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LMIs in Control/Click here to continue/Optimal control systems/H2-Optimal Filtering

For systems that have disturbances, filtering can be used to reduce the effects of these disturbances. Described on this page is a method of attaining a filter that will reduce the effects of the disturbances as completely as possible. To do this, we look to find a set of new coefficient matrices that describe the filtered system. The process to achieve such a new system is described below. The H2-filter tries to minimize the average magnitude of error.

The System
For the application of this LMI, we will look at linear systems that can be represented in state space as

\begin{align} \dot x &= Ax + Bw, x(0) = x_{0}\\ y &= Cx + Dw\\ z &= Lx \end{align}$$

where $$x \in R^{n}, y \in R^{l}, z \in R^{m}$$ represent the state vector, the measured output vector, and the output vector of interest, respectively, $$w \in R^{p}$$ is the disturbance vector, and $$A, B, C, D$$ and $$L$$ are the system matrices of appropriate dimension. To further define: $$x$$ is $$\in R^{n}$$ and is the state vector, $$A $$ is $$\in R^{n*n}$$ and is the state matrix, $$B$$ is $$\in R^{n*r}$$ and is the input matrix, $$w$$ is $$\in R^{r}$$ and is the exogenous input, $$C $$ is $$\in R^{m*n}$$ and is the output matrix, $$D$$ and $$L$$ are $$\in R^{m*r}$$ and are feedthrough matrices, and $$y$$ and $$z$$ are $$\in R^{m}$$ and are the output and the output of interest, respectively.

The Data
The data are $$w$$ (the disturbance vector), and $$A, B, C, D$$ and $$L$$ (the system matrices). Furthermore, the $$A$$ matrix is assumed to be stable

The Optimization Problem
We need to design a filter that will eliminate the effects of the disturbances as best we can. For this, we take a filter of the following form:

\begin{align} \dot \sigma &= A_{f}\sigma + B_{f}y, \sigma(0) = \sigma_{0}\\ \hat z &= C_{f}\sigma, \end{align}$$

where $$\sigma \in R^{n}$$ is the state vector, $$\hat z \in R^{m}$$ is the estimation vector, and $$A_{f}, B_{f}, C_{f}$$ are the coefficient matrices of appropriate dimensions.

Note that the combined complete system can be represented as

\begin{align} \dot x_{e} &= \tilde{A}x_{e} + \tilde{B}w, x_{e}(0) = x_{e0}\\ \tilde{z} &= \tilde{C}x_{e}, \end{align}$$

where $$ \tilde{z} = z - \hat z \in R^{m}$$ is the estimation error,

\begin{align} x_{e} = \begin{bmatrix} x \\ \sigma\end{bmatrix}\\ \end{align}$$ is the state vector of the system, and $$\tilde{A}, \tilde{B}, \tilde{C}$$ are the coefficient matrices, defined as:

\begin{align} \tilde{A} = \begin{bmatrix} A & 0 \\ B_{f}C & A_{f} \end{bmatrix}, \tilde{B} = \begin{bmatrix} B \\ B_{f}D\end{bmatrix},\\ \tilde{C} = \begin{bmatrix} L & -C_{f} \end{bmatrix} \end{align}$$

In other words, for the system defined above we need to find $$A_{f}, B_{f}, C_{f}$$ such that

\begin{align} \end{align}$$ where $$ \gamma $$ is a positive constant, and
 * G_{\tilde{z}w}(s)||_{2} < \gamma,

\begin{align} G_{\tilde{z}w}(s) = \tilde{C}(sI-\tilde{A})^{-1}\tilde{B} \end{align}$$

The LMI: H-2 Filtering
For this LMI, the solution exists if one of the following sets of LMIs hold:

Matrices $$ R, X, M, N, Z, Q$$ exist that obey the following LMIs:

\begin{align} R - X &>0,\\ trace(Q) &< \gamma^{2},\\ \begin{bmatrix} -Q& *&*\\ L^{T} &-R & *\\ -N^{T} & -X & -X\end{bmatrix}&<0,\\ \begin{bmatrix} RA + A^{T}R + ZC + C^{T}Z^{T} & *&*\\ M^{T}+ZC+XA & M^{T}+M &*\\ B^{T}R+D^{T}Z^{T} & B^{T}X+D^{T}Z^{T} & -I\end{bmatrix}&<0.\\ \end{align}$$

or

Matrices $$ \bar{R}, \bar{X}, \bar{M}, \bar{N}, \bar{Z}, \bar{Q}$$ exist that obey the following LMIs:

\begin{align} \bar{R} - \bar{X} &>0,\\ trace(\bar{Q}) &< \gamma^{2},\\ \begin{bmatrix} -\bar{Q}& *&*\\ \bar{R}B+\bar{Z}D &-\bar{R} & *\\ \bar{X}B + \bar{Z}D & -\bar{X} & -I\end{bmatrix}&<0,\\ \begin{bmatrix} \bar{R}A + A^{T}\bar{R} + \bar{Z}C + C^{T}\bar{Z}^{T} & *&*\\ \bar{M}^{T}+\bar{Z}C+\bar{X}A & \bar{M}^{T}+\bar{M} &*\\ L & -\bar{N} & -I\end{bmatrix}&<0.\\ \end{align}$$

Conclusion:
To find the corresponding filter, use the optimized matrices from the first solution to find:

A_{f} = X^{-1}M, B_{f}=X^{-1}Z, C_{f} = N $$

Or the second solution to find:

A_{f} = \bar{X}^{-1}\bar{M}, B_{f}=\bar{X}^{-1}\bar{Z}, C_{f} = \bar{N} $$

These matrices can then be used to produce $$\tilde{A}, \tilde{B}, \tilde{C} $$ to construct the final filter below, that will best eliminate the disturbances of the system.

\begin{align} \dot x_{e} &= \tilde{A}x_{e} + \tilde{B}w, x_{e}(0) = x_{e0}\\ \tilde{z} &= \tilde{C}x_{e}, \end{align}$$

Implementation
This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/H2_Filtering.m

Related LMIs
H-infinity filtering


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LMIs in Control/Click here to continue/Optimal control systems/Hinf-Optimal Filtering

For systems that have disturbances, filtering can be used to reduce the effects of these disturbances. Described on this page is a method of attaining a filter that will reduce the effects of the disturbances as completely as possible. To do this, we look to find a set of new coefficient matrices that describe the filtered system. The process to achieve such a new system is described below. The H-infinity-filter tries to minimize the maximum magnitude of error.

The System
For the application of this LMI, we will look at linear systems that can be represented in state space as

\begin{align} \dot x &= Ax + Bw, x(0) = x_{0}\\ y &= Cx + Dw\\ z &= Lx \end{align}$$

where $$x \in R^{n}, y \in R^{l}, z \in R^{m}$$ represent the state vector, the measured output vector, and the output vector of interest, respectively, $$w \in R^{p}$$ is the disturbance vector, and $$A, B, C, D$$ and $$L$$ are the system matrices of appropriate dimension.

To further define: $$x$$ is $$\in R^{n}$$ and is the state vector, $$A $$ is $$\in R^{n*n}$$ and is the state matrix, $$B$$ is $$\in R^{n*r}$$ and is the input matrix, $$w$$ is $$\in R^{r}$$ and is the exogenous input, $$C $$ is $$\in R^{m*n}$$ and is the output matrix, $$D$$ and $$L$$ are $$\in R^{m*r}$$ and are feedthrough matrices, and $$y$$ and $$z$$ are $$\in R^{m}$$ and are the output and the output of interest, respectively.

The Data
The data are $$w$$ (the disturbance vector), and $$A, B, C, D$$ and $$L$$ (the system matrices). Furthermore, the $$A$$ matrix is assumed to be stable

The Optimization Problem
We need to design a filter that will eliminate the effects of the disturbances as best we can. For this, we take a filter of the following form:

\begin{align} \dot \sigma &= A_{f}\sigma + B_{f}\sigma, \sigma(0) = \sigma_{0}\\ \hat z &= C_{f}\sigma + D_{f}y, \end{align}$$

where $$\sigma \in R^{n}$$ is the state vector, $$\hat z \in R^{m}$$ is the estimation vector of z, and $$A_{f}, B_{f}, C_{f}, and D_{f}$$ are the coefficient matrices of appropriate dimensions.

Note that the combined complete system can be represented as

\begin{align} \dot x_{e} &= \tilde{A}x_{e} + \tilde{B}w, x_{e}(0) = x_{e0}\\ \tilde{z} &= \tilde{C}x_{e} + \tilde{D}w, \end{align}$$

where $$ \tilde{z} = z - \hat z \in R^{m}$$ is the estimation error,

\begin{align} x_{e} = \begin{bmatrix} x \\ \sigma\end{bmatrix}\\ \end{align}$$ is the state vector of the system, and $$\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D} $$ are the coefficient matrices, defined as:

\begin{align} \tilde{A} = \begin{bmatrix} A & 0 \\ B_{f}C & A_{f} \end{bmatrix}, \tilde{B} = \begin{bmatrix} B \\ B_{f}D\end{bmatrix},\\ \tilde{C} = \begin{bmatrix} L-D_{f}C & -C_{f} \end{bmatrix}, \tilde{D} = -D_{f}D \end{align}$$

In other words, for the system defined above we need to find $$A_{f}, B_{f}, C_{f}, and D_{f}$$ such that

\begin{align} \end{align}$$ where $$ \gamma $$ is a positive constant, and
 * G_{\tilde{z}w}(s)|_{\inf} < \gamma,

\begin{align} G_{\tilde{z}w}(s) = \tilde{C}(sI-\tilde{A})^{-1}\tilde{B} + \tilde{D}. \end{align}$$

The LMI: H-inf Filtering
The solution can be obtained by finding matrices $$ R, X, M, N, Z, D_{f}$$ that obey the following LMIs:

\begin{align} X&>0\\ R - X &>0\\ \begin{bmatrix} RA + A^{T}R + ZC + C^{T}Z^{T} & *&*&*\\ M^{T}+ZC+XA & M^{T}+M &*&*\\ B^{T}R+D^{T}Z^{T} & B^{T}X+D^{T}Z^{T} & -\gamma I &*\\ L-D_{f}C & -N & -D_{f}D & -\gamma I\end{bmatrix}&<0\\ \end{align}$$

Conclusion:
To find the corresponding filter, use the optimized matrices from the solution to find:

A_{f} = X^{-1}M, B_{f}=X^{-1}Z, C_{f} = N, D_{f} = D_{f} $$

These matrices can then be used to produce $$\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D}$$ to construct the filter described above, that will best eliminate the disturbances of the system.

Implementation
This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/hinf_filtering.m

Related LMIs
H-2_filtering

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


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Negative Imaginary Lemma

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

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

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

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

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

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

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

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

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

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

https://github.com/yashgvd/ygovada

Related LMIs
Positive Real Lemma


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-time Negative Imaginary Lemma

The System
Given a square, discrete-time LTI system G: L2e --> L2e with state-space realization (Ad, Bd, Cd, Dd) where

$$A_d \in \mathbb{R}^{n x n}$$, $$B_d \in \mathbb{R}^{n x m}$$, $$C_d \in \mathbb{R}^{m x n}$$, and $$D_d \in \mathbb{R}^{m x m}$$.

In this system, $$C_d(z1 - A_d)^{-1} B_d + D_d = B_d^T (z1 - A_d^T)^{-1} C_d^T + D_d^T$$ and $$det(1+A) \neq 0$$  and $$det(1-A) \neq 0$$.

