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Linear matrix inequalities and control theory/pages/Fundamentals of Matrix and LMIs

Fundamentals of Matrix and LMIs

Fundamentals of Matrix and LMI Properties of LMIs
 * 1) Basic Matrix Theory
 * 2) Notion of Matrix Positivity
 * 3) Matrix Inequalities and LMIs
 * 4) Convexity of LMIs
 * 5) Concatenation of LMIs
 * 1) Change of Subject(Variable)
 * 2) Congruence Transformation
 * 3) Young’s Relation (Completion of the Squares)
 * 4) Young’s Relation-Based Properties
 * 5) Special Cases of Young’s Relation
 * 6) Iterative Convex Overbounding
 * 7) Projection Lemma (Matrix Elimination Lemma)
 * 8) Strict Projection Lemma
 * 9) Nonstrict Projection Lemma
 * 10) Reciprocal Projection Lemma
 * 11) Projection Lemma-Based Properties
 * 12) Ellipsoidal Inequality
 * 13) Continuous Time Properties
 * 14) Schur Complement
 * 15) Strict and Nonstrict Schur Complement
 * 16) Schur Complement Lemma-Based Properties
 * 17) Eigenvalue related Problems
 * 18) LMI for Eigenvalue Minimization
 * 19) Eigenvalue Problem
 * 20) LMI for Generalized Eigenvalue Problem
 * 21) LMI for Matrix Norm Minimization
 * 22) LMI for Generalized Eigenvalue Problem
 * 23) LMI for Linear Programming
 * 24) LMI for Feasibility Problem
 * 25) Structured Singular Value
 * 26) LMI for Minimizing Condition Number of Positive Definite Matrix
 * 27) Continuous Quadratic Stability
 * 28) Discrete Time Properties
 * 29) Discrete Time Minimum Gain Lemma
 * 30) Discrete Time Modified Minimum Gain Lemma
 * 31) Finsler’s Lemma
 * 32) Dilation
 * 33) Tangential Nevanlinna Pick
 * 34) Nevanlinna Pick Interpolation with Scaling


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LMIs in Control/Linear matrix inequalities and control theory/pages/Fundamentals of Matrix and LMIs/Notion of Matrix Positivity

Notation of Positivity
A symmetric matrix $$A\in\R^{n\times n}$$ is defined to be:

positive semidefinite, $$(A\ge 0)$$, if $$x^TAx\ge 0 $$ for all $$x\in\R^n, x\neq \mathbf{0} $$.

positive definite, $$(A>0)$$, if $$x^TAx> 0 $$ for all $$x\in\R^n, x\neq \mathbf{0} $$.

negative semidefinite, $$(-A\ge 0)$$.

negative definite, $$(-A>0)$$.

indefinite if $$A$$ is neither positive semidefinite nor negative semidefinite.

Properties of Positive Matricies

 * For any matrix $$M$$, $$M^TM>0$$.
 * Positive definite matricies are invertible and the inverse is also positive definite.
 * A positive definite matrix $$A>0$$ has a square root, $$A^{1/2}>0$$, such that $$A^{1/2}A^{1/2}=A$$.
 * For a positive definite matrix $$A>0$$ and invertible $$M$$, $$M^TAM>0$$.
 * If $$A>0$$ and $$M>0$$, then $$A+M>0$$.
 * If $$A>0$$ then $$\mu A>0$$ for any scalar $$\mu>0$$.

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


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LMIs in Control/Linear matrix inequalities and control theory/pages/Basic Matrix Theory

Basic Matrix Notation
Consider the complex matrix $$A\in \C^{n\times m} $$.
 * $$ A=\begin{bmatrix} a_{11} & \dots & a_{1m} \\

\vdots & \ddots & \vdots \\ a_{n1} & \dots & a_{nm}\end{bmatrix} \in \C^{n\times m} $$

Transpose of a Matrix

The transpose of $$A$$, denoted as $$A^{T}$$ or $$A'$$ is:
 * $$ A^T=\begin{bmatrix} a_{11} & \dots & a_{n1} \\

\vdots & \ddots & \vdots \\ a_{1m} & \dots & a_{nm}\end{bmatrix} \in \C^{m\times n}. $$

Adjoint of a Matrix

The adjoint or hermitian conjugate of $$A$$, denoted as $$A^{*}$$ is:
 * $$ A^*=\begin{bmatrix} a_{11}^* & \dots & a_{n1} ^*\\

\vdots & \ddots & \vdots \\ a_{1m}^* & \dots & a_{nm}^*\end{bmatrix} \in \C^{m\times n}. $$ Where $$a_{nm}^*$$ is the complex conjugate of matrix element $$a_{nm}$$.

Notice that for a real matrix $$A\in\R^{n\times m}$$, $$A^*=A^T$$.

Important Properties of Matricies
Hermitian, Self-Adjoint, and Symmetric Matricies

A square matrix $$A\in \C^{n\times n} $$ is called Hermitian or self-adjoint if $$A=A^*$$.

If $$A\in \R^{n\times n} $$ is Hermitian then it is called symmetric.

Unitary Matricies

A square matrix $$A\in \C^{n\times n} $$ is called unitary if $$A^*=A^{-1}$$ or $$A^*A=I$$.


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Linear matrix inequalities and control theory/pages/Notion of Matrix Positivity

Notation of Positivity
A symmetric matrix $$A\in\R^{n\times n}$$ is defined to be:

positive semidefinite, $$(A\ge 0)$$, if $$x^TAx\ge 0 $$ for all $$x\in\R^n, x\neq \mathbf{0} $$.

positive definite, $$(A>0)$$, if $$x^TAx> 0 $$ for all $$x\in\R^n, x\neq \mathbf{0} $$.

negative semidefinite, $$(-A\ge 0)$$.

negative definite, $$(-A>0)$$.

indefinite if $$A$$ is neither positive semidefinite nor negative semidefinite.

Properties of Positive Matricies

 * For any matrix $$M$$, $$M^TM>0$$.
 * Positive definite matricies are invertible and the inverse is also positive definite.
 * A positive definite matrix $$A>0$$ has a square root, $$A^{1/2}>0$$, such that $$A^{1/2}A^{1/2}=A$$.
 * For a positive definite matrix $$A>0$$ and invertible $$M$$, $$M^TAM>0$$.
 * If $$A>0$$ and $$M>0$$, then $$A+M>0$$.
 * If $$A>0$$ then $$\mu A>0$$ for any scalar $$\mu>0$$.


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LMIs in Control/Linear matrix inequalities and control theory/pages/Fundamentals of Matrix and LMIs/Fundamentals of Matrix and LMI/Basic Matrix Theory

Basic Matrix Notation
Consider the complex matrix $$A\in \C^{n\times m} $$.
 * $$ A=\begin{bmatrix} a_{11} & \dots & a_{1m} \\

\vdots & \ddots & \vdots \\ a_{n1} & \dots & a_{nm}\end{bmatrix} \in \C^{n\times m} $$

Transpose of a Matrix

The transpose of $$A$$, denoted as $$A^{T}$$ or $$A'$$ is:
 * $$ A^T=\begin{bmatrix} a_{11} & \dots & a_{n1} \\

\vdots & \ddots & \vdots \\ a_{1m} & \dots & a_{nm}\end{bmatrix} \in \C^{m\times n}. $$

Adjoint of a Matrix

The adjoint or hermitian conjugate of $$A$$, denoted as $$A^{*}$$ is:
 * $$ A^*=\begin{bmatrix} a_{11}^* & \dots & a_{n1} ^*\\

\vdots & \ddots & \vdots \\ a_{1m}^* & \dots & a_{nm}^*\end{bmatrix} \in \C^{m\times n}. $$ Where $$a_{nm}^*$$ is the complex conjugate of matrix element $$a_{nm}$$.

Notice that for a real matrix $$A\in\R^{n\times m}$$, $$A^*=A^T$$.

Important Properties of Matricies
Hermitian, Self-Adjoint, and Symmetric Matricies

A square matrix $$A\in \C^{n\times n} $$ is called Hermitian or self-adjoint if $$A=A^*$$.

If $$A\in \R^{n\times n} $$ is Hermitian then it is called symmetric.

Unitary Matricies

A square matrix $$A\in \C^{n\times n} $$ is called unitary if $$A^*=A^{-1}$$ or $$A^*A=I$$.


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs

Fundamentals of Matrix and LMIs

Fundamentals of Matrix and LMIs Properties of LMIs
 * 1) /Basic Matrix Theory/
 * 2) /Notion of Matrix Positivity/
 * 3) /Matrix Inequalities and LMIs/
 * 4) /Convexity of LMIs/
 * 5) /Concatenation of LMIs/
 * 1) /Change of Subject/
 * 2) /Congruence Transformation/
 * 3) /Young’s Relation (Completion of the Squares)/
 * 4) /Young’s Relation-Based Properties/
 * 5) /Special Cases of Young’s Relation/
 * 6) /Iterative Convex Overbounding/
 * 7) /Projection Lemma (Matrix Elimination Lemma)/
 * 8) /Strict Projection Lemma/
 * 9) /Non Strict Projection Lemma/
 * 10) /Reciprocal Projection Lemma/
 * 11) /Projection Lemma-Based Properties/
 * 12) /Ellipsoidal inequality/
 * 13) Continuous Time Properties
 * 14) /Schur Complement/
 * 15) /Strict Schur Complement/
 * 16) /Non-Strict Schur Complement/
 * 17) /Schur Complement Lemma-Based Properties/
 * 18) Eigenvalue related Problems
 * 19) /Matrix Eigen Value Minimization/
 * 20) /Continuous Time/Eigenvalue Problem/
 * 21) /LMI for Generalized eigenvalue problem/
 * 22) /Matrix Norm Minimization/
 * 23) /LMI for Generalized Eigenvalue Problem/
 * 24) /LMI for Linear Programming/
 * 25) /LMI for Feasibility Problem/
 * 26) /Continuous Time/Structured Singular Value/
 * 27) /LMI for Minimizing Condition Number of Positive Definite Matrix/
 * 28) /Continuous time quadratic stability/
 * 29) Discrete Time Properties
 * 30) /Discrete Time Minimum Gain Lemma/
 * 31) /Discrete Time Modified Minimum Gain Lemma/
 * 32) /Finsler’s Lemma/
 * 33) /Dilation/
 * 34) /Tangential Nevanlinna Pick/
 * 35) /Nevanlinna Pick Interpolation with Scaling/
 * 36) /Deduced LMI Conditions for Hinf Index/
 * 37) /Deduced LMI Conditions for H2 Index/
 * 38) /Dissipativity of Systems/
 * 39) /Generalized H2 Norm/
 * 40) /Passivity and Positive Realness/
 * 41) /Non-Expansivity and boundedness Realness/
 * 42) /Variable Reduction Lemma/
 * 43) /D-stability Rise Poles/
 * 44) /D-stability Settling time poles/
 * 45) /D-stability Max Percent Overshoot Poles/
 * 46) /Iterative Convex Overbounding/
 * 47) /Frobenius Norm/
 * 48) /Dualization Lemma/


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LMIs in Control/pages/Fundamentals of Matrix and LMIs/Notion of Matrix Positivity

Notation of Positivity
A symmetric matrix $$A\in\R^{n\times n}$$ is defined to be:

positive semidefinite, $$(A\ge 0)$$, if $$x^TAx\ge 0 $$ for all $$x\in\R^n, x\neq \mathbf{0} $$.

positive definite, $$(A>0)$$, if $$x^TAx> 0 $$ for all $$x\in\R^n, x\neq \mathbf{0} $$.

negative semidefinite, $$(-A\ge 0)$$.

negative definite, $$(-A>0)$$.

indefinite if $$A$$ is neither positive semidefinite nor negative semidefinite.

