (1) gives (2).

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

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

Related LMIs
KYP Lemma State Space Stability


text:
User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Modified Exterior Conic Sector Lemma

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

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

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

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

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

 * $$\begin{bmatrix}

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

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

 * $$\begin{bmatrix}

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

Proof, Applying the Schur complement lemma to the $$ r^{2}I $$ terms in (1) gives (2).

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

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

Related LMIs
KYP Lemma State Space Stability


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Generalized KYP Lemma for Conic Sectors

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

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

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

Also consider $$ \pi_{c}(a,b) \in \mathcal{S}^{m} $$, which is defined as

 * $$ \pi_{c}(a,b) = \begin{bmatrix}

-\tfrac{1}{b}I & \tfrac{1}{2}(1+\tfrac{a}{b})I  \\ (\tfrac{1}{2}(1+\tfrac{a}{b})I)^{T} &  -aI \end{bmatrix}$$,

where $$ a \in \mathcal{R}, b \in \mathcal{R}_{>0} $$ and $$ a < b $$.

The Data
The matrices The matrices $$ A,B,C $$ and $$D$$. The values of a and b

LMI : Generalized KYP (GKYP) Lemma for Conic Sectors
The following generalized KYP Lemmas give conditions for $$\mathcal{G}$$ to be inside the cone $$ [a,b] $$ within finite frequency bandwidths.
 * 1. (Low Frequency Range) The system $$\mathcal{G}$$ is inside the cone $$ [a,b] $$ for all $$ \omega \in {\omega \in \mathcal{R} | |\omega| < \omega_{1}, det(j\omega I - A) \neq 0} $$, where $$\omega_{1} \in \mathcal{R}_{>0}, a \in \mathcal{R}, b \in \mathcal{R}_{>0} $$ and $$ a0}$$, where $$Q \ge 0$$, such that

 * $$\begin{bmatrix}

A &  B  \\ I &  0 \end{bmatrix}^{T}\begin{bmatrix} -Q & P \\ P^{T} & (\omega_{1} - \overline{\omega}_{1})^{2}Q \end{bmatrix} \begin{bmatrix} A &  B  \\ I &  0 \end{bmatrix} - \begin{bmatrix} C &  D  \\ 0 &  I \end{bmatrix}^{T} \pi_{c}(a,b) \begin{bmatrix} C &  D  \\ 0 &  I \end{bmatrix} < 0 $$.

 * If $$\omega_{1} \rightarrow \infty, P>0. $$ and Q = 0, then the traditional Conic Sector Lemma is recovered. The parameter $$\overline{\omega}_{1}$$ is incuded in the above LMI to effectively transform $$|\omega| \le (\omega_{1} - \overline{\omega}_{1})$$ into the strict inequality $$|\omega| < \omega_{1}$$

 * 2. (Intermediate Frequency Range) The system $$\mathcal{G}$$ is inside the cone $$ [a,b] $$ for all $$ \omega \in {\omega \in \mathcal{R} | \omega_{1} \le |\omega| < \omega_{2}, det(j\omega I - A) \neq 0} $$, where $$\omega_{1},\omega_{2} \in \mathcal{R}_{>0}, a \in \mathcal{R}, b \in \mathcal{R}_{>0} $$ and $$ a0}$$ and $$\hat{\omega}_{2} = (\omega_{1} + \tfrac{(\omega_{2} - \hat{\omega}_{2})}{2}),$$ where $$ P^{H} = P, Q^{H} = Q $$ and $$Q \ge 0$$, such that

 * $$\begin{bmatrix}

A &  B  \\ I &  0 \end{bmatrix}^{T}\begin{bmatrix} -Q & P + \mathcal{j\hat{\omega}_{2}}Q\\ P - \mathcal{j\hat{\omega}_{2}}Q & \omega_{1}(\omega_{2} - \hat{\omega} - 2)Q \end{bmatrix} \begin{bmatrix} A &  B  \\ I &  0 \end{bmatrix} - \begin{bmatrix} C &  D  \\ 0 &  I \end{bmatrix}^{T} \pi_{c}(a,b) \begin{bmatrix} C &  D  \\ 0 &  I \end{bmatrix} < 0 $$.

 * The parameter $$\overline{\omega}_{2}$$ is incuded in the above LMI to effectively transform $$ \omega_{1} \le |\omega| \le (\omega_{2} - \overline{\omega}_{2})$$ into the strict inequality $$\omega_{1} \le |\omega| < \omega_{2}$$.

 * 3. (High Frequency Range) The system $$\mathcal{G}$$ is inside the cone $$ [a,b] $$ for all $$ \omega \in {\omega \in \mathcal{R} | \omega_{2} < |\omega|, det(j\omega I - A) \neq 0} $$, where $$\omega_{2} \in \mathcal{R}_{>0}, a \in \mathcal{R}, b \in \mathcal{R}_{>0} $$ and $$ a<b $$, if there exist $$ P,Q \in \mathcal{S}^{n} $$, where $$Q \ge 0$$, such that

 * $$\begin{bmatrix}

A &  B  \\ I &  0 \end{bmatrix}^{T}\begin{bmatrix} -Q & P \\ P^{T} & \omega_{2}^{2}Q \end{bmatrix} \begin{bmatrix} A &  B  \\ I &  0 \end{bmatrix} - \begin{bmatrix} C &  D  \\ 0 &  I \end{bmatrix}^{T} \pi_{c}(a,b) \begin{bmatrix} C &  D  \\ 0 &  I \end{bmatrix} < 0 $$.

If (A, B, C, D) is a minimal realization, then the matrix inequalities in all of the above LMI, then it can be nostrict.

Conclusion:
If there exist a positive definite $$q$$ matrix satisfying above LMIs for the given frequency bandwidths then the system $$\mathcal{G}$$ is inside the cone [a,b].

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

Related LMIs
KYP Lemma State Space Stability Exterior Conic Sector Lemma Modified Exterior Conic Sector Lemma


text:
User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Minimum Gain Lemma

Minimum Gain Lemma

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

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

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

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

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

\begin{align}

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

\end{align}$$

Or (Obtained using Schur compliment):

\begin{align}

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

\end{align}$$

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

Implementation

 * Minimum Gain Lemma Example