The Data
$$A_d \in \mathbb{R}^{n x n}$$, $$B_d \in \mathbb{R}^{n x m}$$, $$C_d \in \mathbb{R}^{m x n}$$, and $$D_d \in \mathbb{R}^{m x m}$$

The LMI:
The system G posed above is considered to be negative imaginary under either of the sufficient and necessary conditions:

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

$$A_d^T P A_d - P \leq 0,$$

$$C_d + B_d^T (A_d^T - 1)^{-1} P (A_d +1) = 0$$

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

$$A_d Q A_d^T - Q \leq 0,$$

$$B_d + (A_d -1)^{-1} Q (A_d^T + 1)C_d^T = 0$$

Conclusion
By using the LMI described above, a discrete LTI system can be evaluated for the negative imaginary condition.

Implementation
This LMI can be implemented in any LMI solver such as YALMIP, using an algorithmic solver like MOSEK.

Related LMIs

 * Negative Imaginary Lemma


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Algebraic Riccati Inequalities

Algebraic Riccati Equations are particularly significant in Optimal Control, filtering and estimation problems. The need to solve such equations is common in the analysis and linear quadratic Gaussian control along with general Control problems. In one form or the other, Riccati Equations play significant roles in optimal control of multivariable and large-scale systems, scattering theory, estimation, and detection processes. In addition, closed forms solution of Riccti Equations are intractable for two reasons namely; one, they are nonlinear and two, are in matrix forms.

The System

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

The Data
Following matrices are needed as Inputs:.

 * $$ A,B,N$$.

The Optimization Problem
In control systems theory, many analysis and design problems are closely related to Riccati algebraic equations or inequalities. Find

The LMI: Algebraic Riccati Inequality
Title and mathematical description of the LMI formulation.

\begin{align} P>0, \\Q>0, \\ R>0.\\ \text{ The Algebraic Riccati inequality is given by}\;\\ A^T P+PA - (PB+N^T)R^-1(B^TP+N)+Q & \geq 0\\ \\ \text{ can be written using the Schur complement lemma as}\;\\ \begin{bmatrix} A^TP+PA+Q && PB+N^T\\ \star && R\\ \end{bmatrix}& \geq 0 \end{align}$

Conclusion:
If the solution exists, LMIs give a unique, stabilizing, symmetric matrix P.

Implementation:
Matlab code for this LMI in the Github repository:

 * 1) REDIRECT []- CODE


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-time Algebraic Riccati Inequalities

Algebraic Riccati Equations are particularly significant in Optimal Control, filtering and estimation problems. The need to solve such equations is common in the analysis and linear quadratic Gaussian control along with general Control problems. In one form or the other, Riccati Equations play significant roles in optimal control of multivariable and large-scale systems, scattering theory, estimation, and detection processes. In addition, closed forms solution of Ricatti Equations are intractable for two reasons namely; one, they are nonlinear and two, are in matrix forms.

The System
The system is defined as:

$$y_t = Ay_{t-1} + Bu_{t-1},$$

The Data
A typical specification of the discrete-time linear quadratic control problem is to minimize

 * $$\sum_{t=1}^T (y_t^T Qy_t + u_t^T Ru_t)$$

subject to the state equation

 * $$y_t = Ay_{t-1} + Bu_{t-1},$$

where y is an n × 1 vector of state variables, u is a k × 1 vector of control variables, A is the n × n state transition matrix, B is the n × k matrix of control multipliers, Q (n × n) is a symmetric positive semi-definite state cost matrix, and R (k × k) is a symmetric positive definite control cost matrix.

The LMI: Discrete-Time Algebraic Riccati Inequality
Title and mathematical description of the LMI formulation.

\begin{align} \text{ Consider}\; A_d \in \mathbb{R}^{n \times m}, B_d \in \mathbb{R}^{n \times m}, P, Q \in \mathbb{S}^{n}, \text {where}\;\\ P>0, \\Q \geq 0, \\ R>0.\\ \text{ The Discrete-Time Algebraic Riccati inequality is given by}\;\\ A^T_d PA_d - A^T_d PB_d(R + B^T_d P B_d)^{-1} B^T_d P A_d + Q - P & \geq 0,\\ \\ \text{ can be written using the Schur complement lemma as}\;\\ \begin{bmatrix} A^T_d - P + Q && A^T_d P B_d\\ \star && R+B^T_d P B_d\\ \end{bmatrix}& \geq 0 \\ \end{align}$$

Equivalently, this discrete-time algebraic Riccati inequality is satisfied under any of the following necessary and sufficient conditions.

1. There exist $$P, Q \in \mathbb{S}^{n}, \text {and} \; R \in \mathbb{S}^m,$$ where $$ P>0, Q \geq 0,and R>0, $$ such that

$$ \begin{bmatrix} Q && 0 && A^T_d P && P\\ \star && R && B^T_d P && 0\\ \star && \star && P && 0 \\ \star && \star && \star && P \\ \end{bmatrix}$$ $$\begin{align} \geq 0 \end{align}$$,

2. There exist $$X, Q \in \mathbb{S}^{n}, \text {and} \; R \in \mathbb{S}^m,$$ where $$ X>0, Q \geq 0,and R>0, $$ such that

$$ \begin{bmatrix} Q && 0 && A^T_d && 1\\ \star && R && B^T_d P && 0\\ \star && \star && X && 0 \\ \star && \star && \star && X \\ \end{bmatrix}$$ $$\begin{align} \geq 0 \end{align}$$,

Conclusion:
If the solution exists, LMIs give a unique, stabilizing, symmetric matrix P.

Implementation:
Matlab code for this LMI in the Github repository:


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Continuous-time Stabilizability

Stabilizability LMI

A system is stabilizable if all unstable modes of the system are controllable. This implies that if the system is controllable, it will also be stabilizable. Thus, stabilizability is a essentially a weaker version of the controllability condition. The LMI condition for stabilizability of pair $$(A,B)$$ is shown below.

The System

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

The Data
The matrices necessary for this LMI are $$A$$ and $$B$$. There is no restriction on the stability of A.

The LMI: Stabilizability LMI
$$(A,B)$$ is stabilizable if and only if there exists $$X>0$$ such that
 * $$AX+XA^T+BB^T<<0$$,

where the stabilizing controller is given by
 * $$u(t)=-\frac{1}{2} B^TX^{-1}x(t)$$.

Conclusion:
If we are able to find $$X>0$$ such that the above LMI holds it means the matrix pair $$(A,B)$$ is stabilizable. In words, a system pair $$(A,B)$$ is stabilizable if for any initial state $$x(0)=x_0$$ an appropriate input $$u(t)$$ can be found so that the state $$x(t)$$ asymptotically approaches the origin. Stabilizability is a weaker condition than controllability in that we only need to approach $$x(t)=0$$ as $$t\rightarrow \infty$$ whereas controllability requires that the state must reach the origin in a finite time.

Implementation
This implementation requires Yalmip and Sedumi.

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

Related LMIs
Hurwitz Stability LMI

Detectability LMI

Controllability Grammian LMI

Observability Grammian LMI


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-time Stabilizability

Discrete-Time Stabilizability

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

Discrete-Time LTI systems can be made stable using controller gain K, which can be found using LMI optimization, such that the close loop system is stable.

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

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

The Optimization Problem
The following feasibility problem should be optimized:

Maximize P while obeying the LMI constraints. Then K is found.

The LMI:
Discrete-Time Stabilizability

The LMI formulation

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

K_d = WP^{-1}

\end{align}$$

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

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

Related LMIs
- Continuous Time Stabilizability


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Continuous-time Detectability

Detectability LMI
Detectability is a weaker version of observability. A system is detectable if all unstable modes of the system are observable, whereas observability requires all modes to be observable. This implies that if a system is observable it will also be detectable. The LMI condition to determine detectability of the pair $$(A,C)$$ is shown below.

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$$, $$u(t)\in \R^m$$, at any $$t\in \R$$.

The Data
The matrices necessary for this LMI are $$A$$ and $$C$$. There is no restriction on the stability of $$A$$.

The LMI: Detectability LMI
$$(A,B)$$ is detectable if and only if there exists $$X>0$$ such that
 * $$AX+XA^T-B^TB<0$$.

Conclusion:
If we are able to find $$X>0$$ such that the above LMI holds it means the matrix pair $$(A,C)$$ is detectable. In words, a system pair $$(A,C)$$ is detectable if the unobservable states asymptotically approach the origin. This is a weaker condition than observability since observability requires that all initial states must be able to be uniquely determined in a finite time interval given knowledge of the input $$u(t)$$ and output $$y(t)$$.

Implementation
This implementation requires Yalmip and Sedumi.

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

Related LMIs
Stabilizability LMI

Hurwitz Stability LMI

Controllability Grammian LMI

Observability Grammian LMI


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-time Detectability

Discrete-Time Detectability

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

Discrete-Time LTI systems can be made detectable using observer gain L, which can be found using LMI optimization, such that the close loop system is detectable.

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

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

The Optimization Problem
The following feasibility problem should be optimized:

Maximize P while obeying the LMI constraints. Then L is found.

The LMI:
Discrete-Time Detectability

The LMI formulation

$$ \begin{align} P \in {S^{n}}; W \in {R^{m*n}}\;\\ &P>0\\ \begin{bmatrix}P & A_d^TP+C_d^TW\\
 * & P\end{bmatrix}&>0,\\

L = P^{-1}W

\end{align}$$

Conclusion:
The system is detectable iff there exits a $$P$$, such that $$P>0$$. The matrix $$A_d+LC_d$$ is Schur with $$L = P^{-1}W$$

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

Related LMIs
- Continuous time Detectability


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Continuous-time Static Output Feedback Stabilizability

In view of applications, static feedback of the full state is not feasible in general: only a few of the state variables (or a linear combination of them, $$y= Cx(t)$$, called the output) can be actually measured and re-injected into the system. So, we are led to the notion of static output feedback

The System
Consider the continuous-time LTI system, with generalized state-space realization  $$(A,B,C,0)$$

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

\end{align}$$

The Data

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

 * $$x\in\mathbb{R}^n, y\in\mathbb{R}^p, u\in\mathbb{R}^m$$

The Optimization Problem
This system is static output feedback stabilizable (SOFS) if there exists a matrix F such that the closed-loop system $$ \dot x = (A-BKC)x $$ (obtained by replacing $$u = -Ky$$ which means applying static output feedback) is asymptotically stable at the origin

The LMI: LMI for Continuous Time - Static Output Feedback Stabilizability
The system is static output feedback stabilizable if and only if it satisfies any of the following conditions:

 * There exists a $$K\in\mathbb{R}^{m\times p}$$ and $$P\in\mathbb{S}^n$$, where $$P>0$$, such that

\begin{bmatrix} A^TP + PA - PBB^TP & PB+C^TK^T \\ KC+B^TP & -1 \end{bmatrix}$$$$\begin{align}< 0\end{align}$$

 * There exists a $$K\in\mathbb{R}^{m\times p}$$ and $$Q\in\mathbb{S}^n$$, where $$Q>0$$, such that

\begin{bmatrix} QA^T + AQ - QC^TCQ & BK+QC^T \\ CQ^T+K^TB^T & -1 \end{bmatrix}$$$$\begin{align}< 0\end{align}$$

 * There exists a $$K\in\mathbb{R}^{m\times p}$$ and $$Q\in\mathbb{S}^n$$, where $$Q>0$$, such that

\begin{bmatrix} QA^T + AQ - BB^T & B+QC^TK^T \\ B^T+KCQ^T & -1 \end{bmatrix}$$$$\begin{align}< 0\end{align}$$

 * There exists a $$K\in\mathbb{R}^{m\times p}$$ and $$P\in\mathbb{S}^n$$, where $$P>0$$, such that

\begin{bmatrix} A^TP + PA - C^TC & PBK+C^T \\ K^TB^TP & -1 \end{bmatrix}$$$$\begin{align}< 0\end{align}$$

Conclusion
On implementation and optimization of the above LMI using YALMIP and MOSEK (or) SeDuMi we get 2 output matrices one of which is the Symmeteric matrix $$P$$ (or $$Q$$) and $$K$$

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

https://github.com/yashgvd/LMI_wikibooks

Related LMIs
Discrete time Static Output Feedback Stabilizability Static Feedback Stabilizability


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-time Static Output Feedback Stabilizability

The static output feedback (SOF) problem has been investigated and analyzed by many people and the literature concerning this topic is vast. In practicality, it is not always possible to have full access to the state vector and only a partial information through a measured output is available. This explains why this problem has challenged many researchers in control theory. Here is a systematic approach for the SOF control design for discrete time linear systems.