Properties of Positive Matricies

 * For any matrix $$M$$, $$M^TM>0$$.
 * Positive definite matricies are invertible and the inverse is also positive definite.
 * A positive definite matrix $$A>0$$ has a square root, $$A^{1/2}>0$$, such that $$A^{1/2}A^{1/2}=A$$.
 * For a positive definite matrix $$A>0$$ and invertible $$M$$, $$M^TAM>0$$.
 * If $$A>0$$ and $$M>0$$, then $$A+M>0$$.
 * If $$A>0$$ then $$\mu A>0$$ for any scalar $$\mu>0$$.


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Basic Matrix Theory

Basic Matrix Notation
Consider the complex matrix $$A\in \C^{n\times m} $$.
 * $$ A=\begin{bmatrix} a_{11} & \dots & a_{1m} \\

\vdots & \ddots & \vdots \\ a_{n1} & \dots & a_{nm}\end{bmatrix} \in \C^{n\times m} $$

Transpose of a Matrix

The transpose of $$A$$, denoted as $$A^{T}$$ or $$A'$$ is:
 * $$ A^T=\begin{bmatrix} a_{11} & \dots & a_{n1} \\

\vdots & \ddots & \vdots \\ a_{1m} & \dots & a_{nm}\end{bmatrix} \in \C^{m\times n}. $$

Adjoint of a Matrix

The adjoint or hermitian conjugate of $$A$$, denoted as $$A^{*}$$ is:
 * $$ A^*=\begin{bmatrix} a_{11}^* & \dots & a_{n1} ^*\\

\vdots & \ddots & \vdots \\ a_{1m}^* & \dots & a_{nm}^*\end{bmatrix} \in \C^{m\times n}. $$ Where $$a_{nm}^*$$ is the complex conjugate of matrix element $$a_{nm}$$.

Notice that for a real matrix $$A\in\R^{n\times m}$$, $$A^*=A^T$$.

Important Properties of Matricies
Hermitian, Self-Adjoint, and Symmetric Matricies

A square matrix $$A\in \C^{n\times n} $$ is called Hermitian or self-adjoint if $$A=A^*$$.

If $$A\in \R^{n\times n} $$ is Hermitian then it is called symmetric.

Unitary Matricies

A square matrix $$A\in \C^{n\times n} $$ is called unitary if $$A^*=A^{-1}$$ or $$A^*A=I$$.


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Notion of Matrix Positivity

Notation of Positivity
A symmetric matrix $$A\in\R^{n\times n}$$ is defined to be:

positive semidefinite, $$(A\ge 0)$$, if $$x^TAx\ge 0 $$ for all $$x\in\R^n, x\neq \mathbf{0} $$.

positive definite, $$(A>0)$$, if $$x^TAx> 0 $$ for all $$x\in\R^n, x\neq \mathbf{0} $$.

negative semidefinite, $$(-A\ge 0)$$.

negative definite, $$(-A>0)$$.

indefinite if $$A$$ is neither positive semidefinite nor negative semidefinite.

Properties of Positive Matricies

 * For any matrix $$M$$, $$M^TM>0$$.
 * Positive definite matricies are invertible and the inverse is also positive definite.
 * A positive definite matrix $$A>0$$ has a square root, $$A^{1/2}>0$$, such that $$A^{1/2}A^{1/2}=A$$.
 * For a positive definite matrix $$A>0$$ and invertible $$M$$, $$M^TAM>0$$.
 * If $$A>0$$ and $$M>0$$, then $$A+M>0$$.
 * If $$A>0$$ then $$\mu A>0$$ for any scalar $$\mu>0$$.


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Matrix Inequalities and LMIs

Matrix Inequality
Definition-1

A Matrix Inequality, $$G : \mathbb{R}^{m} \to \mathbb{S}^{n}$$, in the variable $$x\in\mathbb{R}^{m}$$ is an expression of the form

 * $$\begin{align}

G(x)=G_0+\sum_{i=1}^p f_i(x)G_i\leq0 \end{align}$$,

where $$x^T=[x_1\cdots x_m], G_0\in\mathbb{S}^{n}$$ and $$ G_i \in\mathbb{R}^{n \times n}$$, $$i=1,\ldots,p.$$

Linear Matrix Inequality
Definition-2

A Linear Matrix Inequality, $$F: \mathbb{R}^{m}\to\mathbb{S}^{n}$$, in the variable $$x\in\mathbb{R}^{m}$$ is an expression of the form
 * $$\begin{align}

F(x)=F_0+\sum_{i=1}^mx_iF_i\leq0 \end{align}$$,

where $$x^T=[x_1\ldots x_m]$$ and $$F_i\in\mathbb{S}^{n}$$, $$i=0\ldots, m.$$

Bilinear Matrix Inequality
Definition-3

A Bilinear Matrix Inequality (BMI), $$H:\mathbb{R}^{m}\to \mathbb{S}^{n}$$, in the variable $$x\in \mathbb{R}^{m}$$ is an expression of the form

 * $$\begin{align}

H(x)=H_0+\sum_{i=1}^mx_iH_i+\sum_{i=1}^m\sum_{j=1}^mx_ix_jH_{i,j}\leq0, \end{align}$$

where $$x^T=[x_1\cdots x_m]$$, and $$H_i$$, $$H_{i,j}\in\mathbb{S}^{n},$$ $$i=0,\ldots ,m$$, $$j=0 \ldots,m.$$

Example
Consider the matrices $$A\in\mathbb{R}^{n\times n}$$ and $$Q\in\mathbb{S}^{n}$$, where $$Q>0$$. It is desired to find a symmetric matrix $$P\in\mathbb{S}^{n}$$ satisfying the inequality

 * $$\begin{align}

PA+A^TP+Q<0,   \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad \qquad(1) \end{align}$$

where $$P>0$$. The elements of $$P$$ are the design variables in this problem, and although equation $$(1)$$ is indeed an LMI in the matrix $$P$$, it does not look like the LMI in definition 3. For simplicity, let us consider the case of $$n=2$$ so that each matrix is of dimension $$2 \times 2$$, and $$x=[p_1\quad p_2 \quad p_3]^T.$$ Writing the matrix $$P$$ in terms of a basis $$E_i\in\mathbb{S}^{2},$$ $$i=1,2,3$$, yields

 * $$\begin{align}

P=\begin{bmatrix}p_1&p_2\\p_2&p_3\end{bmatrix} = p_1\underbrace{\begin{bmatrix}1&0\\0&0\end{bmatrix}}_{E_1}+p_2\underbrace{\begin{bmatrix}0&1\\1&0\end{bmatrix}}_{E_2}+p_3\underbrace{\begin{bmatrix}0&0\\0&1\end{bmatrix}}_{E_3} \end{align}$$

Note that the matrices $$E_i$$ are linearly independent and symmetric, thus forming a basis for the symmetric matrix $$P$$. The matrix inequality in equation $$(1)$$ can be written as

 * $$\begin{align}

p_1(E_1A+A^TE_1)+p_2(E_2A+A^TE_2)+p_3(E_3A+A^TE_3). \end{align}$$

Defining $$F_0=Q$$ and $$F_i=E_iA+A^TE_i,$$ $$i=1,2,3,$$ yields

 * $$\begin{align}

F_0+\sum{i=1}^3p_iF_i<0, \end{align}$$

which now resembles the definition of LMI given in definition 2. Through out this wiki book, LMIs are typically written in the matrix form of equation $$(1)$$ rather than the scalar form of definition 2.


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Convexity of LMIs

Definition
A set, $$\mathcal{S}$$, in a real inner product space is convex if for all $$x, y\in\mathcal{S}$$ and $$\alpha\in\mathbb{R}$$, where $$0\leq\alpha\leq1$$, it holds that $$\alpha x+(1-\alpha)y\in\mathcal{S}$$.

Lemma 1.1
The set of solutions to an LMI is convex.

That is, the set $$\mathcal{S}=\{x\in\mathbb{R}^{m}\mid F(x)\leq0\}$$ is a convex set, where $$F:\mathbb{R}^{m}\longrightarrow\mathbb{S}^{n}$$ is an LMI.

Lemma 1.2
An LMI, $$F:\mathbb{R}^{m}\longrightarrow\mathbb{S}^{n}$$, in the variable $$x\in\mathbb{R}^{m}$$ is an expression of the form

$$F(x)=F_{0}+\sum_{i=1}^m x_{i}F_{i}\leq0$$

where $$x^{T}=[x_{1}\cdots x_{m}]$$ and $$F_{i}\in\mathbb{S}^{n}$$, $$i=0,\ldots,m$$.

Proof
Consider $$x,y\in\mathbb{R}^{m}$$ and $$\alpha\in[0,1]$$, and suppose that $$x$$ and $$y$$ satisfy Lemma 1.2.

The LMI $$F:\mathbb{R}^{m}\longrightarrow\mathbb{S}^{n}$$ is convex, since

$$F(\alpha x+(1-\alpha)y)=F_{0}+\sum_{i=1}^m (\alpha x_{i}+(1-\alpha)y_{i})F_{i}$$

$$\begin{alignat}{2} F(\alpha x+(1-\alpha)y) & = F_{0}+\sum_{i=1}^m (\alpha x_{i}+(1-\alpha)y_{i})F_{i}

\\ & = F_{0}-\alpha F_{0}+\alpha F_{0}+\alpha\sum_{i=1}^m x_{i}F_{1}+(1-\alpha)\sum_{i=1}^m y_{i}F_{i} \\ & = \alpha F_{0}+\alpha\sum_{i=1}^m x_{i}F_{i}+(1-\alpha)F_{0}+(1-\alpha)\sum_{i=1}^m y_{i}F_{i} \\ & = \alpha F(x)+(1-\alpha)F(y) \\ \end{alignat}$$

Convexity of LMI
From Lemma 1.1, it is known that an optimization problem with a convex objective function and LMI constraints is convex.

The following is a non-exhaustive list of scalar convex objective functions involving matrix variables that can be minimized in conjunction with LMI constraints to yield a semi-definite programming (SDP) problem.

 * $$\mathcal{J}(x)=\frac{1}{2}x^{T}PX+q^{T}x+r$$, where $$x,q\in\mathbb{R}^{n}$$, $$P\in\mathbb{S}^{n}$$, $$P>0$$, and $$r\in\mathbb{R}$$.

 * 1) Special case when $$q=0$$ and $$r=0:\mathcal{J}(x)=\frac{1}{2}x^{T}Px$$, where$$x\in\mathbb{R}^{n}$$, $$P\in\mathbb{S}^{n}$$, and $$P>0$$.
 * 2) Special case when $$P=$$2$$\cdot1$$, $$q=0$$, and $$r=0:\mathcal{J}(x)=x^{T}x=\|x\|^{2}_{2}$$, where $$x\in\mathbb{R}^{n}$$.

 * $$\mathcal{J}(X)=tr(X^{T}PX+Q^{T}X+X^{T}R+S)$$, where $$X$$, $$Q$$, $$R\in\mathbb{R}^{n \times m}$$, $$P\in\mathbb{S}^{n}$$, $$S\in\mathbb{R}^{n \times n}$$, and $$P\geq0$$.

 * 1) Special case when $$Q=R=0$$ and $$S=0:\mathcal{J}(X)=tr(X^{T}PX)$$, where $$X\in\mathbb{R}^{n \times m}$$, $$P\in\mathbb{S}^{n}$$, and $$P\geq0$$.
 * 2) Special case when $$P=1$$, $$Q=R=0$$ and $$S=0:\mathcal{J}(X)=tr(X^{T}X)=\|X\|^{2}_{F}$$, where $$X\in\mathbb{R}^{n \times m}$$.
 * 3) Special case when $$P=0$$, $$R=0$$ and $$S=0:\mathcal{J}(X)=tr(Q^{T}X)$$, where $$X$$, $$Q\in\mathbb{R}^{n \times m}$$.
 * 4) Special case when $$P=1$$, $$Q=R=0$$, $$S=0$$, and $$X\in\mathbb{S}^{n}:\mathcal{J}(X)=tr(X^{2})$$, where $$X\in\mathbb{S}^{n}$$.

 * $$\mathcal{J}(X)=\log(\det(X^{ -1}))=\log(\det(X))$$, where $$X\in\mathbb{S}^{n}$$ and $$X>0$$.

Relative Definition of a Matrix
The definiteness of a matrix can be found relative to another matrix.

For example,

Consider the matrices $$A\in\mathbb{S}^{n}$$ and $$B\in\mathbb{S}^{n}$$. The matrix inequality $$A0$$, when we know that $$B>0$$.

This follows from $$0<A<B$$.


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Concatenation of LMIs

Introduction
Matrix concatenation is the process of joining one or more matrices to make a new matrix. This process is similar for concatenating the LMIs as well.

Concatenation of LMIs
A useful property of LMIs is that multiple LMIs can be concatenated together to form a single LMI. For example, satisfying the LMIs $$ A < 0$$ and, $$B < 0$$ is equivalent to satisfying the concatenated LMI:

$$ \begin{bmatrix} A && 0 \\ 0 && B \end{bmatrix}$$ $$\begin{align} <0 \end{align}$$,

More generally, satisfying the LMIs $$Ai < 0$$, where $$i = 1,. . ., n $$ is equivalent to satisfying the concatenated LMI:

$$ \begin{bmatrix} A && 0 && 0 &&. && . && 0 \\ 0 && B && 0 &&. && . && 0 \\ 0 && 0 && C &&. && . && 0 \\ . && . && . && . && . && 0\\ . && . && . && . && . && 0 \\ 0 && 0 && 0 && 0 && 0 && A_n\\ \end{bmatrix}$$ $$\begin{align} <0 \end{align}$$,


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Change of Subject

LMIs in Control/Matrix and LMI Properties and Tools/Change of Subject

A Bilinear Matrix Inequality (BMI) can sometimes be converted into a Linear Matrix Inequality (LMI) using a change of variables. This is a basic mathematical technique of changing the position of variables with respect to equal signs and the inequality operators. The change of subject will be demonstrated by the example below.

Example
Consider $$A \in \mathbb{R}^{n\times n}, B \in \mathbb{R}^{n\times m},K \in \mathbb{R}^{m\times n}$$, and $$Q \in \mathbb{S}^{n}$$, where $$Q > 0$$.

The matrix inequality given by:

$$QA^{T} + AQ - QK^{T}B{T} - BKQ < 0 $$ is bilinear in the variables $$Q$$ and $$K$$.

Defining a change of variable as $$ F = KQ $$ to obtain

$$QA^{T} + AQ + - F^{T} B^{T} - BF < 0$$,

which is an LMI in the variables $$Q$$ and $$F$$.

Once this LMI is solved, the original variable can be recovered by $$ K = FQ^{-1}$$.

Conclusion
It is important that a change of variables is chosen to be a one-to-one mapping in order for the new matrix inequality to be equivalent to the original matrix inequality. The change of variable $$F = KQ$$ from the above example is a one-to-one mapping since $$Q^{-1}$$ is invertible, which gives a unique solution for the reverse change of variable $$ K = FQ^{-1}$$.


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Congruence Transformation

LMIs in Control/Matrix and LMI Properties and Tools/Congruence Transformation

This methods uses change of variable and some matrix properties to transform Bilinear Matrix Inequalities to Linear Matrix Inequalities. This method preserves the definiteness of the matrices that undergo the transformation.

Theorem
Consider $$ Q \in \mathbb{S}^{n}, W \in \mathbb{R}^{n\times n}$$, where $$\operatorname{rank}(W) = n$$. The matrix inequality $$Q < 0$$ is satisfied if and only if $$WQW^{T} < 0$$ or equivalently, $$W^{T}QW < 0$$.

Example
Consider $$ A \in \mathbb{R}^{n\times n}, B \in \mathbb{R}^{n\times m}, K \in \mathbb{R}^{m\times p}, C^{T} \in \mathbb{R}^{n\times p}, P \in \mathbb{S}^{n}$$ and $$ V \in \mathbb{S}^{p} $$, where $$P > 0$$ and $$V > 0 $$. The matrix inequality given by

$$

\begin{align} \ Q &= \begin{bmatrix} \ A^{T}P + PA & -PBK + C^{T}V \\ \ * & -2V \\ \end{bmatrix} \ < 0, \end{align} $$

is linear in variable $$V$$ and bilinear in the variable pair $$(P.K)$$. Choose the matrix $$\operatorname{diag}(P^{-1},V^{-1})$$ to obtain the equivalent BMI given by

$$

\begin{align} \ WQW^{T} &= \begin{bmatrix} \ P^{-1}A^{T} + AP^{-1} & -BKV^{-1} + P^{-1}C^{T} \\ \ * & -2V^{-1} \\ \end{bmatrix} \ < 0, \end{align} $$

Using a change of variable $$X = P^{-1}, U = V^{-1}$$ and $$F = KV^{-1}$$, the above equation becomes

$$

\begin{align} \ WQW^{T} &= \begin{bmatrix} \ XA^{T} + AX & -BF + XC^{T} \\ \ * & -2U \\ \end{bmatrix} \ < 0, \end{align} $$

which is an LMI of variables $$X, U$$ and $$F$$. The original variable $$K$$ is recovered by doing a reverse change of variable $$K = FU^{-1}$$.

Conclusion
A congruence transformation preserves the definiteness of a matrix by ensuring that $$Q < 0$$ and $$W^{T}QW < 0$$ are equivalent. A congruence transformation is related, but not equivalent to a similarity transformation $$TQT^{-1} $$, which preserves not only the definiteness, but also the eigenvalues of a matrix. A congruence transformation is equivalent to a similarity transformation in the special case when $$W^{T} = W^{-1} $$.


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Young’s Relation (Completion of the Squares)

This method is used to solve quadratic equations that can't be factorized.

Matrix inequality
Consider $$ X,Y \in \mathbb{R}^{n\times m}$$ and $$ S \in \mathbb{S}^{n\times n}$$ , where $$ S $$ >0, The matrix inequality given by

\begin{align} \ X^{T}Y + Y^{T}X \leq X^{T}S^{-1}X + Y^{T}SY,\\ \end{align}$$ which is named Young’s relation or Young’s inequality.

Derivation
Young’s relation can be derived from a completion of the squares as follows.

\begin{align} 0 \leq (X-SY)^{T}S^{-1}(X-SY) \\ 0 \leq X^{T}S^{-1}X + Y^{T}SY - X^{T}Y - Y^{T}X\\ X^{T}Y + Y^{T}X \leq X^{T}S^{-1}X + Y^{T}SY, \end{align}$$ which is named Young’s relation.

Reformulation of Young’s Relation
Consider $$ X,Y \in \mathbb{R}^{n\times m}$$ and $$ S \in \mathbb{S}^{n\times n}$$ , where $$ S $$ >0, The matrix inequality given by

\begin{align} \ X^{T}Y + Y^{T}X \leq \frac12(X+SY)^{T}S^{-1}(X+SY) ,\\ \end{align}$$ is a reformulation of Young’s relation.


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Young’s Relation-Based Properties

Young’s Relation-Based Properties
1. Consider $$X$$,$$Y\in\mathbb{R}^{n\times m}$$ and $$Z\in\mathbb{S}^m$$. The matrix inequality given by

 * $$\begin{align}

Z + XTY + YTX>0, \end{align}$$

 * is satisfied if and only if there exist $$Q\in \mathbb{S}^m$$, $$P\in\mathbb{S}^n$$,$$ G1\in\mathbb{R}^{n\times n}$$,$$

G2\in\mathbb{R}^{n\times m}$$,$$ F\in\mathbb{R}^{m\times n}$$, and $$H\in \mathbb{R}^{m\times m}$$, where $$Q > 0$$

 * and $$P > 0$$, such that

\begin{bmatrix}P&Y\\*&Q\end{bmatrix}>0 $$ and $$\begin{bmatrix}Z+Q+X^TPX & F-X^TG_2&H-X^TG_1\\*&G_1+G_1^T-P&F^T+G_2-Y\\*&*&H^T+H-Q\end{bmatrix} $$ 2. Consider $$X$$,$$Y\in\mathbb{R}^{n\times n}$$ and $$W\in\mathbb{S}^m$$, where $$X$$ is full rank and $$W>0$$. The matrix inequality given by

 * $$\begin{align}

X^T-W>0, \end{align}$$

 * is satisfied if there exist $$\lambda\in \mathbb{R}_{>0}$$ such that

\begin{bmatrix}\lambda\mathbf{1}&\lambda\mathbf{1}&\mathbf{0}\\*&\mathbf{X}+\mathbf{X}^T&\mathbf{W}^{\frac{1}{2}}\\*&*&\lambda\mathbf{1}\end{bmatrix}>0 $$


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Iterative Convex Overbounding

Iterative convex overbounding is a technique based on Young’s relation that is useful when solving an optimization problem with a BMI constraint.

The System
Consider the matrices $$ Q=Q^{T}\in \mathbb{R}^{n\times n}, B \in \mathbb{R}^{n\times m}, R \in \mathbb{R}^{m\times p}, D \in \mathbb{R}^{p\times q},S \in \mathbb{R}^{q\times r} and C \in \mathbb{R}^{r\times n}$$, where S and R are design variables in the BMI given by

\begin{align} \qquad Q + BRDSC + C^{T}S^{T}D^{T}R^{T}B^{T} < 0      \qquad    (1)\\ \end{align}                              $$

The LMI:Iterative Convex Overbounding
Suppose that S0 and R0 are known to satisfy (1). The BMI of (1) is implied by the LMI
 * $$\begin{bmatrix}

Q+\phi(R,S)+\phi^{\text{T}}(R,S) & B(R-R0)U &C^{\text{T}}(S-S0)^{\text{T}}V^{\text{T}}\\ \end{bmatrix} < 0 \qquad   (2) $$ where Φ(R,S) = B(RDS0+R0DS-R0DS0)C, W>0 is an arbitrary matrix, D=UV, and the matrices U and VT have full column rank. The LMI of (2) is equivalent to the BMI of (1) when R = R0 and S = S0, and is therefore non-conservative for values of R and S and are close to the previously known solutions R0 and S0.
 * &W^{\text{-1}}&0\\
 * &*&-W\\

Alternatively, the BMI of (1) is implied by the LMI
 * $$\begin{bmatrix}

Q+\phi(R,S)+\phi^{\text{T}}(R,S) &Z^{\text{T}}U^{\text{T}}(R-R0)^{\text{T}}B^{\text{T}}+V(S-S0)C\\
 * &Z\\

\end{bmatrix} < 0 \qquad   (3) $$ where Z > 0 is an arbitrary matrix, D = UV, and the matrices U and VT have full column rank. Again, the LMI of (4) is equivalent to the BMI of (2) when R = R0 and S = S0, and is therefore non-conservative for values of R and S and are close to the previously known solutions R0 and S0.