The System
Consider a discrete-time LTI system, with state-space realization $$(Ad, Bd,Cd, 0)$$,

 * $$\begin{align}

x_{k+1} = A_dx_k+B_du_k\\ y_k=C_dx_k \end{align}$$

$$x_k\in\mathbb{R}^n$$ is the state, $$y_k\in\mathbb{R}^p$$ is the measured output, $$u_k\in\mathbb{R}^m$$ is the control input.

The Data
$$A_d\in\mathbb{R}^{n\times n}, B_d\in\mathbb{R}^{n\times m}, C_d\in\mathbb{R}^{p\times n}$$ are the constant matrices of appropriate dimensions.

The Optimization Problem
The full state is not measurable and only partial information is available through $$y_k$$ which can used for control purposes.

We have to find a static output feedback gain with respect to
 * $$u_k = -K_dy_k,$$

where $$K_d\in\mathbb{R}^{m\times p}$$ is the output feedback gain such that the final closed loop system is asymptotically stable.

The LMI: LMI for Discrete-Time Static Output Feedback Stabilizability
The discrete time system considered is static output feedback stabilizable under any of the following equivalent necessary or sufficient conditions.

 * There exists a $$K_d\in\mathbb{R}^{m\times p}$$ and $$P \in\mathbb{S}^n$$ where $$P>0$$ such that

\begin{bmatrix} -P & (A_d+B_dK_dC_d)P \\ ((A_d+B_dK_dC_d)P)' & -P \end{bmatrix}$$$$\begin{align}< 0\end{align}.$$

 * There exists a $$K_d\in\mathbb{R}^{m\times p}$$ and $$P \in\mathbb{S}^n$$ where $$P>0$$ such that

\begin{bmatrix} -A_dPPA_d^T & A_dP+B_dK_dC_d & A_dP \\ (A_dP+B_dK_dC_d)' & -1 & 0 \\ PA_d^T & 0 & -P \end{bmatrix}$$$$\begin{align}< 0\end{align}.$$

Conclusion
If it is feasible we obtain a output feedback gain matrix $$K_d$$ such that the closed loop system is asymptotically stable. While implementing the optimization problem the following conditions are assumed to be satisfied
 * The Transfer matrix and its inverse are both analytical a s=0
 * The matrix $$C_dA_d^{-1}B_d$$ is non-singular
 * The triple $$(A_d,B_d,C_d)$$ are reachable and observable.

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

https://github.com/yashgvd/LMI_wikibooks

Related LMIs
Continuous-time Static Output Feedback Stabilizability


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Continuous-time strong stabilizability

The System
Consider the continous-time LTI system, $$G:L_{2e}\rightarrow L_{2e}$$ with state-space realization (A,B,C,0)

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

where $$A\in \R^{n\times n}$$, $$B\in \R^{n\times m}$$, $$C\in \R^{p\times n}$$, and it and it is assumed that (A, B) is stabilizable, (A, C) is detectable, and the transfer matrix $$G(s)=C(s1-A^{-1})B$$ has no poles on the imaginary axis.

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

The Optimization Problem
The system G is strongly stabilizable if there exist $$P \in \mathbb S^n$$, $$Z\in \R^{n\times p}$$, and $$\gamma\in \R_{>0}$$, where $$P > 0$$, such that

\begin{align} PA+A^T+ZC+C^TZ^T<0\\ {\displaystyle {\begin{aligned}{\begin{bmatrix}-P(A+BF)+(A+BF)^TP+ZC+C^TZ^T&&-Z&&-XB\\ \end{align}$$
 * &&-\gamma I && 0\\
 * && * && -\gamma I \end{bmatrix}}<0\end{aligned}}}\\

Conclusion:
where $$F=-B^TX$$ and $$X \in S_n$$, $$X \geq 0$$ is the solution to the Lyapunov equation given by

\begin{align} XA+A^TX-XBB^TX=0 \end{align}$$ Moreover, a controller that strongly stabilizes G is given by the state-space realization

\begin{align} \dot x_c = (A+BF+P^{-1}ZC)x(t)-P^{-1}Zu(t)\\ y_C=-B^TXx(t) \end{align}$$

Implementation

 * Example Code

Related LMIs

 * https://en.wikibooks.org/wiki/LMIs_in_Control/Stability_Analysis/Discrete_Time/DiscreteTimeStrongStabilizability - Discrete Time Strong Stabilizability


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-time strong stabilizability

The System
Consider the continous-time LTI system, $$G:L_{2e}\rightarrow L_{2e}$$ with state-space realization ($$A_d,B_d,C_d,0$$)

\begin{align} \dot x(k)&=A_dx(k)+B_du(k),\\ y(k)&=C_dx(k), \end{align}$$

where $$A_d\in \R^{n\times n}$$, $$B_d\in \R^{n\times m}$$, $$C_d\in \R^{p\times n}$$, and it and it is assumed that ($$A_d,B_d $$),is stabilizable, ($$A_d, C_d$$) is detectable, and the transfer matrix $$G(s)=C_d(s1-A_d)^{-1})B_d$$ has no poles on the imaginary axis.

The Data
The matrices $$ A_d,B_d,C_d $$.

The Optimization Problem
The system G is strongly stabilizable if there exist $$P \in \mathbb S^n$$, $$Z\in \R^{n\times p}$$, and $$\gamma\in \R_{>0}$$, where $$P > 0$$, such that

\begin{align}

{\displaystyle {\begin{aligned}{\begin{bmatrix}A^T_dPA_d-P-A_d^TZC_d-C^T_dZ^TA_d && C_d^TZ^T\\ {\displaystyle {\begin{aligned}{\begin{bmatrix}N_{11} &&(A_d +B_dF)^TZ&& XB_d &&C_d^TZ^T\\ \end{align}$$
 * &&-P\\
 * && * && -\gamma I \end{bmatrix}}<0\end{aligned}}}\\
 * && -\gamma I && 0 && Z^T\\
 * && * && -\gamma I && 0\\
 * &&*&&*&&-P\end{bmatrix}}<0\end{aligned}}}\\

Conclusion:
where $$N_{11}=(A_d+B_dF)^TP(A_d+B_dF)-P+(A_d+B_dF)^TZC_d^TZ^T(A_d+B_dF), F=-B_d^TX,X=Y$$ and $$X \in S_n$$, $$X \geq 0$$ is the solution to the discrete-time Lyapunov equation given by

\begin{align} A_dXA^T_d+X-B_dB_d^TX=0 \end{align}$$ Moreover, a controller that strongly stabilizes G is given by the state-space realization

\begin{align} \dot x_{c,k+1} = (A_d+B_dF+P^{-1}ZC_d)x_k-P^{-1}Zu_k\\ y_{c,k}=-B_d^TXx_k \end{align}$$

Implementation

 * -example code

Related LMIs

 * https://en.wikibooks.org/wiki/LMIsinControl/StabilityAnalysis/ContinuousTime/StrongStabilizability - Continuous Time Strong Stabilizability


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/System Zeros without Feedthrough

Let's say we have a transfer function defined as a ratio of two polynomials: $$ \begin{align} H(s) = \frac{N(s)}{D(s)} \\ \end{align}$$ Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting $$ N(s) = 0 $$ and solving for s.The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Similarly, the system zeros are either real or appear in complex conjugate pairs. In the case of system zeros without feedthrough, we take the assumption that $$ D=0 $$.

The System
Consider a continuous-time LTI system, $$ G $$, with minimal statespace representation $$ (A,B,C,0) $$

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

The Data
The matrices:

\begin{align} A \in \mathbb{R}^{n \times n} \\ M \in \mathbb{R}^{n \times q} \\ N \in \mathbb{R}^{q \times n} \\ \end{align}$$

The LMI: System Zeros without feedthrough
The transmission zeros of $$ G(s) = C(sI-A)^-1 B $$ are the eigenvalues of $$ NAM $$, where $$ N \in \mathbb{R}^{q \times n}, M \in \mathbb{R}^{n \times q}, CM = 0, NB = 0, NM = 1. $$. Therefore, $$ G(s) $$ is a minimum phase if and only if there exists $$ P \in \mathbb{S}^{q} $$, where $$ P>0 $$ such that

\begin{align} PNAM + M^T A^T N^T P <0 \end{align}$$

Conclusion:
If P exists, it ensures non-minimum phase. Eigenvalues of NAM then gives the zeros of the system.

Implementation
https://github.com/Ricky-10/coding107/blob/master/Systemzeroswithoutfeedthrough


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/System Zeros with Feedthrough

Let's say we have a transfer function defined as a ratio of two polynomials: $$ \begin{align} H(s) = \frac{N(s)}{D(s)} \\ \end{align}$$ Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting $$ N(s) = 0 $$ and solving for s.The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Similarly, the system zeros are either real or appear in complex conjugate pairs. In the case of system zeros with feedthrough, we take $$ D $$ as full rank.

The System
Consider a continuous-time LTI system, $$ G $$, with minimal statespace representation $$ (A,B,C,D) $$

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

The Data
The matrices needed as inputs are:

\begin{align} A \in \mathbb{R}^{n \times n} \\ B \in \mathbb{R}^{n \times m} \\ N \in \mathbb{R}^{p \times n} \\ \end{align}$$ In this case, $$ m \leq p $$

The LMI: System Zeros with feedthrough
The transmission zeros of $$ G(s) = C(sI-A)^-1 B + D $$ are the eigenvalues of $$ A-B(D^T D)^{-1} D^T C $$. Therefore, $$ G(s) $$ is a minimum phase if and only if there exists $$ P \in \mathbb{S}^{q} $$, where $$ P>0 $$ such that

\begin{align} P(A-B(D^T D)^{-1} D^T C) + (A-B(D^T D)^{-1} D^T C)^T P < 0 \\ \end{align}$$

Conclusion:
If P exists, it ensures non-minimum phase. Eigenvalues of $$ A-B(D^T D)^{-1} D^T C $$ then gives the zeros of the system.

Related LMIs
LMIs_in_Controls/pages/systemzeroswithoutfeedthrough

Implementation
https://github.com/Ricky-10/coding107/blob/master/systemzeroswithfeedthrough


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Conic Sector Region Stability via the Dilation Lemma

Definition
Consider $$A\in\mathbb{R}^{n \times n}$$. The matrix $$A$$ is $$\mathcal{D}$$-stable if and only if there exists $$P\in\mathbb{S}^{n}$$, where $$P>0$$, such that

$$[\lambda_{kl}P+\phi_{kl}AP+\phi_{lk}PA^{T}]_{1<k,l<m}<0$$,

or equivalent

$$\Lambda\otimes P+\Phi\otimes (AP)+\Phi^{T}\otimes (PA^{T})<0$$,

where $$\otimes$$ is the Kroenecker product,

The eigenvalues of a $$\mathcal{D}$$-stable matrix lie within the LMI region $$\mathcal{D}$$, which is defined as

$$\mathcal {D}=\{z\in\mathbb{C}:f_{D}(z)<0\}$$, where

$$f_{D}(z):=\Lambda+z\Phi+\bar{z}\Phi^{T}= \{\lambda_{kl}+\phi_{kl}z+\phi_{lk}\bar{z}\}_{1\leq k,l\leq m}$$,

$$\Lambda\in\mathbb{S}^{m}$$, $$\Phi\in\mathbb{R}^{m \times m}$$, and $$\bar{z}$$ is the complex conjugate of $$z$$.

Conic Sector Region Stability via the Dilation Lemma
Consider $$A\in\mathbb{R}^{n \times n}$$ and $$k\in\mathbb{R}_{>0}$$.

The matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{P}(k)$$, where $$\mathcal{P}(k):=\{\lambda\in\mathbb{C}:\left\vert Im( \lambda) \right\vert0}$$, where $$X>0$$, such that

$$\begin{bmatrix} k(AX+XA^{T}) & AX-XA^{T} \\ * & k(AX+XA^{T}) \end{bmatrix}<0$$.

Equivalently, the matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{P}(k)$$ if and only if there exist $$X\in\mathbb{S}^{n}$$ and $$\epsilon\in\mathbb{R}_{>0}$$, and $$F\in\mathbb{R}^{n \times n}$$, where $$X>0$$, such that

$$\begin{bmatrix} 0 & -kX & X & 0 \\ * & 0 & 0 & -X \\ * & * & 0 & -kX \\ +He\begin{Bmatrix} \begin{bmatrix} A &0 \\ 1 & 0 \\ 0 & 1 \\ 0 & A \end{bmatrix} \begin{bmatrix} F &0 \\ 0 & F \end{bmatrix} \begin{bmatrix} k1 & -\epsilon k1 & \epsilon1 & 1 \\ -1 & -\epsilon1 & \epsilon k1 & k 1 \end{bmatrix}\end{Bmatrix}<0$$.
 * & * & * & 0\end{bmatrix}

Moreover, for every $$X$$ that satisfies

$$\begin{bmatrix} k(AX+XA^{T}) & AX-XA^{T} \\ * & k(AX+XA^{T}) \end{bmatrix}<0$$,

$$X$$ and $$F=-\epsilon^{ -1}(A-\epsilon^{ -1}1)^{ -1}X$$ are solutions to

$$\begin{bmatrix} 0 & -kX & X & 0 \\ * & 0 & 0 & -X \\ * & * & 0 & -kX \\ +He\begin{Bmatrix} \begin{bmatrix} A &0 \\ 1 & 0 \\ 0 & 1 \\ 0 & A \end{bmatrix} \begin{bmatrix} F &0 \\ 0 & F \end{bmatrix} \begin{bmatrix} k1 & -\epsilon k1 & \epsilon1 & 1 \\ -1 & -\epsilon1 & \epsilon k1 & k 1 \end{bmatrix}\end{Bmatrix}<0$$
 * & * & * & 0\end{bmatrix}


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/alpha-Region via the Dilation Lemma

Definition
Consider $$A\in\mathbb{R}^{n \times n}$$. The matrix $$A$$ is $$\mathcal{D}$$-stable if and only if there exists $$P\in\mathbb{S}^{n}$$, where $$P>0$$, such that

$$[\lambda_{kl}P+\phi_{kl}AP+\phi_{lk}PA^{T}]_{1<k,l<m}<0$$,

or equivalent

$$\Lambda\otimes P+\Phi\otimes (AP)+\Phi^{T}\otimes (PA^{T})<0$$,

where $$\otimes$$ is the Kroenecker product,

The eigenvalues of a $$\mathcal{D}$$-stable matrix lie within the LMI region $$\mathcal{D}$$, which is defined as

$$\mathcal {D}=\{z\in\mathbb{C}:f_{D}(z)<0\}$$, where

$$f_{D}(z):=\Lambda+z\Phi+\bar{z}\Phi^{T}= \{\lambda_{kl}+\phi_{kl}z+\phi_{lk}\bar{z}\}_{1\leq k,l\leq m}$$,

$$\Lambda\in\mathbb{S}^{m}$$, $$\Phi\in\mathbb{R}^{m \times m}$$, and $$\bar{z}$$ is the complex conjugate of $$z$$.

α-Region Stability via the Dilation Lemma
Consider $$A\in\mathbb{R}^{n \times n}$$ and $$\alpha\in\mathbb{R}_{>0}$$. The matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{H}(\alpha)$$, where $$\mathcal{H}(\alpha):=\{\lambda\in\mathbb{C}:Re(\lambda)<-\alpha\}$$ if and only if there exist $$X\in\mathbb{S}^{n}$$ and $$\epsilon\in\mathbb{R}_{>0}$$, where $$X>0$$, such that

$$AX+XA^{T}+2\alpha X<0$$.

Equivalently, the matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{H}(\alpha)$$ if and only if there exist $$X\in\mathbb{S}^{n}$$, $$\epsilon\in\mathbb{R}_{>0}$$, and $$F\in\mathbb{R}^{n \times n}$$, where $$X>0$$, such that

$$\begin{bmatrix} 0 & -X & X \\ * & 0 & 0 \\ * & * & -\frac{1}{2}\alpha^{ -1}X \end{bmatrix} +He\begin{Bmatrix} \begin{bmatrix} A \\ 1 \\ 0 \end{bmatrix}F \begin{bmatrix} 1 & -\epsilon1 & \epsilon1 \end{bmatrix} \end{Bmatrix}<0$$.

Moreover, for every $$X$$ that satisfies

$$AX+XA^{T}+2\alpha X<0$$

$$X$$ and $$F=-\epsilon^{ -1}(A-\epsilon^{ -1}1)^{ -1}X$$ are solutions to

$$\begin{bmatrix} 0 & -X & X \\ * & 0 & 0 \\ * & * & -\frac{1}{2}\alpha^{ -1}X \end{bmatrix} +He\begin{Bmatrix} \begin{bmatrix} A \\ 1 \\ 0 \end{bmatrix}F \begin{bmatrix} 1 & -\epsilon1 & \epsilon1 \end{bmatrix} \end{Bmatrix}<0$$.


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Circular Region Stability via the Dilation Lemma

Definition
Consider $$A\in\mathbb{R}^{n \times n}$$. The matrix $$A$$ is $$\mathcal{D}$$-stable if and only if there exists $$P\in\mathbb{S}^{n}$$, where $$P>0$$, such that

$$[\lambda_{kl}P+\phi_{kl}AP+\phi_{lk}PA^{T}]_{1<k,l<m}<0$$,

or equivalent

$$\Lambda\otimes P+\Phi\otimes (AP)+\Phi^{T}\otimes (PA^{T})<0$$,

where $$\otimes$$ is the Kroenecker product,

The eigenvalues of a $$\mathcal{D}$$-stable matrix lie within the LMI region $$\mathcal{D}$$, which is defined as

$$\mathcal {D}=\{z\in\mathbb{C}:f_{D}(z)<0\}$$, where

$$f_{D}(z):=\Lambda+z\Phi+\bar{z}\Phi^{T}= \{\lambda_{kl}+\phi_{kl}z+\phi_{lk}\bar{z}\}_{1\leq k,l\leq m}$$,

$$\Lambda\in\mathbb{S}^{m}$$, $$\Phi\in\mathbb{R}^{m \times m}$$, and $$\bar{z}$$ is the complex conjugate of $$z$$.

Circular Region Stability via the Dilation Lemma
Consider $$A\in\mathbb{R}^{n \times n}$$, $$r\in\mathbb{R}_{>0}$$, and $$c\in\mathbb{R}_{<0}$$, where $$c<-r$$. The matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{G}(c,r)$$, where $$\mathcal{G}(c,r):=\{\lambda\in\mathbb{C}:\left\vert \lambda-c \right\vert0}$$, where $$X>0$$, such that

$$AX+XA^{T}-\frac{c^{2}-r^{2}}{c}X-\frac{1}{c}AXA^{T}<0$$.

Equivalently, the matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{G}(c,r)$$ if and only if there exist $$X\in\mathbb{S}^{n}$$, $$\epsilon\in\mathbb{R}_{>0}$$, and $$F\in\mathbb{R}^{n \times n}$$, where $$X>0$$, such that

$$\begin{bmatrix} 0 & -X & X & 0 \\ * & 0 & 0 & -X \\ * & * & \frac{c}{c^{2}-r^{2}}X & 0 \\ +He\begin{Bmatrix} \begin{bmatrix} A \\ 1 \\ 0 \\ 0 \end{bmatrix}F \begin{bmatrix} 1 & -\epsilon1 & \epsilon1 & 1 \end{bmatrix} \end{Bmatrix}<0$$
 * & * & * & cX \end{bmatrix}

Moreover, for every $$X$$ that satisfies

$$AX+XA^{T}-\frac{c^{2}-r^{2}}{c}X-\frac{1}{c}AXA^{T}<0$$

$$X$$ and $$F=-\epsilon^{ -1}(A-\epsilon^{ -1}1)^{ -1}X$$ are solutions to

$$\begin{bmatrix} 0 & -X & X & 0 \\ * & 0 & 0 & -X \\ * & * & \frac{c}{c^{2}-r^{2}}X & 0 \\ +He\begin{Bmatrix} \begin{bmatrix} A \\ 1 \\ 0 \\ 0 \end{bmatrix}F \begin{bmatrix} 1 & -\epsilon1 & \epsilon1 & 1 \end{bmatrix} \end{Bmatrix}<0$$
 * & * & * & cX \end{bmatrix}


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/DC Gain of a Transfer Matrix

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

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

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

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

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

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

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

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

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

https://github.com/yashgvd/LMI_wikibooks


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Kharitonov-Bernstein-Haddad(KBH) Theorem

WIP, Description in progress

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

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

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

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

where

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

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

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

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

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

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

WIP, additional references to be added


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Delay-Independent Condition

Stabilization of Time-Delay Systems - Delay Independent Case

Suppose, for instance, there was a system where a time-delay was introduced. In that instance, stabilization would have to be done in a different manner. The following example demonstrates how one can stabilize such a system independent of the delay.

The System
For this particular problem, suppose that we were given the time-delayed system in the form of:

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

where

\begin{align} &A\, A_d\in\mathbb{R}^{n\times n}, B\in\mathbb{R}^{nxr}\text{ are the system coefficient matrices,}\\ &\phi(t)\text{ is the initial condition,}\\ &d\text{ represents the time-delay, and}\\ &\bar{d}\text{ is a known upper-bound of }d\\ \end{align}$$ Then the LMI for determining the Time-Delay System for the Delay-Independent case would be obtained as described below.

The Data
In order to obtain the LMI, we need the following 3 matrices: $$A, A_d,$$ and $$B$$.

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

\begin{align} \begin{cases} \dot x(t)&=(A+BK)x(t)+A_d(t-d),\\ x(t)&=\phi(t), t\in[0,d], 0 < d \leq \bar{d},\\ \end{cases} \end{align}$$ is simultaneously both uniform and asymptotically stable, then the system would be stabilized as follows.