Conclusion:
A benefit of convex overbounding compared to a linearization approach, is that in addition to ensuring conservatism or error is reduced in the neighborhood of R = R0 and S = S0, the LMIs of (2) and (3) imply (1). Iterative convex overbounding is particularly useful when used to solve an optimization problem with BMI constraints.


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Projection Lemma (Matrix Elimination Lemma)

WIP, Description in progress

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

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

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

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

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

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

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

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

if and only if

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

WIP, additional references to be added


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Strict Projection Lemma

Introduction
Projection Lemma is also known as Matrix Elimination Lemma. Strict Projection Lemma is one of the characteristics of the Projection Lemma.

Strict Projection Lemma
Consider $$\Psi\in \mathbb{S}^n$$, $$ G\in \mathbb{R}^{n\times m}$$, $$\Lambda\in \mathbb{R}^{m\times p}$$, and $$ H\in \mathbb{R}^{n\times p}.$$ There exists $$\Lambda $$ such that:

$$ \begin{align} \ \Psi + G \Lambda H^T + H \Lambda^T G^T<0, \end{align} $$

if and only if,

$$ \begin{align} \ N_G^T \Psi N_G <0 \end{align} $$,

$$ \begin{align} \ N_H^T \Psi N_H <0 \end{align} $$,

where $$\mathcal{R} (N_G) =\mathcal{N} (G^T)$$ and $$\mathcal{R} (N_H) = \mathcal{N} (H^T)$$.


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Non Strict Projection Lemma

Introduction
Projection Lemma is also known as Matrix Elimination Lemma. Strict Projection Lemma is one of the characteristics of the Projection Lemma.

Non Strict Projection Lemma
Consider $$\Psi\in \mathbb{S}^n$$, $$ G\in \mathbb{R}^{n\times m}$$, $$\Lambda\in \mathbb{R}^{m\times p}$$, and $$ H\in \mathbb{R}^{n\times p}.$$, where $$ \mathcal{R} (G)$$ and $$ \mathcal{R} (H)$$ is linear;y independent. There exists $$\Lambda $$ such that:

$$ \begin{align} \ \Psi + G \Lambda H^T + H \Lambda^T G^Tleq0, \end{align} $$

if and only if,

$$ \begin{align} \ N_G^T \Psi N_G \leq0 \end{align} $$,

$$ \begin{align} \ N_H^T \Psi N_H \leq0 \end{align} $$,

where $$\mathcal{R} (N_G) =\mathcal{N} (G^T)$$ and $$\mathcal{R} (N_H) = \mathcal{N} (H^T)$$.


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Reciprocal Projection Lemma

WIP, Description in progress

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

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

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

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

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

WIP, additional references to be added


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Ellipsoidal inequality

Introduction
The simplex algorithm was the first algorithm proposed for linear programming, and although the algorithm is quite fast in practice, no variant of it is known to be polynomial time. The Ellipsoid algorithm is the first polynomial-time algorithm discovered for linear programming. The Ellipsoid algorithm was proposed by the Russian mathematician Shor in 1977 for general convex optimization problems and applied to linear programming by Khachyan in 1979.

The Ellipsoidal inequality constraints are a type of Lyapunov Function. They are important in process identification, parameter estimation, and statistics. Applications include the crystallization process, polymer film extrudes, and paper machines.

The Data
Formulation of the inequality depends upon the coordinates of the ellipsoid $$ (x_i) $$ and the centre of the ellipsoid $$ (x_c) $$.

The Ellipsoidal Inequality
An Ellipsoid is described by:

$$ \begin{align} \ (x-x_c)^{T} P^{-1} (x-x_c) < 1,\\ \ where, P=P^{T} > 0\\ \end{align} $$

It can be expressed as an LMI using the Schur complement lemma with $$ Q(x)=1, R(x)=P, and S(x)= (x-x_c)^{T} : $$ $$ \begin{bmatrix} 1 && (x-x_c)^{T}\\ (x-x_c) && P \end{bmatrix}$$ $$\begin{align} \geq 0\end{align}$$

Conclusion
Using this inequality, as the algorithm advances to the next step; if the value of P is positive; the volume will keep on shrinking until the LMI is fulfilled.

Implementation

 * https://github.com/Margav06/LMI-for-Ellipsoidal-Inequality

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


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Lentis/Chloramination of Drinking Water

Introduction
Chloramine, also known as monochloramine, is the chemical compound NH2Cl formed when ammonia is added to chlorine. It is used as a secondary disinfectant for water treatment. Secondary disinfectants keep drinking water clean as it is transported through the pipes from the treatment plant to the user, whereas primary ones remove contaminants at the plant. Chloramine is used as secondary disinfectant because it is more stable, lasts longer than free chlorine, and does not form harmful byproducts like trihalomethanes. The goal of chloramine use is promotion of public health by preventing disease causing germs from contaminating water and being ingested (e.g. salmonella and norovirus). The Environmental Protection Agency (EPA) allows chloramine use up to concentrations of 4 mg/L and studies suggest there are no health effects from drinking chloramine with concentration less than 50 mg/L.

History
Chemical water disinfection with chlorine began near the beginning of the 20th century. In 1929 chloramine use began in American cities (e.g. Cleveland, OH; Springfield, IL; and Lansing, MI). 1998 EPA studies suggested 68 million Americans were drinking water disinfected with chloramine. Major US cities including Washington DC, Philadelphia, and San Francisco began shifting to chloramine treatment. More than 20% of Americans were being served by chloramine treated water by 2009. As of 2011, analysis of EPA data found population percentages being supplied with chloramine treated drinking water, which revealed Texas and Florida leading with over 50 percent. One-third of all US public water systems use chloraminated water. Chloramine is not expected to overtake chlorine usage, but is predicted to reach 40 percent of the population in future years.

Pushback
Though chloramines have been used since the 1930’s, the transition to chloramine has not been as smooth as planned. Activists have “called for more scientific study” on the usage of chloramines, claiming many potential dangers. These issues range from health effects to environmental damages. Activists have had success in preventing the transition notably in Charlottesville with Transition Charlottesville/Albemarle, a grassroots campaign. Despite the opposition, water utilities continue to push change in order to save money and meet EPA standards.

Health Issues
According to the EPA and Centers for Disease Control and Prevention (CDC), there are no health effects of 50 mg/L of chloramine. Both organizations claim “the normal level for drinking water can range from 1.0 to 4.0 mg/L”. The EPA, however, has had no follow up studies. This remains a source of tension for critics who dispute these claims and argue for more research. Critics like Friends of Water claim “it is suggested that anyone with chemical sensitivities will react badly to it,” which can include severe skin rashes and breathing problems. Critics are also concerned with the production of toxic disinfection byproducts (DBP). Research shows “chloramine produces the same types of halogenated by-products as chlorine, but they were few in number and lower in concentration,” allowing the DBP’s to be dealt with easier.

Environmental Effects
When chloramines escape into water sources, they are toxic to aquatic life. Aquatic life absorbs water and associated disinfectants like chloramines directly into the bloodstream, which kills them. Chloramines are also “harmful to humans and animals that consume the contaminated fish or other aquatic animals”. Two chloramine escape incidents have been documented by Citizens Concerned About Chloramine (CCAC) to warn and “raise the public's level of awareness about chloramine”. Water suppliers are required to prevent escapes because of known dangers. Suppliers failing to comply have been “slapped with a $199,350 fine”.

Implementation
Switching to chloramine requires water utilities to refurbish existing distribution pipes because chloramines cause lead in pipes to become soluble, which releases lead into the water system. This issue occurred in Washington D.C. in 2000 and led to lead contaminants in the water system, resulting in a public health crisis. Many children received lifelong health issues from exposure. Many cities have made the transition before and others continue to follow. However, “Washington's experience had a chilling effect; many have put off switching disinfectants, fearing their own lead crisis”. Activists like the Environmental Working Group (EWG) often cite examples like D.C. and others to promote knowledge of effects of switching, stating “unfortunately, this situation is not unique: similar results have been reported in Greenville, North Carolina”. Suppliers must prepare for the transition beforehand by investing money to refurbish and test pipes to avoid public health issues.

Advocacy Groups
Stakeholders drive the implementation or de-escalation of chloramine use. The EPA regularly updates their page on chloramine research. Chloramine has been chosen as a secondary disinfectant option because the, “EPA and CDC believe the benefits of drinking water disinfection outweigh the potential risks from disinfection byproducts.” It is also used for its chemical stability compared to chlorine, wherein it produces fewer harmful regulated disinfection byproducts and it lasts longer than chlorine. Other groups question the use of chloramine as a whole, like Citizens Concerned About Chloramine (CCAC) who are a non profit group based in the San Francisco Bay Area. The CCAC researches and publicizes the effects of chloramination on people, animals, and the environment. Transition Charlottesville/Albemarle is another advocacy group, “dedicated to creating more resilient and self-reliant communities in and around the Charlottesville, Virginia area. They have held “teach-ins” to inform local citizens about the potential hazards of chloramine usage in Albemarle county.

Advocacy Group Goals
The EPA’s notes on chloramines comments, “both chlorine and monochloramine are effective secondary disinfectants.” but does not state any side effects or disadvantages of it’s usage. If anything, the Center for Disease Control (CDC) states chloramine creates fewer byproducts than chlorine due to it lasting longer so it won’t break down as quickly as chlorine does. The EPA, government, and local water agencies goal is to use more efficient means of water treatment, and chloramines fit that description. The CCAC’s goal is to “raise the public's level of awareness about chloramine and its health effects when used as a disinfectant in the water.” This includes publicizing the harmful effects of chloramine and the damage it does to the environment and the San Francisco Bay Area as a whole. Transition Charlottesville/Albemarle's goal is, “to inspire and support individuals, organizations and communities working toward sustainable local systems of food, goods, energy, communication, and culture.” They have done so by asking and educating the public on what chloramine is, how it can be helpful, and how it can be harmful.

Advocacy Group Differences
Each group differs in their research, methods, and distribution of information. The EPA publishes their work on their websites citing expert research or their own findings. They promote research by answering questions pertaining to chloramines through publishing documents of their research. The EPA does not list drawbacks of chloramines, just comparisons to chlorine usage. The CCAC polarizes the use of chloramines through publicizing research and promoting the risks of it’s usage. This is done via fliers, photographs of rashes, and eye catching article tabs to discuss chloramine’s worrying effects. Transition Charlottesville/Albemarle pursues public advocacy as a path towards impacting chloramine usage. They do it by holding forums with researchers, concerned citizens, and local boards during their discussions of the impacts of chloramine use in Albemarle county.

Conclusion
Chloramine is an example of a technical development that was put into practice before its impacts were fully understood. As a result of changing EPA standards for drinking water disinfection research was expanded beyond chlorine to find a substance that would have desired impacts and be safe. The problem is solved from the technical side by chloramination for secondary disinfection, but little attention was paid to the downstream impacts. With little conclusive research on chloramine's impacts on public health, the environment, and pipe degradation advocacy groups have raised concerns over its widespread adoption. Further exploration into issues with chloramine could focus on the alternatives water treatment plants could take (e.g. replacing pipes), the reason for lacking scientific research, and additional motivations of social groups resisting chloramine adoption.

title: Lentis/Chloramination of Drinking Water


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Schur Complement

An important tool for proving many LMI theorems is the Schur Compliment. It is frequently used as a method of LMI linearization.

The Schur Compliment
Consider the matricies $$Q$$, $$M$$, and $$R$$ where $$Q$$ and $$M$$ are self-adjoint. Then the following statements are equivalent:
 * $$Q>0$$ and $$M-RQ^{-1}R^*>0$$ both hold.
 * $$M>0$$ and $$Q-R^*M^{-1}R>0$$ both hold.
 * $$\begin{bmatrix} M & R \\

R^* & Q\end{bmatrix} > 0 $$ is satisfied.

More concisely:
 * $$\begin{bmatrix} M & R \\ R^* & Q\end{bmatrix} > 0 \iff \begin{bmatrix} M & 0 \\ 0 & Q-R^*M^{-1}R\end{bmatrix} > 0 \iff \begin{bmatrix} M-RQ^{-1}R^* & 0 \\ 0 & Q\end{bmatrix} > 0

$$


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Strict Schur Complement

Strict Schur Complement
Consider $$A \in \mathbb{S}^n, B \in \mathbb{R}^{n \times m},$$ and $$C \in \mathbb{S}^m.$$ The following statements are equivalent.

a) $$\begin{bmatrix}A &B\\ B^T& C \end{bmatrix}< 0$$

b) $$A-BC^{-1}B^T<0,C<0$$

c) $$C-B^TA^{-1}B<0,A<0$$


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Non-Strict Schur Complement

Nonstrict Schur Complement
Consider $$A \in \mathbb{S}^n, B \in \mathbb{R}^{n \times m},$$ and $$C \in \mathbb{S}^m.$$ The following statements are equivalent.

a) $$\begin{bmatrix}A &B\\ B^T& C \end{bmatrix} \leq 0$$

b) $$A-BC^{+}B^T <0,C \leq 0, B(\mathbf{1}-CC^+)=\mathbf{0},$$ where $$C^+$$ is the Moore-Penrose inverse of $$C.$$

c) $$C-B^TA^{+}B<0,A \leq 0,B^T(\mathbf{1}-AA^+)=\mathbf{0}$$ where $$A^+$$ is the Moore-Penrose inverse of $$A.$$


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Continuous Time/Eigenvalue Problem

The maximum eigenvalue of a matrix is going to have the most impact on system performance. This LMI allows for minimization of the maximum eigenvalue by minimizing $$ \gamma $$.

The System

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

The Data
The matrices $$ A \in R^{n \times n}, B \in R^{n \times m}, C \in R^{o \times n} $$.

The Optimization Problem
$$ \text{ Minimize } \gamma \text{ subject to the LMI below.} $$

The LMI:

\begin{align} \text{Find} \; &P>0:\\ \begin{bmatrix} -A^TP-PA-C^TC & PB \\ B^TP & \gamma I\end{bmatrix} > 0\\ \end{align}$$

Conclusion:
The eigenvalue problem can be utilized to minimize the maximum eigenvalue of a matrix that depends affinely on a variable.

Implementation
https://github.com/mcavorsi/LMI

Related LMIs
Structured Singular Value


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/LMI for Generalized eigenvalue problem

LMI for Generalized Eigenvalue Problem

Technically, the generalized eigenvalue problem considers two matrices, like $$A$$ and $$B$$, to find the generalized eigenvector, $$x$$, and eigenvalues, $$\lambda $$, that satisfies $$ Ax = \lambda Bx $$. If the matrix $$ B $$ is an identity matrix with the proper dimension, the generalized eigenvalue problem is reduced to the eigenvalue problem.

The System
Assume that we have three matrice functions which are functions of variables $$ x = [x_{1} \quad x_{2} \quad ... \quad x_{n}]^{\text{T}} \in \mathbb{R}^{n} $$ as follows:

$$ A(x) = A_{0}+A_{1}x_{1}+ ... + A_{n}x_{n} $$

$$ B(x) = B_{0}+B_{1}x_{1}+ ... + B_{n}x_{n} $$

$$ C(x) = C_{0}+C_{1}x_{1}+ ... + C_{n}x_{n} $$

where are $$A_{i}$$, $$B_{i}$$, and $$C_{i}$$ ($$i=1, 2, ..., n$$) are the coefficient matrices.

The Data
The $$A(x)$$, $$B(x)$$, and $$C(x)$$ are matrix functions of appropriate dimensions which are all linear in the variable $$x$$ and $$A_{i}$$, $$B_{i}$$, $$C_{i}$$ are given matrix coefficients.

The Optimization Problem
The problem is to find $$\begin{align} x = [x_{1} \quad x_{2} ... x_{n}] \end{align}$$ such that:

$$A(x) < \lambda B(x) $$, $$ B(x) >0 $$, and $$ C(x) <0 $$ are satisfied and $$\lambda$$ is a scalar variable.

The LMI: LMI for Schur stabilization
A mathematical description of the LMI formulation for the generalized eigenvalue problem can be written as follows:

$$ \begin{align} &\text{min} \quad \lambda \\ & \text{s.t.} \quad A(x) < \lambda B(x) \\ & \quad \quad B(x) > 0 \\ & \quad \quad C(x) <0 \end{align} $$

Conclusion:
The solution for this LMI problem is the values of variables $$x$$ such that the scalar parameter, $$\lambda$$, is minimized. In practical applications, many problems involving LMIs can be expressed in the aforementioned form. In those cases, the objective is to minimize a scalar parameter that is involved in the constraints of the problem.

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

https://github.com/asalimil/LMI-for-Schur-Stability

Related LMIs
LMI for Generalized Eigenvalue Problem

LMI for Matrix Norm Minimization

LMI for Maximum Singular Value of a Complex Matrix

Return to Main Page
LMIs in Control/Tools


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Matrix Eigen Value Minimization

LMI for Minimizing Eigenvalue of a Matrix

Synthesizing the eigenvalues of a matrix plays an important role in designing controllers for linear systems. The eigenvalues of the state matrix of a linear time-invariant system determine if the system is stable or not. The system is stable if all the eigenvalues of the state matrix are located in the left half of the complex plane. Thus, we may desire to minimize the maximal eigenvalue of the state matrix such that the minimized eigenvalue is placed in the left half-plane, which guarantees that the system is stable.

The System
Assume that we have a matrix function of variables $$x$$:

$$ \begin{align} A(x) = A_{0}+A_{1}x_{1}+ ... + A_{n}x_{n} \end{align}$$

where $$ \begin{align} A_{i}, \quad i=1, 2, ..., n \end{align}$$ are symmetric matrices.

The Data
The symmetric matrices $$A_{i}$$ ($$ \begin{align} A_{0}, A_{1}, ..., A_{n}\end{align}$$) are given.

The Optimization Problem
The optimization problem is to find the variables $$\begin{align} x = [x_{1} \quad x_{2} ... x_{n}] \end{align}$$ to minimize the following cost function:

$$\begin{align} J(x) = \lambda_{\text{max}}(A(x)) \end{align}$$

where $$J(x)$$ is the cost function and $$\lambda_{\text{max}}(.)$$ indicates the maximim eigenvalue of a matrix.

According to Lemma 1.1 in LMI in Control Systems Analysis, Design and Applications (page 10), the following statements are equivalent

$$\begin{align} \lambda_{max}(A(x)) \leq t \iff A(x)-tI \leq 0 \end{align}$$

where $$t$$ is defined as the maximim eigenvalue of the matrix $$A$$.

The LMI: LMI for eigenvalue minimization
This optimization problem can be converted to an LMI problem.

The mathematical description of the LMI formulation can be written as follows:

$$ \begin{align} &\text{min} \quad t\\ &\text{s.t.} \quad A(x)-tI \leq 0 \end{align}$$

Conclusion:
As a result, the variables $$ x_{i}, \quad i =1, 2, ..., n $$ after solving this LMI problem. Moreover, we obtain the maximum eigenvalue, $$ t $$, of the matrix $$A(x)$$.

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

https://github.com/asalimil/LMI-for-Minimizing-the-Maximum-Eigenvalue-of-Matrix

Related LMIs
LMI for Generalized Eigenvalue Problem

LMI for Matrix Norm Minimization

LMI for Maximum Singular Value of a Complex Matrix

LMI for Matrix Positivity

Return to Main Page
LMIs in Control/Tools


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Matrix Norm Minimization

LMI for Matrix Norm Minimization

This problem is a slight generalization of the eigenvalue minimization problem for a matrix. Calculating norm of a matrix is necessary in designing an $$H_{2}$$ or an $$H_{\infty}$$ optimal controller for linear time-invariant systems. In those cases, we need to compute the norm of the matrix of the closed-loop system. Moreover, we desire to design the controller so as to minimize the closed-loop matrix norm.

The System
Assume that we have a matrix function of variables $$x$$:

$$ \begin{align} A(x) = A_{0}+A_{1}x_{1}+ ... + A_{n}x_{n} \end{align}$$

where $$ \begin{align} A_{i}, \quad i=1, 2, ..., n \end{align}$$ are symmetric matrices.

The Data
The symmetric matrices $$A_{i}$$ ($$ \begin{align} A_{0}, A_{1}, ..., A_{n}\end{align}$$) are given.

The Optimization Problem
The optimization problem is to find the variables $$\begin{align} x = [x_{1} \quad x_{2} ... x_{n}] \end{align}$$ in order to minimize the following cost function:

$$\begin{align} J(x) = ||A(x)||_{2} \end{align}$$

where $$J(x)$$ is the cost function and $$||.||_{2}$$ indicates the norm of the matrix function $$A$$.

According to Lemma 1.1 in LMI in Control Systems Analysis, Design and Applications (page 10), the following statements are equivalent:

$$\begin{align} A^{T}A - t^{2}I \leq 0 \iff \begin{bmatrix} -tI & A \\ A^{T} & -tI\end{bmatrix} \leq 0\\ \end{align}$$

The LMI: LMI for matrix norm minimization
This optimization problem can be converted to an LMI problem.

The mathematical description of the LMI formulation can be written as follows:

$$ \begin{align} &\text{min} \quad t&\\ &\text{s.t.} \quad \begin{bmatrix} -tI & A(x) \\ A(x)^{T} & -tI\end{bmatrix} \leq 0\\ \end{align}$$

Conclusion:
As a result, the variables $$ x_{i}, \quad i =1, 2, ..., n $$ after solving this LMI problem and we obtain $$ t $$ that is the norm of matrix function $$A(x)$$.

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

https://github.com/asalimil/LMI-for-Matrix-Norm-Minimization

Related LMIs
LMI for Matrix Norm Minimization

LMI for Generalized Eigenvalue Problem

LMI for Maximum Singular Value of a Complex Matrix

LMI for Matrix Positivity

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LMIs in Control/Tools


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/LMI for Generalized Eigenvalue Problem

LMI for Generalized Eigenvalue Problem

Technically, the generalized eigenvalue problem considers two matrices, like $$A$$ and $$B$$, to find the generalized eigenvector, $$x$$, and eigenvalues, $$\lambda $$, that satisfies $$ Ax = \lambda Bx $$. If the matrix $$ B $$ is an identity matrix with the proper dimension, the generalized eigenvalue problem is reduced to the eigenvalue problem.

The System
Assume that we have three matrice functions which are functions of variables $$ x = [x_{1} \quad x_{2} \quad ... \quad x_{n}]^{\text{T}} \in \mathbb{R}^{n} $$ as follows:

$$ A(x) = A_{0}+A_{1}x_{1}+ ... + A_{n}x_{n} $$

$$ B(x) = B_{0}+B_{1}x_{1}+ ... + B_{n}x_{n} $$

$$ C(x) = C_{0}+C_{1}x_{1}+ ... + C_{n}x_{n} $$

where are $$A_{i}$$, $$B_{i}$$, and $$C_{i}$$ ($$i=1, 2, ..., n$$) are the coefficient matrices.