The LMI: The Delay-Independent Stabilization of Time-Delay Systems
From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a matrix$$W\in\mathbb{R}^{r\times n}$$ and 2 symmetric matrices $$X>0$$ and $$Y>0$$ that satisfy the following:

\begin{align} \begin{bmatrix}X{A^T}+AX+BW+{W^T}{B^T}+Y&&{A_d}X\\X{A_d}^T&&-Y\end{bmatrix}&<0\\ \end{align}$$

Conclusion:
Given the resulting feedback gain matrix $$K=WX^{-1}$$, it can be observed that the matrix is asymptotically stable while simultaneously ensuring that the solution is delay-independent from the time-delay system where this gain matrix was derived.

Implementation

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

Related LMIs

 * ../Delay Dependent Time-Delay Stabilization/ - Equivalent LMI for delay-dependent time-delay stabilization.


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Delay-Dependent Condition

Stabilization of Time-Delay Systems - Delay Dependent Case

Suppose, for instance, there was a system where a time-delay was introduced. In that instance, stabilization would have to be done in a different manner. The following example demonstrates how one can stabilize such a system while being dependent on the delay.

The System
For this particular problem, suppose that we were given the time-delayed system in the form of:

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

where

\begin{align} &A\text{, }{A_d}\in\mathbb{R}^{n\times n}\text{, }{B}\in\mathbb{R}^{n\times r}\text{ are the system coefficient matrices,}\\ &\phi(t)\text{ is the initial condition,}\\ &d\text{ represents the time-delay, and}\\ &\bar{d}\text{ is a known upper-bound of }d\\ \end{align}$$ Then the LMI for determining the Time-Delay System for the Delay-Dependent case would be obtained as described below.

The Data
In order to obtain the LMI, we need the following 3 matrices: $$A\text{, }{A_d}\text{, and }B$$.

The Optimization Problem
Suppose - for the time-delayed system given above - we were asked to design a memoryless feedback control law where $$u(t)=Kx(t)$$ such that the closed-loop system:

\begin{align} \begin{cases} \dot x(t)&=(A+BK)x(t)+A_d(t-d),\\ x(t)&=\phi(t), t\in[0,d], 0 < d \leq \bar{d},\\ \end{cases} \end{align}$$ is simultaneously both uniform and asymptotically stable, then the system would be stabilized as follows.

The LMI: The Delay-Dependent Stabilization of Time-Delay Systems
From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a scalar $$0<\beta<1$$, a symmetric matrix $$X>0$$ and a matrix $$W$$ that satisfy the following:

\begin{align} \begin{bmatrix}\Phi(X,W)&&\bar{d}(X{A^T}+{W^T}{B^T})&&\bar{d}X{A^T_d}\\ \bar{d}(AX+BW)&&-\bar{d}{\beta}I&&0\\ \bar{d}{A_d}X&&0&&-\bar{d}(1-\beta)I\end{bmatrix}&<0\\ \end{align}$$

where$$\Phi(X,W)=X{(A+{A_d})^T}+(A+{A_d})X+BW+{W^T}{B^T}+\bar{d}{A_d}{A^T_d}$$

Conclusion:
Given the resulting stabilizing control gain matrix $$K=WX^{-1}$$, it can be observed that the matrix is asymptotically stable from the time-delay system where this gain matrix was derived.

Implementation

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

Related LMIs

 * ../Delay Independent Time-Delay Stabilization/ - Equivalent LMI for delay-independent time-delay stabilization.


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/mu-Analysis

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

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

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

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

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

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

\begin{align}

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

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

Implementation

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


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Delay Dependent Time-Delay Stabilization

Stabilization of Time-Delay Systems - Delay Dependent Case

Suppose, for instance, there was a system where a time-delay was introduced. In that instance, stabilization would have to be done in a different manner. The following example demonstrates how one can stabilize such a system while being dependent on the delay.

The System
For this particular problem, suppose that we were given the time-delayed system in the form of:

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

where

\begin{align} &A\text{, }{A_d}\in\mathbb{R}^{n\times n}\text{, }{B}\in\mathbb{R}^{n\times r}\text{ are the system coefficient matrices,}\\ &\phi(t)\text{ is the initial condition,}\\ &d\text{ represents the time-delay, and}\\ &\bar{d}\text{ is a known upper-bound of }d\\ \end{align}$$ Then the LMI for determining the Time-Delay System for the Delay-Dependent case would be obtained as described below.

The Data
In order to obtain the LMI, we need the following 3 matrices: $$A\text{, }{A_d}\text{, and }B$$.

The Optimization Problem
Suppose - for the time-delayed system given above - we were asked to design a memoryless feedback control law where $$u(t)=Kx(t)$$ such that the closed-loop system:

\begin{align} \begin{cases} \dot x(t)&=(A+BK)x(t)+A_d(t-d),\\ x(t)&=\phi(t), t\in[0,d], 0 < d \leq \bar{d},\\ \end{cases} \end{align}$$ is simultaneously both uniform and asymptotically stable, then the system would be stabilized as follows.

The LMI: The Delay-Dependent Stabilization of Time-Delay Systems
From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a scalar $$0<\beta<1$$, a symmetric matrix $$X>0$$ and a matrix $$W$$ that satisfy the following:

\begin{align} \begin{bmatrix}\Phi(X,W)&&\bar{d}(X{A^T}+{W^T}{B^T})&&\bar{d}X{A^T_d}\\ \bar{d}(AX+BW)&&-\bar{d}{\beta}I&&0\\ \bar{d}{A_d}X&&0&&-\bar{d}(1-\beta)I\end{bmatrix}&<0\\ \end{align}$$

where$$\Phi(X,W)=X{(A+{A_d})^T}+(A+{A_d})X+BW+{W^T}{B^T}+\bar{d}{A_d}{A^T_d}$$

Conclusion:
Given the resulting stabilizing control gain matrix $$K=WX^{-1}$$, it can be observed that the matrix is asymptotically stable from the time-delay system where this gain matrix was derived.

Implementation

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

Related LMIs

 * ../Delay Independent Time-Delay Stabilization/ - Equivalent LMI for delay-independent time-delay stabilization.


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Delay Independent Time-Delay Stabilization

Stabilization of Time-Delay Systems - Delay Independent Case

Suppose, for instance, there was a system where a time-delay was introduced. In that instance, stabilization would have to be done in a different manner. The following example demonstrates how one can stabilize such a system independent of the delay.

The System
For this particular problem, suppose that we were given the time-delayed system in the form of:

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

where

\begin{align} &A\, A_d\in\mathbb{R}^{n\times n}, B\in\mathbb{R}^{nxr}\text{ are the system coefficient matrices,}\\ &\phi(t)\text{ is the initial condition,}\\ &d\text{ represents the time-delay, and}\\ &\bar{d}\text{ is a known upper-bound of }d\\ \end{align}$$ Then the LMI for determining the Time-Delay System for the Delay-Independent case would be obtained as described below.

The Data
In order to obtain the LMI, we need the following 3 matrices: $$A, A_d,$$ and $$B$$.

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

\begin{align} \begin{cases} \dot x(t)&=(A+BK)x(t)+A_d(t-d),\\ x(t)&=\phi(t), t\in[0,d], 0 < d \leq \bar{d},\\ \end{cases} \end{align}$$ is simultaneously both uniform and asymptotically stable, then the system would be stabilized as follows.

The LMI: The Delay-Independent Stabilization of Time-Delay Systems
From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a matrix$$W\in\mathbb{R}^{r\times n}$$ and 2 symmetric matrices $$X>0$$ and $$Y>0$$ that satisfy the following:

\begin{align} \begin{bmatrix}X{A^T}+AX+BW+{W^T}{B^T}+Y&&{A_d}X\\X{A_d}^T&&-Y\end{bmatrix}&<0\\ \end{align}$$

Conclusion:
Given the resulting feedback gain matrix $$K=WX^{-1}$$, it can be observed that the matrix is asymptotically stable while simultaneously ensuring that the solution is delay-independent from the time-delay system where this gain matrix was derived.

Implementation

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

Related LMIs

 * ../Delay Dependent Time-Delay Stabilization/ - Equivalent LMI for delay-dependent time-delay stabilization.


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Parametric, Norm-Bounded Uncertain System Quadratic Stability

Given a system with matrices A,M,N,Q the quadratic stability of the system with parametric, norm-bounded uncertainty can be determined by the following LMI. The feasibility of the LMI tells if the system is quadratically stable or not.

The System

\begin{align} \dot x(t)&=Ax(t)+Mp(t),&&p(t) = \Delta (t)q(t),\\ q(t) &= Nx(t)+Qp(t),&&\Delta \in \mathbf{\Delta} \; := \{\Delta \in \mathbb{R}^{n \times n} : \| \Delta \| \leq 1 \}\\ \end{align}$$

The Data
The matrices $$ A,M,N,Q $$.

The LMI:

\begin{align} \text{Find} \; &P>0,\mu \geq 0:\\ \begin{bmatrix} AP+PA^T & PN^T \\ NP & 0\end{bmatrix} + \mu \begin{bmatrix} MM^T & MQ^T \\ QM^T & QQ^T - I \end{bmatrix} < 0\\ \end{align}$$

Conclusion:
The system above is quadratically stable if and only if there exists some mu >= 0 and P > 0 such that the LMI is feasible.

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

Related LMIs
Stability of Structured, Norm-Bounded Uncertainty

Stability under Arbitrary Switching

Quadratic Stability Margins


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Stability of Structured, Norm-Bounded Uncertainty

Given a system with matrices A,M,N,Q with structured, norm-bounded uncertainty, the stability of the system can be found using the following LMI. The LMI takes variables P and $$ \Theta $$ and checks for a feasible solution.

The System

\begin{align} \dot x(t)&=Ax(t)+Mp(t),&&p(t) = \Delta (t)q(t),\\ q(t) &= Nx(t)+Qp(t),&&\Delta \in \bf{\Delta} \;, || \Delta || \leq 1 \\ \end{align}$$

The Data
The matrices $$ A,M,N,Q $$.

The LMI:

\begin{align} \text{Find} \; &P>0:\\ \begin{bmatrix} AP+PA^T & PN^T \\ NP & 0\end{bmatrix} + \begin{bmatrix} M \Theta M^T & M \Theta Q^T \\ Q \Theta M^T & Q \Theta Q^T - \Theta \end{bmatrix} < 0\\ \end{align}$$

Conclusion:
$$ \begin{align} \text{The system above is quadratically stable if and only if there exists some } \Theta \in P \Theta \text{ and } P > 0 \text{ such that the LMI is feasible.} \end{align} $$

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

Related LMIs
Parametric, Norm-Bounded Uncertain System Quadratic Stability

Stability under Arbitrary Switching

Quadratic Stability Margins


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Stability under Arbitrary Switching

LMIs in Control/Stability Analysis/Continuous Time/Stability under Arbitrary Switching

Using the LMI below, find a P matrix that fits the constraints. If there exists one, then the system can switch between subsystems $$ A_1 $$ and $$ A_2 $$ arbitrarily and remain stable.

The System

\begin{align} \dot x(t) \in \{ A_1x(t),A_2x(t) \} \end{align}$$

The Data
The matrices $$ A_1 \in R^{n \times n} ,A_2 \in R^{n \times n} $$.

The LMI

\begin{align} \text{Find} \; &P>0:\\ A_1^TP+PA_1 < 0 \text{ and }\\ A_2^TP+PA_2 < 0 \end{align}$$

Conclusion
The switched system is stable under arbitrary switching if there exists some P > 0 such that the LMIs hold.