The Data
The $$A(x)$$, $$B(x)$$, and $$C(x)$$ are matrix functions of appropriate dimensions which are all linear in the variable $$x$$ and $$A_{i}$$, $$B_{i}$$, $$C_{i}$$ are given matrix coefficients.

The Optimization Problem
The problem is to find $$\begin{align} x = [x_{1} \quad x_{2} ... x_{n}] \end{align}$$ such that:

$$A(x) < \lambda B(x) $$, $$ B(x) >0 $$, and $$ C(x) <0 $$ are satisfied and $$\lambda$$ is a scalar variable.

The LMI: LMI for Schur stabilization
A mathematical description of the LMI formulation for the generalized eigenvalue problem can be written as follows:

$$ \begin{align} &\text{min} \quad \lambda \\ & \text{s.t.} \quad A(x) < \lambda B(x) \\ & \quad \quad B(x) > 0 \\ & \quad \quad C(x) <0 \end{align} $$

Conclusion:
The solution for this LMI problem is the values of variables $$x$$ such that the scalar parameter, $$\lambda$$, is minimized. In practical applications, many problems involving LMIs can be expressed in the aforementioned form. In those cases, the objective is to minimize a scalar parameter that is involved in the constraints of the problem.

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

https://github.com/asalimil/LMI-for-Schur-Stability

Related LMIs
LMI for Generalized Eigenvalue Problem

LMI for Matrix Norm Minimization

LMI for Maximum Singular Value of a Complex Matrix

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LMIs in Control/Tools


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/LMI for Linear Programming

LMI for Linear Programming

Linear programming has been known as a technique for the optimization of a linear objective function subject to linear equality or inequality constraints. The feasible region for this problem is a convex polytope. This region is defined as a set of the intersection of many finite half-spaces which are created by the inequality constraints. The solution for this problem is to find a point in the polytope of existing solutions where the objective function has its extremum (minimum or maximum) value.

The System
We define the objective function as:

$$ c_{1}x_{1} + c_{2}x_{2} + ... c_{n}x_{n} $$

and constraints of the problem as:

$$ a_{11}x_{1} + a_{12}x_{2} + ... + a_{1n}x_{n} < b_{1} $$

$$ a_{21}x_{1} + a_{22}x_{2} + ... + a_{2n}x_{n} < b_{2} $$

.

.

.

$$ a_{m1}x_{1} + a_{m2}x_{2} + ... + a_{mn}x_{n} < b_{m} $$

The Data
Suppose that $$ c_{j} \in \mathbb{R}^{n} $$, $$ a_{ij} \in \mathbb{R}^{n}$$, and $$ b_{i} \in \mathbb{R}^{m} $$ are given parameters where $$ i =1, 2, ..., m $$ and $$ j =1, 2, ... ,n $$. Moreover, $$ x = [x_{1} \quad x_{2} \quad ... \quad x_{n}]^{\text{T}} $$ is an $$ n\times 1 $$ vector of positive variables.

The Optimization Problem
The optimization problem is to minimize the objective function, $$ c^{\text{T}}x $$ when the aforementioned linear constraints are satisfied.

The LMI: LMI for linear programming
The mathematical description of the optimization problem can be readily written in the following LMI formulation:

$$ \begin{align} &\text{min} \quad c^{\text{T}}x&\\ &\text{s.t.} \quad a_{i}^{\text{T}}x \leq b_{i}x \\ \end{align}$$

Conclusion:
Solving this problem results in the values of variables $$x$$ which minimize the objective function. It is also worthwhile to note that if $$ m \geq n $$, the computational cost for solving this problem would be $$ mn^{2} $$.

There does not exist an analytical formulation to solve a general linear programming problem. Nonetheless, there are some efficient algorithms, like the Simplex algorithm, for solving a linear programming problem.

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

https://github.com/asalimil/LMI-for-Linear-Programming

Related LMIs
LMI for Feasibility Problem

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LMIs in Control/Tools


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/LMI for Feasibility Problem

LMI for Feasibility Problem in Optimization

The feasibility problem is to find any feasible solutions for an optimization problem without regard to the objective value. This problem can be considered as a special case of an optimization problem where the objective value is the same for all the feasible solutions. Many optimization problems have to start from a feasible point in the range of all possible solutions. One way is to add a slack variable to the problem in order to relax the feasibility condition. By adding the slack variable the problem any start point would be a feasible solution. Then, the optimization problem is converted to find the minimum value for the slack variable until the feasibility is satisfied.

The System
Assume that we have two matrices as follows:

$$ \begin{align} A(x) = A_{0}+A_{1}x_{1}+ ... + A_{n}x_{n} \quad i=1, 2, ..., n \end{align}$$

$$ \begin{align} B(x) = B_{0}+B_{1}x_{1}+ ... + B_{n}x_{n} \quad i=1, 2, ..., n \end{align}$$

which are matrix functions of variables $$ x = [x_{1} \quad x_{2} \quad ... \quad x_{n}]^{\text{T}} \in \mathbb{R}^{n} $$.

The Data
Suppose that the matrices $$ \begin{align} A_{0}, A_{1}, ..., A_{n} \quad \end{align}$$ and $$ \begin{align} B_{0}, B_{1}, ..., B_{n} \quad \end{align}$$ are given.

The Optimization Problem
The optimization problem is to find variables $$\begin{align} x = [x_{1} \quad x_{2} ... x_{n}] \end{align}$$ such that the following constraint is satisfied:

$$\begin{align} A(x) < B(x) \end{align}$$

The LMI: LMI for Feasibility Problem
This optimization problem can be converted to a standard LMI problem by adding a slack variable, $$t$$.

The mathematical description for this problem is to minimize $$ t $$ in the following form of the LMI formulation:

$$ \begin{align} & \text{min} \quad t\\ & \text{s.t.} \quad A(x) < B(x) + tI \\ \end{align}$$

Conclusion:
In this problem, $$ x $$ and $$ t $$ are decision variables of the LMI problem.

As a result, these variables are determined in the optimization problem such that the minimum value of $$t$$ is found while the inequality constraint is satisfied.

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

https://github.com/asalimil/LMI-for-Feasibility-Problem-of-Convex-Optimization

Related LMIs
LMI for Linear Programming

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LMIs in Control/Tools


text:
User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Continuous Time/Structured Singular Value

The LMI can be used to find a $$ \Theta $$ that belongs to the set of scalings $$ P\Theta $$. Minimizing $$ \gamma $$ allows to minimize the squared norm of $$ \Theta M \Theta^-1 $$.

The System

\begin{align} M \text{ with transfer function } \hat M(s) = C(sI-A)^{-1}B+D,&& \hat M \in H_\infty \end{align}$$

The Data
The matrices $$ A \in R^{n \times n}, B \in R^{n \times m}, C \in R^{o \times n}, D \in R^{o \times m} $$.

The Optimization Problem
$$ \begin{align} \text{There exists } \Theta \in \Theta \text{ such that } ||\Theta M \Theta ^{-1} ||^2 < \gamma. \end{align} $$

The LMI:

\begin{align} \text{Find} \; &X>0:\\ \begin{bmatrix} A^TX+XA & XB \\ B^TX & -\Theta \end{bmatrix} + \gamma ^{-2} \begin{bmatrix} C^T \\ D^T \end{bmatrix} \Theta \begin{bmatrix} C & D \end{bmatrix} < 0\\ \end{align}$$

Conclusion:
$$ \text{The optimization problem and the LMI are equivalent. } \gamma \text{ must be optimized using bisection.} $$

Implementation
https://github.com/mcavorsi/LMI

Related LMIs
Eigenvalue Problem


text:
User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/LMI for Minimizing Condition Number of Positive Definite Matrix

The System:
A related problem is minimizing the condition number of a positive-defnite matrix $$M$$ that depends affinely on the variable $$x$$, subject to the LMI constraint $$F(x)$$ > 0. This problem can be reformulated as the GEVP.

The Optimization Problem:
The GEVP can be formulated as follows:

minimize $$ \gamma $$

subject to $$ F(x)$$ > 0;

$$\mu$$>0;

$$\mu I $$< $$M(x)$$ < $$\gamma \mu I $$.

We can reformulate this GEVP as an EVP as follows. Suppose,

$$M(x)$$= $$M_0$$ +$$\sum_{n=1}^{m}$$$$x_i M_i $$, $$F(x)$$= $$F_0$$+ $$\sum_{n=1}^{m}$$$$x_i F_i $$

The LMI:
Defining the new variables $$\nu$$=$$1/\mu$$, $$ \tilde{x}$$=$$x/\mu$$ we can express the previous formulation as the EVP with variables $$ \tilde{x},\nu$$ and $$\gamma$$:

miminize$$ \gamma $$

subject to $$\nu F_0$$+ $$\sum_{n=1}^{m}$$$$x_i F_i $$ >0; $$I$$ < $$\nu M_0$$+ $$\sum_{n=1}^{m}$$$$x_i M_i $$ < $$ \gamma I $$

Conclusion:
The LMI is feasible.


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Discrete Time Minimum Gain Lemma

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

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

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

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

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

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

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

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

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

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

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

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

 * $$\begin{bmatrix}

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

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

 * $$\begin{bmatrix}

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

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

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

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

Related LMIs
KYP Lemma State Space Stability KYP Lemma without Feedthrough


text:
User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Discrete Time Modified Minimum Gain Lemma

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

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

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

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

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

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

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

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

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

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

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

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

 * $$\begin{bmatrix}

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

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

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

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

 * $$\begin{bmatrix}

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

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

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

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

Related LMIs
KYP Lemma State Space Stability KYP Lemma without Feedthrough Discrete Time Minimum Gain Lemma


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Finsler’s Lemma

LMIs in Control/Matrix and LMI Properties and Tools/Finsler's Lemma

This method It states equivalent ways to express the positive definiteness of a quadratic form Q constrained by a linear form L. It is equivalent to other lemmas used in optimization and control theory, such as Yakubovich's S-lemma, Finsler's lemma and it is wedely used in Linear Matrix Inequalities

Theorem
Consider $$\Psi \in \mathbb{S}^{n}, G \in \mathbb{R}^{n\times m}, \Alpha \in \mathbb{R}^{m\times p}, H \in \mathbb{R}^{n\times p} $$and $$ \sigma \in \mathbb{R} $$. There exists $$\Alpha$$ such that

$$\Psi + G \Alpha H^{T} + H \Alpha^{T} G^{T} < 0,$$

if and only if there exists $$\sigma$$ such that

$$\Psi - \sigma G G^{T} < 0$$

$$\Psi - \sigma H H^{T} < 0$$

Alternative Forms of Finsler's Lemma
Consider $$\Psi \in \mathbb{S}^{n}, Z \in \mathbb{R}^{p\times n}, x \in \mathbb{R}^{n} $$and $$ \sigma \in \mathbb{R}_{>0} $$. If there exists $$Z$$ such that

$$x^{T} \Psi x, 0$$

holds for all $$x$$ ≠ $$0$$ satisfying $$Zx = 0$$, then there exists $$\sigma$$ such that

$$\Psi - \sigma Z^{T} Z < 0$$

Modified Finsler's Lemma
Consider $$\Psi \in \mathbb{S}^{n}, G \in \mathbb{R}^{n\times m}, \Alpha \in \mathbb{R}^{m\times p}, H \in \mathbb{R}^{n\times p} $$and $$ \epsilon \in \mathbb{R}_{>0} $$, where $$ \Alpha^{T} \Alpha$$ is less that on equal to $$\R$$, and $$R > 0$$. There exists $$ \Alpha $$ such that

$$ \Psi + G \Alpha H^{T} + H \Alpha^{T} G^T{T} < 0,$$

there exists $$\epsilon$$ such that

$$ \Psi + \epsilon^{-1} G G^{T} + \epsilon H R H^{T} < 0.$$

Conclusion
In summary, a number of identical methods have been stated above to determine the positive definiteness of LMIs.


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Dilation

Dilation
Matrix inequalities can be dilated in order to obtain a larger matrix inequality. This can be a useful technique to separate design variables in a BMI (bi-linear matrix inequality), as the dilation often introduces additional design variables.

A common technique of LMI dilation involves using the projection lemma in reverse, or the "reciprocal projection lemma." For instance, consider the matrix inequality

$$ \begin{bmatrix} \bold{PA}+\bold{A^TP}-\bold{P} & \bold{P}\\ * & \bold{-P}\end{bmatrix} <0, $$

where $$\bold{P}\in\S^{n\times m}$$, $$\bold{A}\in\R^{n \times n}$$, with $$\bold{P}>0.$$This can be rewritten as

$$ \begin{bmatrix} \bold{A^T} & \bold{1} & \bold{0} \\ \bold{1} & \bold{0} & \bold{1}\end{bmatrix} \begin{bmatrix} \bold{0} & \bold{P} & \bold{0} \\ \begin{bmatrix} \bold{A} & \bold{1}\\ \bold{1} & \bold{0} \\ \bold{0} & \bold{1} \end{bmatrix}<0. $$                  (1)
 * & \bold{-P} & \bold{0}\\
 * &*&\bold{-P}\end{bmatrix}

Then since $$\bold{P}>0,$$

$$ \begin{bmatrix} \bold{-P} & \bold{0}\\ * & \bold{-P}\end{bmatrix}<0, $$

which is equivalent to

$$                           \begin{bmatrix} \bold{0} & \bold{1} & \bold{0} \\ \bold{0} & \bold{0} & \bold{1}\end{bmatrix} \begin{bmatrix} \bold{0} & \bold{P} & \bold{0} \\ \begin{bmatrix} \bold{0} & \bold{0}\\ \bold{1} & \bold{0} \\ \bold{0} & \bold{1} \end{bmatrix}<0. $$                       (2)
 * & \bold{-P} & \bold{0}\\
 * &*&\bold{-P}\end{bmatrix}

These expanded inequalities (1) and (2) are now in the form of the strict projection lemma, meaning they are equivalent to

$$\bold{\Phi}(\bold{P})+\bold{G}(\bold{A})\bold{VH^T}+\bold{HV^TG^T}(\bold{A}),$$                                    (3)

where $$N(\bold{G^T}(\bold{A}))=R(\bold{N}_G(\bold{A})), N(\bold{H^T})=R(\bold{N}_H),$$and $$V\in\R^{n\times n}.$$ By choosing

$$                           \bold{G}(\bold{A})=\begin{bmatrix}\bold{-1}\\ \bold{A^T}\\ \bold{1} \end{bmatrix}, \bold{H}=\begin{bmatrix} \bold{1} \\ \bold{0} \\ \bold{0} \end{bmatrix}, $$

we can now rewrite the inequality (3) as

$$ \begin{bmatrix} -(\bold{V}+\bold{V^T}) & \bold{V^TA}+\bold{P} & \bold{V^T}\\ \end{bmatrix}<0, $$
 * & \bold{-P} & \bold{0}\\
 * & * & \bold{-P}

which is the new dilated inequality.

Examples
Some useful examples of dilated matrix inequalities are presented here.

Example 1

Consider matrices $$\bold{A,G}\in \R^{n\times n}, \bold{\Delta} \in \R^{m\times m}, \bold{P}\in \S^n, \delta_1, \delta_2, a, b \in \R_{>0},$$where $$\bold{P}>0$$and $$b=a^{-1}.$$ The following matrix inequalities are equivalent:

$$\bold{AP}+\bold{PA^T}+\delta_1\bold{P}+\delta_2\bold{APA^T}+\bold{P\Delta^T\Delta P}<0;$$

$$ \begin{bmatrix} \bold{0}&\bold{-P}&\bold{P}&\bold{0}&\bold{P\Delta^T}\\ \end{bmatrix}+He( \begin{bmatrix} \bold{A}\\ \bold{1}\\ \bold{0}\\ \bold{0}\\ \bold{0}\\ \end{bmatrix} \bold{G} \begin{bmatrix} \bold{1} & -b\bold{1} & b\bold{1} & \bold{1} & b\bold{\Delta}^T \end{bmatrix})<0. $$
 * &\bold{0}&\bold{0}&\bold{-P}&\bold{0}\\
 * &*&-\delta_1^{-1}\bold{P}&\bold{0}&\bold{0}\\
 * &*&*&-\delta_2^{-1}\bold{P}&\bold{0}\\
 * &*&*&*&\bold{-1}\\

Example 2

Consider matrices $$\bold{A, V} \in \R^{n\times n}, \bold{P, X} \in \S^n, \bold{B} \in \R^{n\times m}, C\in \R^{p\times n}, \bold{D}\in\R^{p\times m}, \bold{R}\in\S^m,$$and $$\bold{S}\in\S^p,$$where $$\bold{P, R, S, X}>0.$$The matrix inequality

$$ \begin{bmatrix} -\bold{V}-\bold{V^T} & \bold{VA}+\bold{P} & \bold{VB} & \bold{0} & \bold{V}\\ \end{bmatrix}<0 $$
 * & -2\bold{P}+\bold{X} & \bold{0} & \bold{C^T} & \bold{0}\\
 * & * & -\bold{R} & \bold{D^T} & \bold{0}\\
 * & * & * & -\bold{S} & \bold{0}\\
 * & * & * & * & -\bold{X}\\

implies the inequality

$$ \begin{bmatrix} \bold{PA}+\bold{A^TP} & \bold{PB} & \bold{C^T}\\ \end{bmatrix} $$
 * & -\bold{R} & \bold{D^T}\\
 * & * & -\bold{S}

Example 3

Consider matrices $$\bold{A, V} \in \R^{n\times n}, \bold{Q, X} \in \S^n, \bold{B} \in \R^{n\times m}, C\in \R^{p\times n}, \bold{D}\in\R^{p\times m}, \bold{R}\in\S^m,$$and $$\bold{S}\in\S^p,$$where $$\bold{Q, R, S, X}>0.$$The matrix inequality

$$ \begin{bmatrix} -\bold{V}-\bold{V^T} & \bold{V^TA^T}+\bold{Q} & \bold{0} & \bold{V^TC} & \bold{V^T}\\ \end{bmatrix}<0 $$
 * & -2\bold{Q}+\bold{X} & \bold{B} & \bold{0} & \bold{0}\\
 * & * & -\bold{R} & \bold{D^T} & \bold{0}\\
 * & * & * & -\bold{S} & \bold{0}\\
 * & * & * & * & -\bold{X}\\

implies the inequality

$$ \begin{bmatrix} \bold{AQ}+\bold{QA^T} & \bold{B} & \bold{QC^T}\\ \end{bmatrix} $$
 * & -\bold{R} & \bold{D^T}\\
 * & * & -\bold{S}

Related Pages

 * Projection Lemma - The projection lemma.
 * Reciprocal Projection Lemma - The reciprocal projection lemma.


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Tangential Nevanlinna Pick

Tangential Nevanlinna-Pick
The Tangential Nevanlinna-Pick arises in multi-input, multi-output (MIMO) control theory, particularly $$ H_{\infty} $$ robust and optimal control.

The problem is to try and find a function $$ H: C \to C^{p x q} $$ which is analytic in $$ C_{+} $$ and satisfies $$ H({\lambda}_{i})u_{i} = v_{i}, $$     $$ i = 1,...,m $$      with $$ ||H||_{\infty} \leq 1 $$    $$ (1) $$

The System
$$ N_{(ij)} $$ is a set of $$ p x q $$ matrix valued Nevanlinna functions. The matrix valued function H({\lambda}) analytic on the open upper half plane is a Nevanlinna function if $$ Im(H({\lambda})) \geq 0 $$ $$ ({\lambda} \in {\pi}^{+}) $$.

The Data
Given: Initial sequence of data points on real axis $$ {\lambda}_{1},...,{\lambda}_{m} $$ with $$ {\lambda}_{i} \in C_{+} \widehat{=} (s | Re(s) > 0) $$, And two sequences of row vectors containing distinct target points $$ u_{1},....,u_{m}$$ with $$ u_{i} \in C^{q} $$, and $$ v_{1},...,v_{m} $$ with $$ v_{i} \in C^{p}, i = 1,...,m $$.

The LMI: Tangential Nevanlinna- Pick
Problem (1) has a solution if and only if the following is true:

Nevanlinna-Pick Approach
$$ N_{ij} = $$ $$ \frac{u_{i}^{\ast}u_{j} - v_{i}^{\ast}v_{j}}{\ {\lambda}_{i}^{\ast} + {\lambda}_{j}}$$

Lyapunov Approach
N can also be found using the Lyapunov equation:

$$ A^{\ast}N + NA - (U^{\ast}U -V^{\ast}V) = 0 $$

where $$ A = diag({\lambda}_{1},...,{\lambda}_{m}), U = [u_{1} ... u_{m}], V = [v1 ... vm] $$

The tangential Nevanlinna-Pick problem is then solved by confirming that $$ N \ge 0 $$.

Conclusion:
If $$ N_{(ij)} $$ is positive (semi)-definite, then there exists a norm-bounded analytic function, $$ H $$ which satisfies $$ H({\lambda}_{i})u_{i} = v_{i}, $$    $$ i = 1,...,m $$      with $$ ||H||_{\infty} \leq 1 $$

Implementation
Implementation requires YALMIP and a linear solver such as sedumi. - MATLAB code for Tangential Nevanlinna-Pick Problem.

Related LMIs
Nevalinna-Pick Interpolation with Scaling

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/Fundamentals of Matrix and LMIs/Nevanlinna Pick Interpolation with Scaling

Nevanlinna-Pick Interpolation with Scaling
The Nevanlinna-Pick problem arises in multi-input, multi-output (MIMO) control theory, particularly $$ H_{\infty} $$ robust and optimal controller synthesis with structured perturbations.

The problem is to try and find $$ {\gamma}_{opt} = inf(||DHD^{-1}||_{\infty}) $$ such that $$ H $$ is analytic in $$ C_{+}, $$      $$ D = D^{\ast} > 0, $$ and $$ D \in\mathbb{D} $$ define the scaling, and finally,      $$ H({\lambda}_{i})u_{i} = v_{i} $$      $$ i = 1,...,m $$       $$ (1) $$

The System
The scaling factor $$ \mathbb{D} $$ is given as a set of $$ m x m $$ block-diagonal matrices with specified block structure. The matrix valued function H({\lambda}) analytic on the open upper half plane is a Nevanlinna function if $$ Im(H({\lambda})) \geq 0 $$ $$ ({\lambda} \in {\pi}^{+}) $$. The Nevanlinna LMI matrix $$ N $$ is defined as $$ N = G_{in} - G_{out} $$. The matrix $$ A $$ is a diagonal matrix of the given sequence of data points $$ {\lambda}_{i} \in\mathbb{C} (A = diag({\lambda}_{1},...,{\lambda}_{m}) $$

The Data
Given: Initial sequence of data points in the complex plane $$ {\lambda}_{1},...,{\lambda}_{m} $$ with $$ {\lambda}_{i} \in C_{+} \widehat{=} (s | Re(s) > 0) $$. Two sequences of row vectors containing distinct target points $$ u_{1},....,u_{m}$$ with $$ u_{i} \in C^{q} $$, and $$ v_{1},...,v_{m} $$ with $$ v_{i} \in C^{p}, i = 1,...,m $$.