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

Related LMIs
Parametric, Norm-Bounded Uncertain System Quadratic Stability

Stability of Structured, Norm-Bounded Uncertainty

Quadratic Stability Margins


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Quadratic Stability Margins

$$ \begin{align} \text{The quadratic stability margin of the system is defined as the largest } \alpha \geq 0 \text{ for which the system is quadratically stable.} \text{This LMI applies for systems with norm-bounded uncertainty.} \end{align} $$

The System

\begin{align} \dot x(t)&=Ax(t)+B_pp(t),&&p^Tp \leq \alpha ^2 x^TC_q^TC_qx\\ \end{align}$$

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

The Optimization Problem
$$ \text{Maximize } \beta = \alpha^2 \text{ subject to the LMI constraint.} $$

The LMI:

\begin{align} \text{Find} \; &P,\lambda, \text{ and } \beta = \alpha ^2:\\ \begin{bmatrix} A^TP+PA+ \beta \lambda C_q^TC_q & PB_p \\ B_p^TP & -\lambda I \end{bmatrix} < 0\\ \end{align}$$

Conclusion:
If there exists an $$ \alpha \geq 0 $$ then the system is quadratically stable, and the stability margin is the largest such $$ \alpha $$.

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

Related LMIs
Parametric, Norm-Bounded Uncertain System Quadratic Stability

Stability of Structured, Norm-Bounded Uncertainty

Stability under Arbitrary Switching


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Stability of Linear Delayed Differential Equations

The System

\begin{align} \dot x(t)&=Ax(t)+\sum_{i=1}^{L}A_ix(t-\tau_i),\\ \end{align}$$ where $$ x(t) \in \mathbb{R}^n $$ and $$ \tau_i >0$$.

The Data
The matrices $$ A,\{A_i.\tau_{i}\}_{i=1}^L $$.

The LMI:
Solve the following LMIP

\begin{align} &\text{Find} \{P \succ 0, P_1 \succ 0, \dots,P_L \succ 0\} :\\ &\quad  s.t. \begin{bmatrix} A^\top P +PA+\sum_{i=1}^L P_i & PA_1 & \dots & PA_L \\ A^\top_1P &  -P_1 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ A^\top_L P & 0 & \dots & -P_L \end{bmatrix} \prec 0,P_1 \succ 0, \dots, P_L \succ 0. \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/50fc71737b69f2cf57d15634f2f19d091bf37d02

Conclusion
The stability of the above linear delayed differential equation is proved, using Lyapunov functionals of the form $$ V(x,t)=x(t)^\top P x(t)+\sum_{i=1}^{L}\int_{0}^L x^\top(t-s)P_i x(t-s) \ ds $$, if the provided LMIP is feasible.

Remark
The techniques for proving stability of norm-bound LDIs [Boyd, ch.5] can also be used.


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/H infinity Norm for Affine Parametric Varying Systems

The System

\begin{align} \dot x(t)&=Ax(t)+B_w w(t),\\ z(t)&=C_z(\theta)x(t)+D_{zw}(\theta)w(t), \end{align}$$ where $$ C_z $$ and $$ D_{zw}$$ depend affinity on parameter $$ \theta \in \mathbb{R}^p $$.

The Data
The matrices $$ A,B_w,C_z(.),D_{zw}(.) $$.

The Optimization Problem:
Solve the following semi-definite program

\begin{align} &\min_{\{P \succ 0,\gamma \geq 0\}} \gamma \\ &\quad  s.t. \begin{bmatrix} A^\top P +PA & PB_w \\ B^\top_wP &  -\gamma^2 I \end{bmatrix}+ \begin{bmatrix} C^\top_z(\theta) \\ D^\top_{zw}(\theta) \end{bmatrix}\begin{bmatrix} C_z(\theta) & D_{zw}(\theta) \end{bmatrix}\preceq 0. \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/5462bc1dc441bc298d50a2c35075e9466eba8355

Conclusion
The value function of the above semi-definite program returns the $$ \mathcal{H}_{\infty}$$ norm of the system.

Remark
It is assumed that $$ A $$ is stable and $$ (A,B_w) $$ is controllable and the semi-infinite convex constraint $$ \|H_{\theta}(j\omega)\| < \gamma $$ for all $$ \omega \in \mathbb{R}$$, is converted into a finite-dimensional convex LMI.


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Entropy Bond for Affine Parametric Varying Systems

The System

\begin{align} \dot x(t)&=Ax(t)+B_w w(t),\\ z(t)&=C_z(\theta)x(t)+D_{zw}(\theta)w(t), \end{align}$$ where $$ C_z $$ and $$ D_{zw}$$ depend affinely on parameter $$ \theta \in \mathbb{R}^p $$.

The Data
The matrices $$ A,B_w,C_z(.),D_{zw}(.) $$.

The Optimization Problem:
Solve the following semi-definite program

\begin{align} &\min_{\{P \succ 0,\gamma^2,\lambda,\theta\}} \gamma^2 \\ &\quad  s.t.\quad D_{zw}(\theta)=0, \begin{bmatrix} A^\top P +PA & PB_w & C_z(\theta)^\top \\ B^\top_wP &  -\gamma^2 I & 0 \\ C_z(\theta) & 0 & -I \end{bmatrix}\preceq 0, \quad \rm{Tr}(B^\top_w P B_w) \leq \lambda. \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/02f31a2d7a22b2464dfe9212eb76409bda9439b1

Conclusion
The value function of the above semi-definite program returns a bound for $$ \gamma $$-entropy of the system, which is defined as

\begin{align} I_{\gamma}(H_{\theta}) \triangleq \begin{cases} \frac{-\gamma^2}{2\pi}\int_{-\infty}^{\infty}\log\det(I-\gamma^2 H_{\theta}(i\omega)H_{\theta}(i\omega)^*)d\omega, \quad \text{if} \ \|H_{\theta}\|_{\infty} < \gamma \\ \infty, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text{otherwise.}\end{cases} \end{align} $$

Remark
When it is finite, $$ I_{\gamma}(H_{\theta}) $$ is given by $$ \rm{Tr}(B^\top_wPB_w)$$ where $$ P$$, is asymmetric matrix with the smallest possible maximum singular value among all solutions of a Riccati equation.


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Dissipativity of Affine Parametric Varying Systems

The System

\begin{align} \dot x(t)&=Ax(t)+B_w w(t),\\ z(t)&=C_z(\theta)x(t)+D_{zw}(\theta)w(t), \end{align}$$ where $$ C_z $$ and $$ D_{zw}$$ depend affinely on parameter $$ \theta \in \mathbb{R}^p $$.

The Data
The matrices $$ A,B_w,C_z(.),D_{zw}(.) $$.

The Optimization Problem:
Solve the following semi-definite program

\begin{align} &\min_{\{P \succ 0,\gamma,\theta\}} \gamma \\ &\quad  s.t.\quad \begin{bmatrix} A^\top P +PA & PB_w- C_z(\theta)^\top \\ B^\top_wP-C_z(\theta) &  2\gamma I-D_{zw}(\theta)-D_{zw}(\theta)^\top \end{bmatrix}\preceq 0. \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/b6cd6b81f75be4a2052ba3fa76cad1a2f9c49caa

Conclusion
The dissipativity of $$ H_{\theta}$$ (see [Boyd,eq:6.59]) exceeds $$ {\gamma}$$ if and only if the above LMI holds and the value function returns the minimum provable dissipativity.

Remark
It is worth noticing that passivity corresponds to zero dissipativity.


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Hankel Norm of Affine Parameter Varying Systems

The System

\begin{align} \dot x(t)&=Ax(t)+B_w w(t),\\ z(t)&=C_z(\theta)x(t)+D_{zw}(\theta)w(t), \end{align}$$ where $$ C_z $$ and $$ D_{zw}$$ depend affinely on parameter $$ \theta \in \mathbb{R}^p $$.

The Data
The matrices $$ A,B_w,C_z(.),D_{zw}(.) $$.

The Optimization Problem:
Solve the following semi-definite program

\begin{align} &\min_{\{Q \succeq 0,\gamma^2,\theta\}} \gamma^2 \\ &\quad  s.t.\quad D_{zw}(\theta)=0, \quad  A^\top Q +QA + C_z(\theta)C_z(\theta)\preceq 0,\quad \gamma^2 I-W^{1/2}_c Q W^{1/2}_c \succeq 0, \end{align}$$ where $$ W_c$$ is the controllability Gramian, i.e., $$ W_c \triangleq \int_{0}^{\infty} e^{At}B_wB^\top_w e^{A^\top t}dt $$.

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/0faedcdd9fba92bc27a318d80159c04a0b342f35

Conclusion
The Hanakel norm (i.e., the square root of the maximum eigenvalue) of $$ H_{\theta}$$ is less than $$ {\gamma}$$ if and only if the above LMI holds and the value function returns the maximum provable Hankel norm.

Remark
$$ D_{zw}$$ is assumed to be zero.


text:
LMIs in Control/Click here to continue/LMIs in system and stability Theory/Positive Orthant Stabilizability

Positive Orthant Stabilizability

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

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

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

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

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

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

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

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

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

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

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

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

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


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/LMI For Stabilization Condition for Systems With Unsymmetrical Saturated Control

LMI For Stabilization Condition for Systems With Unsymmetrical Saturated Control

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

The System

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

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

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

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

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

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

The LMI: The Stabilization Feasibility Condition

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

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

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

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


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/LMI Condition For Exponential Stability of Linear Systems With Interval Time-Varying Delays

LMI Condition For Exponential Stability of Linear Systems With Interval Time-Varying Delays

For systems experiencing time-varying delays where the delays are bounded, the feasibility LMI in this section can be used to determine if the system is $$ \alpha $$-exponentially stable.

The System

\begin{align} \dot x(t)&=Ax(t)+Dx(t-h(t)), & t \in \R^+, \\ x(t)&=\phi(t), & t \in [-h_2,0], \end{align}$$

where $$ x(t) \in \R^n $$ is the state, $$ A,D \in \R^{n \times n} $$ are the matrices of delay dynamics, and $$ \phi(t) \in \R^n $$ is the initial function with norm $$ \| \phi \| = sup_{-\bar{h} \leq t \leq 0} \{ \| \phi (t) \|, \| \dot{\phi}(t) \| \} $$ and it is continuously differentiable function on $$ [-h_2, 0] $$. The tyime-varying delay function $$ h(t) $$ satisfies:

0 \leq h_1 \leq h(t) \leq h_2, t \in \R^+, $$

The Data
The matrices $$ (A,D) $$ are known, as well as the bounds $$ (h_1,h_2) $$ of the time-varying delay.

The Optimization Problem
For a given $$ \alpha > 0 $$, the zero solution of the system described above is $$ \alpha $$-exponentially stable if there exists a positive number $$ N > 0 $$ such that every solution $$ x(t,\phi) $$ satisfies the following condition:

$$ \|x(t,\phi)\| \leq Ne^{-\alpha t} \| \phi \|, \forall t \in \R^+ $$

The LMI: $$ \alpha $$-Stability Condition
The following feasibility LMI can be used to check if the system is $$ \alpha $$-exponentially stable or not for a given $$ \alpha > 0 $$:

\begin{align} & \text{Find} \; P,Q,R,U,S_i, \text{ where } i = 1,2,...,5 :\\ & \quad \quad \begin{bmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ *     & M_{22} &    0   & M_{24} & S_2    \\ *     &   *    & M_{33} & M_{34} & S_3    \\ *     &   *    &    *   & M_{44} & S_4 - S_5D \\ *     &   *    &    *   &   *    & M_{55} \end{bmatrix}<0 \\ & \text{where:} \\ & \quad \quad M_{11} = A^\top P + PA + 2\alpha P - (e^{-2\alpha h_1} + e^{-2\alpha h_2})R + 0.5S_1(I-A) + 0.5(I-A^\top)S^\top_1 + 2Q, \\ & \quad \quad M_{12} = e^{-2\alpha h_1}R - S_2A, \quad M_{13} = e^{-2\alpha h_2}R - S_3A, \\ & \quad \quad M_{14} = PD - S_1D -S_4A, \quad M_{15} = S_1 - S_5A, \\ & \quad \quad M_{22} = -e^{-2\alpha h_1}(Q+R), \quad M_{24} = S_2D + e^{-2 \alpha h_2}U, \\ & \quad \quad M_{33} = -e^{-2\alpha h_1}(Q+R+U), \quad M_{34} = -S_3D + e^{-2 \alpha h_2}U, \\ & \quad \quad M_{44} = 0.5(S_4D + D^\top S_4^\top) - e^{-2\alpha h_2}U, \\ & \quad \quad M_{55} = S_5 + S_5^\top + (h_1^2 + h_2^2)R + (h_2 - h_1)^2U, \end{align}$$

The above LMI can be combined with the bisection method to find $$ \alpha $$.

Conclusion:
For systems with time-varying delays with intervals, the LMI in this section can be used to check if the system is exponentially stable with a certain $$ \alpha $$. The bisection algorithm can be additionally used to compute $$ \alpha $$.

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

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


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Polytopic Quadratic Stability

An important result to determine the stability of the system with uncertainties

The System:
Consider the system with Affine Time-Varying uncertainty (No input)

\begin{align} \dot x(t)&=(A_0+\Delta A(t))x(t)\\ \end{align}$$

where

\begin{align} \Delta A(t) = A_1 \delta_1 (t)+....+A_k \delta_k (t)\\ \end{align}$$ where $$

\delta_i (t) $$ lies in either the intervals

\begin{align} \delta_i \in [\delta_i^-,\delta_i^+] \\ \end{align} $$ or the simplex

\begin{align} \delta(t) \in { \delta : \Sigma \alpha_i = 1, \alpha \geq 0 } \end{align} $$

where $$x\in\mathbb{R}^{m}$$ and $$A\in\mathbb{R}^{mxm}$$

The Data
The matrix A and $$\Delta_{A(t)}$$ are known

The Optimization
The Definitions: Quadratic Stability for Dynamic Uncertainty

The system

\begin{align} \dot x(t)&=(A_0+\Delta A(t))x(t)\\ \end{align}$$ is Quadraticallly Stable over $$ \Delta $$ if there exists a P > 0

Theorem $$ (A+\Delta,\boldsymbol{\Delta}) $$ is quadratically stable over $$ \boldsymbol{\Delta} :=Co(A_1,...,A_k) $$ if and only if there exists a P > 0 such that

\begin{align} (A+A_i)^TP + P(A+A_i) < 0 \quad

for \quad all \quad i = 1,....,k \end{align} $$ The theorem says the LMI only needs to hold at the EXTREMAL POINTS or VERTICES of the polytope.
 * Quadratic Stability MUST be expressed as an LMI

The LMI

\begin{align} (A+\Delta)^TP + P(A+\Delta ) < 0 \quad for \quad all \quad

\Delta \in \boldsymbol{\Delta} \end{align} $$

Conclusion:
Quadratic Stability Implies Stability of trajectories for any $$ \Delta $$ with $$ \Delta \in \boldsymbol{\Delta} $$ for all $$ t \geq 0 $$ Quadratic Stability is CONSERVATIVE. There are Stable System which are not Quadratically stable. Quadratic Stability is sometimes referred to as an "infinite-dimensional LMI"
 * Meaning it represents an infinite number of LMI constraints.
 * One for each possible value $$ \Delta $$ with $$ \Delta \in \boldsymbol{\Delta} $$
 * Also called a parameterized LMI
 * Such LMIs are not tractable.
 * For polytopic sets, the LMI can be made finite.

Implementation
A link to implementation of the LMI https://github.com/JalpeshBhadra/LMI/blob/master/polytopicstability.m

Related LMIs

 * Parametric Norm Bounded Uncertain System Quadratic Stability


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Optimization Over Affine Family of Linear Systems

Optimization over an Affine Family of Linear Systems
Presented in this page is a general framework for optimizing various convex functionals for a system which depends affinely, or linearly, on a parameter using linear matrix inequalities. The optimization problem presented on this page generalizes an LMI which can be applied to various problems within linear systems and control. Some examples of these applications are finding the $$ H_{2} $$ and $$ H_{\infty} $$ norms, entropy, dissipativity, and the Hankel norm of an affinely parametric system.

The System
Consider a family of linear systems     $$ \dot x = Ax + B_{w}w, $$      $$ z = C_{z}({\theta})x + D_{zw}({\theta})w $$    with state space realization $$ (A,B_{w},C_{z},D_{zw}) $$ where $$ C_{z} $$ and $$D_{zw}$$ depend affinely on the parameter $$ {\theta}\in\mathbb{R}^{p} $$.

We assume $$ A $$ is stable and $$ (A,B_{w}) $$ is controllable.

The transfer function, $$ H_{\theta}(s) \widehat{=} C_{z}({\theta})(sI-A)^{-1}B_{w}+D_{zw}({\theta}) $$ depends affinely on $$ {\theta} $$.

The Data
The transfer function $$H$$, and system matrices $$A$$, $$B_{w}$$, $$C_{z}$$, $$ D_{zw} $$ are known. $$ {\varphi} $$ represents the convex functionals, $${\alpha} $$ and $${\psi}$$ represent some auxiliary variables dependent on the problem being solved.

The LMI:Generalized Optimization for Affine Linear Systems
Several control theory problems, mentioned earlier, take the following form: minimize $${\psi}_{0}(H_{\theta}) $$ subject to $${\psi}_{i}(H_{\theta}) < {\alpha}_{i}, i = 1,. . ., p $$

Problems of this nature can be formulated as an LMI by representing $${\psi}_{i}(H_{\theta}) < {\alpha}_{i} $$ as an LMI in $$ {\theta},{\alpha}_{i}, $$ and possibly $${\psi}_{i}$$ such that $$ F_{i}({\theta},{\alpha}_{i},{\psi}_{i}) > 0 $$

Thus, the general optimization problem to be applied to an affine family of linear systems is as follows: minimize $$ {\alpha}_{0} $$ subject to $$ F_{i}({\theta},{\alpha}_{i},{\psi}_{i}) > 0, i = 0,1,....,P $$

Conclusion:
The LMI for this generalized optimization problem may be extended to various convex functionals for affine parametric systems. For extensions of this LMI, see the related LMIs section.

Implementation
Implementation of LMI's of this form require Yalmip and a linear solver such as Sedumi or SDPT3.

$ H_{\infty} $ Norm for Affine Parametric Systems - MATLAB code for an extension of this generalized LMI.

Entropy Bond for Affine Parametric Systems - MATLAB code for an extension of this generalized LMI.

LMI can be applied to other extensions in stability and controller analysis. Please see the related LMI pages in the section below.

Related LMIs
$ H_{\infty} $ Norm for Affine Parametric Systems

Entropy Bond for Affine Parametric Systems

Dissipativity for Affine Parametric Systems

for Affine Parametric Systems

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


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Hurwitz Stabilizability

This section studies the stabilizability properties of the control systems.

The System
Given a state-space representation of a linear system

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

Where $$ \rho $$ represents the differential operator ( when the system is continuous-time) or one-step forward shift operator ( Discrete-Time system). $$ x \in \mathbb{R}^{n}, y \in \mathbb{R}^{m}, u \in \mathbb{R}^{r} $$ are the state, output and input vectors respectively.

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

Definition
The system, or the matrix pair $$ (A,B) $$ is Hurwitz Stabilizable if there exists a real matrix $$ K $$ such that $$ (A+BK) $$ is Hurwitz Stable. The condition for Hurwitz Stabilizability of a given matrix pair (A,B) is given by the PBH criterion:

The PBH criterion shows that the system is Hurwitz stabilizable if all uncontrollable modes are Hurwitz stable.

LMI Condition
The system, or matrix pair $$ (A,B) $$ is Hurwitz stabilizable if and only if there exists symmetric positive definite matrix $$ P $$ and $$ W $$  such that:

Following definition of Hurwitz Stabilizability and Lyapunov Stability theory, the PBH criterion is true if and only if, a matrix $$ K $$ and a matrix $$ P > 0 $$ satisfying:

Letting

Putting (4) in (3) gives us (2).

Implementation
This implementation requires Yalmip and Mosek.
 * https://github.com/ShenoyVaradaraya/LMI--master/blob/main/Hurwitz_Stabilizability.m

Conclusion
Compared with the second rank condition, LMI has a computational advantage while also maintaining numerical reliability.


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Quadratic Hurwitz Stabilization for Polytopic Systems

This section studies the Quadratic Hurwitz stabilization for polytopic systems.

The System
Given a state-space representation of a linear system

LMI Condition
With $$ \Delta = \Delta_{p} $$, the quadratic Hurwitz Stabilization problem has a solution if and only if there exists a symmetric positive definite matrix $$ P $$ and a matrix $$ W $$ satisfying the below LMI :

In this case, a solution to the problem is given by

Conclusion
Stability of a system does not guarantee quadratic stability. Since quadratic stability can represent infinite LMI constraints, it is not tractable. Therefore, to make it feasible and tractable, polytopic sets are helpful.


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Transient State Bound for Non-Autonomous LTI Systems

The System
For a continuous-time LTI system with a state-space representation of:

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

where $$A \in \mathbb{R}^{n x n}$$, $$B \in \mathbb{R}^{n x m}                               $$ and x(0) = x0,

the transient bound can be evaluated with the following LMI.

The Data
$$A \in \mathbb{R}^{n x n}$$, $$B \in \mathbb{R}^{n x m}                               $$ and x(0) = x0.

The LMI:
The Euclidean norm of the state satisfies:

$$\lVert x(T) \rVert _2^2 \leq \gamma^2 (\lVert x_0 \rVert _2^2 + \lVert u \rVert _{2T}^2), \forall T \in \mathbb{R}_{\geq 0}$$

if there exists $$P \in \mathbb{S}^{n} $$ and $$\gamma \in \mathbb{R}_{> 0}$$, where P > 0, such that:

 * $$P - \gamma 1 \leq 0, $$
 * $$\begin{bmatrix} P & 1 \\ * & \gamma 1 \end{bmatrix} \geq 0, $$
 * $$\begin{bmatrix} PA + A^T P & PB \\ * & -\gamma 1 \end{bmatrix} \leq 0 . $$

if x0 = 0 and u is a unit-energy input ($$\lVert u \rVert _{2T} \leq 1, \forall T \in \mathbb{R} _{\geq 0} $$), then the above LMIs ensure that $$\lVert x(T) \rVert _{2} \leq  \gamma , \forall T \in \mathbb{R} _{\geq 0} $$

Conclusion
By using this LMI the transient state bound can be analyzed for a given non-autonomous LTI system.

Implementation
The LMI given above can be implemented and solved using a tool such as YALMIP, along with an LMI solver such as MOSEK.

Related LMIs

 * Output Energy Bound for Autonomous LTI Systems


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Controller D-Stability

In some control problems, people are still interested in designing a controller whose poles are located in specific regions of a complex plane, also known as D-stability.