The LMI: Nevanlinna- Pick Interpolation with Scaling
First, implement a change of variables for $$ P = D^{\ast}D $$ and $$ N = G_{in} - G_{out} $$.

From this substitution it can be concluded that $$ {\gamma}_{opt} $$ is the smallest positive $$ {\gamma} $$ such that there exists a $$ P > 0, P \in\mathbb{D} $$ such that the following is true:

$$ A^{\ast}G_{in} + G_{in}A - U^{\ast}PU = 0 $$,

$$ A^{\ast}G_{out}+ G_{out}A - V^{\ast}PV = 0 $$,

$$ {\gamma}^{2}G_{in} - G_{out} \geq 0 $$

Conclusion:
If the LMI constraints are met, then there exists a $$ H_{\infty} $$ norm-bounded optimal gain $$ {\gamma} $$ which satisfies the scaled Nevanlinna-Pick interpolation objective defined above in Problem (1).

Implementation
Implementation requires YALMIP and Mosek. - MATLAB code for Nevanlinna-Pick Interpolation.

Related LMIs
Nevalinna-Pick Interpolation

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


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Final Fantasy VI/Bosses/Phunbaba

Introduction
Phunbaba is a boss in Final Fantasy VI. It is one of ten powerful monsters sealed away in ancient times, released when Kefka brought the end of the world (the others are Doom Gaze and the Eight Dragons.)

Story
When the party finds Mobliz in the World of Ruin, Phunbaba attacks it, and Terra goes out to fight it. She is quickly defeated, forcing the party to take up the fight. It eventually escapes, leaving the Magicite Fenrir, but Terra is wounded, and unable (also unwilling) to rejoin the party.

Later, when the party returns, Phunbaba does as well, and blows two party members away with BabaBreath. Terra joins the fight in her Morph form, and together with the other characters, Phunbaba is finally destroyed, and Terra learns what love is.

Mythological Inspiration

Phunbaba may be loosely based on the creature Humbaba from Babylonian mythology, an adversary of Gilgamesh.

title: Final Fantasy VI/Bosses/Phunbaba


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User:Margav06/sandbox/Click here to continue/Software for LMIs and Optimization

Software for LMIs and Optimization

We will be Majorly using these softwares for solving the LMIs' problems:

Gurobi :

 * -You can download the program from this link https://www.gurobi.com/.
 * -Try logging in using the University's email id.
 * -If you cannot proceed with downloads using your personal wifi, it may be required for you to download the program using University's wifi.
 * -Once downloaded, install it in the path where your MATLAB is installed.
 * -Upon successful installation, you should be able to run a demo code from the Gurobi dome code files.

Sedumi :

 * -You can download the program from this link https://sedumi.ie.lehigh.edu/?page_id=58.
 * -Try logging in using the University's email id.
 * -Once downloaded, install it in the path where your MATLAB is installed.
 * -Upon successful installation, you should be able to run a demo code from the Sedumi dome code files.
 * -This is the most commonly used solver for the LMIs in Control course.
 * -You can use this solver to solve linear programming problems as well.

Cplex :

 * -This is the solver from IBM
 * -You can download the program from this link https://www.ibm.com/academic/topic/data-science
 * -Try logging in using the University's email id.
 * -It may take a day or two to get your license approved. Once approved you can use the link to download the program.
 * -Once downloaded, install it in the path where your MATLAB is installed.
 * -Upon successful installation, you should be able to run a demo code from the IBM Cplex dome code files.

MOSEK:

 * -You can download the program from this link https://www.mosek.com/products/academic-licenses/
 * -Try logging in using the University's email id.
 * -Once downloaded, install it in the path where your MATLAB is installed.
 * -Upon successful installation, you should be able to run a demo code from the Mosek dome code files.

SOSTOOLS:

 * -This tool helps us to solve non-linear systems
 * -The download link for the tool is provided here: http://control.asu.edu/sostools/
 * -Once downloaded, install the tool in your MATLAB path.
 * -Try running a demo code from the list of demo codes in the sostools folder.


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User:Margav06/sandbox/Click here to continue/Linear Systems

Linear Systems

 * 1) /Introduction to Linear Systems/
 * 2) Kalman-Yakubovich-Popov (KYP) Lemma
 * 3) Continuous Time KYP Lemmas
 * 4) /Bounded Real Lemma/
 * 5) /Positive Real Lemma/
 * 6) /KYP Lemma for QSR Dissipative System/
 * 7) /KYP Lemma without Feedthrough/
 * 8) /KYP Lemma with Feedthrough/
 * 9) /KYP Lemma for Descriptor Systems/
 * 10) Discrete Time KYP Lemmas
 * 11) /Discrete Time Bounded Real Lemma/
 * 12) /Discrete Time KYP Lemma for QSR Dissipative System/
 * 13) /Discrete Time KYP Lemma with Feedthrough/
 * 14) Deduced LMI Conditions for H∞ Index
 * 15) /H2 Norm of Linear Systems/
 * 16) /Continuous-Time H2 Norm/
 * 17) / Discrete-Time H2 Norm/
 * 18) /Generalized H2 Norm/
 * 19) /Deduced LMI Conditions for H2 Index/
 * 20) /Conic Sectors/
 * 21) /Conic Sector Lemma/
 * 22) /Exterior Conic Sector Lemma/
 * 23) /Modified Exterior Conic Sector Lemma
 * 24) Generalized KYP(GKYP) 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) Generalized Negative Imaginary Lemma
 * 33) Algebraic Riccati Inequalities
 * 34) Algebraic Riccati Inequality
 * 35) Discrete-Time Algebraic Riccati Inequality
 * 36) Variable Reduction Lemma
 * 37) System Zeros
 * 38) System Zeros without Feedthrough
 * 39) System Zeros with Feedthrough
 * 40) DC Gain of a Transfer Matrix
 * 41) Dissipativity of Linear Systems
 * 42) Small Gain Theorem
 * 43) Passivity and Positive Realness
 * 44) Non-expansivity and Bounded Realness
 * 45) Stability Theory
 * 46) Lyapunov Inequalities
 * 47) Lyapunov stability
 * 48) Discrete-time Lyapunov Stability
 * 49) Algebric Ricatti Inequalities
 * 50) Discrete-time Algebric Ricatti Inequality
 * 51) Schur Stabilizability
 * 52) Quadratic Stability
 * 53) Continuous-Time Quadratic Stability
 * 54) Discrete-Time Quadratic Stability
 * 55) Quadratic Schur Stability
 * 56) D-Stabilization
 * 57) D-Stability Rise Time Poles
 * 58) D-Stability Settling Time Poles
 * 59) D-Stability Max Percent Overshoot Poles


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User:Margav06/sandbox/Click here to continue/Linear Systems/Introduction to Linear Systems

The Wikipedia page for the following topic is given below:

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


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

KYP Lemma (Bounded Real Lemma)

The Kalman–Popov–Yakubovich (KYP) Lemma is a widely used lemma in control theory. It is sometimes also referred to as the Bounded Real Lemma. The KYP lemma can be used to determine the $$H_\infty$$ norm of a system and is also useful for proving many LMI results.

The System

\begin{align} \dot x(t)&=Ax(t)+Bu(t)\\ y(t)&=Cx(t)+Du(t)\\ x(0)&=x_0 \end{align}$$ where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$, $$u(t)\in \R^q$$, at any $$t\in \R$$.

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

The Optimization Problem
The following optimization problem must be solved.

 * $$\begin{align}

&\underset{\gamma,\;X}{\operatorname{minimize}}\quad \gamma\\ &\operatorname{subject\;to} & X>0\\ & & \begin{bmatrix} A^TX+XA & XB \\B^TX & -\gamma I\end{bmatrix}+\frac{1}{\gamma}\begin{bmatrix} C^T \\D^T \end{bmatrix}\begin{bmatrix} C & D \end{bmatrix}<0\\ \end{align}$$

The LMI: The KYP or Bounded Real Lemma
Suppose $$\hat G(s)(A,B,C,D)$$ is the system. Then the following are equivalent.
 * $$ 1)\quad \left\| G \right\|_{H_\infty} \leq \gamma $$
 * $$ 2)\quad \text{There exists a}\; X>0 \;\text{such that}$$
 * $$ \begin{bmatrix} A^TX+XA & XB \\B^TX & -\gamma I\end{bmatrix}+\frac{1}{\gamma}\begin{bmatrix} C^T \\D^T \end{bmatrix}\begin{bmatrix} C & D \end{bmatrix}<0

$$

Conclusion:
The KYP Lemma can be used to find the bound $$\gamma$$ on the $$H_\infty$$ norm of a system. Note from the (1,1) block of the LMI we know that $$A$$ is Hurwitz.

Implementation
Since the KYP lemma shown above is nonlinear in gamma, in order to implement it in MATLAB we must first linearize it by using the Schur Complement to arrive at the dual formulation below:
 * $$ \begin{bmatrix} A^TX+XA & XB & C^T \\B^TX & -\gamma I & D^T \\ C & D & -\gamma I \end{bmatrix} <0

$$. This dual KYP LMI is now linear in both $$X$$ and $$\gamma$$.

This implementation requires the use of Yalmip and Sedumi.

https://github.com/eoskowro/LMI/blob/master/KYP_Lemma_LMI.m

Related LMIs
Positive Real Lemma


text:
User:Margav06/sandbox/Click here to continue/Linear Systems/Positive Real Lemma

Positive Real Lemma

The Positive Real Lemma is a variation of the Kalman–Popov–Yakubovich (KYP) Lemma. The Positive Real Lemma can be used to determine if a system is passive (positive real).

The System

\begin{align} \dot x(t)&=Ax(t)+Bu(t)\\ y(t)&=Cx(t)+Du(t)\\ x(0)&=x_0 \end{align}$$ where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$, $$u(t)\in \R^q$$, at any $$t\in \R$$.

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

The LMI: The Positive Real Lemma
Suppose $$\hat G(s)(A,B,C,D)$$ is the system. Then the following are equivalent.
 * $$ 1)\quad G \; \text{is passive, i.e.} \; \left\langle u, Gu \right\rangle _{L_2} \ge 0 \; (\hat G(s)+\hat G(s)^*\ge 0) $$
 * $$ 2)\quad \text{There exists a}\; X>0 \;\text{such that}$$
 * $$ \begin{bmatrix} A^TX+XA & XB-C^T \\B^TX-C & -D^T-D\end{bmatrix}\le 0

$$

Conclusion:
The Positive Real Lemma can be used to determine if the system $$G$$ is passive. Note from the (1,1) block of the LMI we know that $$A$$ is Hurwitz.

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

Related LMIs
KYP Lemma (Bounded Real Lemma)


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User:Margav06/sandbox/Click here to continue/Linear Systems/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 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/Linear Systems/KYP Lemma without Feedthrough

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

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

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

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

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

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

 * $$\begin{align}

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

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

 * $$\begin{align}

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

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

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

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

 * $$\begin{align}

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

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

 * $$\begin{align}

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

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

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

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

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


text:
User:Margav06/sandbox/Click here to continue/Linear Systems/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