The System
Suppose we were given the continuous-time system

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

whose stability was not known. Then the controller that simultaneously stabilizes the above system while ensuring that the poles are at their desired location can be achieved by the controller $$u(t)=Kx(t)$$.

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

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

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

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

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

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

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

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

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

This then allows us to modify our inequality constraints as:

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

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

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

The LMI: The Continuous-Time Controller D-Stability
Keeping the above inequalities in mind, we observe the following:

Suppose there now exists a symmetric matrix $$X>0$$ and matrix $$Z$$, we can now determine the controller the following LMIs:

\begin{align} \begin{bmatrix} -rP&&AP+BZ\\(AP+BZ)^T&&-rP\end{bmatrix}&<0\\ AP+BZ+(AP+BZ)^T +2\alpha P&<0, and\\ \begin{bmatrix} AP+BZ+(AP+BZ)^T&&c(AP+BZ-(AP+BZ)^T)\\c((AP+BZ)^T -(AP+BZ))&&AP+BZ+(AP+BZ)^T\end{bmatrix}&<0 \end{align}$$

where the resulting controller matrix is $$K=ZP^{-1}$$.

Conclusion:
Given the resulting controller $$K$$, we can now determine that the pole locations $$z\in\mathbb{C}$$ of $$A+BK$$ satisfies the inequality constraints $$|x|{\leq}r$$, $$Re(x){\leq}-{\alpha}$$ and $$z+z^{*}{\leq}-c|z-z^{*}|$$.

Implementation

 * CodeOcean Link - An implementation of the LMI.

Related LMIs

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


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Observer D-stability

This is a WIP Page.

The System

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

The Data
In order to properly define the acceptable region of the poles in the complex plane, we need the following three pieces of data: rise time ($$t_r$$), settling time ($$t_s$$), and percent overshoot ($$M_p$$). From this, we have to then define the acceptable region in the complex plane that the poles can lie on using the following inequality constraints:

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

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

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

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

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

This then allows us to modify our inequality constraints as:

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

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

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

The Optimization Problem
A description of the Problem to be solved, if appropriate.

The LMI: The Observer D-Stability
Title and mathematical description of the LMI formulation.

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

Conclusion:
Interpretation of the results of the LMI

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

Related LMIs

 * ../Controller D-Stability/


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-Time Lyapunov Stability

Discrete-Time Lyapunov Stability

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

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

The System
Discrete-Time System

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

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

The Optimization Problem
The following feasibility problem should be optimized:

Find P obeying the LMI constraints.

The LMI:
Discrete-Time Bounded Real Lemma

The LMI formulation

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

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

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

Related LMIs
Continuous_Time_Lyapunov_Inequality - Lyapunov_Inequality


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/LMI for Schur Stabilization

LMI for Schur Stabilization

Similar to the stability of continuous-time systems, one can analyze the stability of discrete-time systems. A discrete-time system is said to be stable if all roots of its characteristic equation lie in the open unit disk. This provides a condition for the stability of discrete-time linear systems and a linear time-invariant system with this property is called a Schur stable system.

The System
We consider the following system:

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

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

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

The state feedback control law is defined as follows:

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

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

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

The Data
The matrices $$ A $$ and $$B $$ are given.

We define the scalar as $$ \gamma $$ with the range of $$ 0 < \gamma \leq 1 $$.

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

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

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

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

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

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

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

$$ \begin{align} & \text{min} \quad \gamma\\ & \text{s.t.} \quad \begin{bmatrix} -\gamma I & (A + BK) \\ (A + BK)^{T} & -\gamma I\end{bmatrix} < 0\\ \end{align}$$

Conclusion:
After solving the LMI problem, we obtain the controller gain $$K$$ and the minimized parameter $$\gamma$$. This problem is a special case of Intensive Disk Region Design (page 230 in [1]). This problem may not have a solution even when the system is stabilizable. In other words, once there exists a solution, the solution is robust in the sense that when there are parameter perturbations, the closed-loop system's eigenvalues are not easy to go outside of a circle region within the unit circle [1].

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 Hurwitz stability

Return to Main Page
LMIs in Control/Tools


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/L2-Gain of Systems with Multiplicative noise

The System

\begin{align} x(k+1)&=Ax(k)+B_ww(k)+\sum_{i=1}^{L}(A_ix(k)+B_{w,i}w(k))p_i(k),\quad x(0)=0,\\ z(k)&=C_zx(k)+D_{zw}w(k)+\sum_{i=1}^{L}(C_{z,i}x(k)+D_{zw,i}w(k))p_i(k), \end{align}$$ where $$ p(0), p(1),\dots $$, are independent, identically distributed random variables with $$ Ep(k) = 0, Ep(k)p^\top(k) = \Sigma = diag(\sigma_1,\dots,\sigma_L)$$ and $$x(0)$$ is independent of the process $$p$$.

The Data
The matrices $$ A,B_w,\{A_i.B_{w,i}\}_{i=1}^L,C_z,D_{zw},\{C_{z,i},D_{zw,i}\}_{i=1}^L,\{\sigma_i\}_{i=1}^L $$.

The LMI:

\begin{align} &\min_{\{P \succ 0, \gamma^2\}}\gamma^2 \\ &\quad  s.t. \begin{bmatrix} A & B_w \\ C_z &  D_{zw}\end{bmatrix}^\top \begin{bmatrix} P & 0 \\ 0 &  I \end{bmatrix} \begin{bmatrix} A & B_w \\ C_z &  D_{zw}\end{bmatrix}-\begin{bmatrix} P & 0 \\ 0 &  \gamma^2 I \end{bmatrix}+\sum_{i=1}^{L}\sigma^2_i\begin{bmatrix} A_i & B_{w,i} \\ C_{z,i} &  D_{zw,i}\end{bmatrix}^\top \begin{bmatrix} P & 0 \\ 0 &  I \end{bmatrix} \begin{bmatrix} A_i & B_{w,i} \\ C_{z,i} &  D_{zw,i}\end{bmatrix}^\top \preceq 0 \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/a34713575cd8ae9831cb5b7eb759d0fd45a8c37f

Conclusion
The optimal $$ \gamma $$ returns an upper bound on the $$ \mathcal{L}_2 $$ gain of the system. .

Remark
It is straightforward to apply scaling method [Boyd, sec.6.3.4] to obtain component-wise results.


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-Time Quadratic Stability

Discrete-Time Quadratic Stability
Stability is an important property, stability analysis is necessary for control theory. For robust control, this criterion is applicable for the uncertain discrete-time linear system. It is based on the Discrete Time Lyapunov Stability.

The System

\begin{align} x_{k+1}&=A_d(\alpha) x_k \\ Where: \\ & A_d(\alpha) = A_{d} + \Delta A_d (\delta(t)) \\ & \Delta A_d (\delta(t)) = \sum_{k = 1}^n \delta_k(t) A_{d;k} \in \mathrm{R}^{n\times n} \\ & \delta(t) = [\delta_1(t), ... \delta_n(t)] - \text{The set of perturbation parameters} \\ & \delta(t) \in \mathrm{R} \;\;\; A_{d;i} \in \mathrm{R}^{n\times n} \end{align} $$

The Data
The matrices $$ A \in R^{n\times n}\; A_{d;i}\in R^{n\times n} $$.

The Optimization Problem
The following feasibility problem should be solved:

\begin{align} \text{Find} \; &P >0:\\ & (A_{d;0} + \Delta A_d (\delta(t)) )^T P (A_{d;0} + \Delta A_d (\delta(t))) - P < 0 \text{ for all } \delta \end{align} $$

Where $$ P \in R^{n,n} $$.

In case of polytopic uncertainty:

\begin{align} \text{Find} \; &P >0:\\ & (A_{d;0} + A_{d;i})^T P (A_{d;0} + A_{d;i}) - P < 0 \text{ for all } i = 1,... n \end{align} $$

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

Implementation:

 * - Matlab implementation using the YALMIP framework and Mosek solver

Related LMIs:

 * - Discrete Time Stabilizability
 * Polytopic stability for continuous time case
 * Quadratic polytopic stabilization
 * Discrete Time Lyapunov Stability

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 * Main Page.


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Output Energy Bound for Discrete-Time Autonomous LTI Systems

Output Energy Bound for Autonomous LTI Systems

Autonomous systems are initialized under a given set of initial conditions, and then run without any additional inputs. It is useful to know ahead of time the bounds such a system will operate within. This analysis can be used to determine the upper bound on the output of a given autonomous LTI system.

The System
Consider the continuous-time, LTI autonomous system with state space representation

$$ \bold{\dot x} = \bold{Ax}$$,

$$ \bold{y}=\bold{Cx}$$,

where $$ \bold{A} \in \R^{n\times n}, \bold{C} \in \R^{p\times n}$$ and $$ \bold{x}(0)=\bold{x}_0.$$

Determining an Upper Bound
The output of this system will satisfy

$$ \sqrt{\int_{0}^{T} \bold{y^Ty}dt}=\left\vert\left\vert \bold{y} \right\vert\right\vert_{2T} \leq \gamma \left\vert\left\vert \bold{x}_0 \right\vert\right\vert_2 ,\forall T \in \R_{\geq0}$$

if there exists some matrix $$ \bold{P} \in \S^p$$and scalar $$ \gamma \in \R_{>0}$$ such that

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

$$ \bold{P} - \gamma\bold{I} \leq0$$,

$$ \begin{bmatrix} \bold{PA}+\bold{A^TP} & \bold{C^T} \\ \end{bmatrix} \leq0$$.
 * & -\gamma\bold{I}

Conclusion
Given an autonomous system with an initial operating condition, the parameter $$ \gamma$$ can be used to determine the largest feasible bound on the output of that system.


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LMIs in Control/Click here to continue/LMIs in system and stability Theory/Transient Bound for Discrete-Time Non-Autonomous LTI Systems

The System
For a discrete-time LTI system with a state-space representation of:

$$x_{k+1} = A_d x_k + B_d u_k, $$

where $$A_d \in \mathbb{R}^{n x n}$$, and $$B_d \in \mathbb{R}^{n x m}$$,

the transient bound can be analyzed with the LMI below.

The Data
$$A_d \in \mathbb{R}^{n x n}$$, and $$B_d \in \mathbb{R}^{n x m}$$

The LMI:
The Euclidean norm of the state satisfies:

$$\lVert x_k \rVert _2^2 \leq \gamma^2 (\lVert x_0 \rVert _2^2 + \lVert u \rVert _{2k}^2), \forall k \in \mathbb{Z}_{\geq 0}$$

if there exists some $$P \in \mathbb{S}^{n}$$ and $$\gamma \in \mathbb{R}_{> 0}$$, where P > 0 such that:

 * $$P - \gamma 1 \leq 0, $$
 * $$\begin{bmatrix} P & 1 \\ * & \gamma 1 \end{bmatrix} \geq 0, $$
 * $$\begin{bmatrix} A_d^T P A_d - P & A_d^T P B_d \\ * & B_d^T B_d- \gamma 1 \end{bmatrix} \leq 0. $$

if x0 = 0 and u is a unit-energy input ($$\lVert u \rVert _{2k} \leq 1, \forall k \in \mathbb{Z} _{\geq 0} $$), then the above LMIs ensure that $$\lVert x_k \rVert _{2} \leq  \gamma , \forall k \in \mathbb{Z} _{\geq 0} $$

Conclusion
By using this LMI the transient state bound can be analyzed for a given non-autonomous LTI system.

Implementation
The LMI given above can be implemented and solved using a tool such as YALMIP, along with an LMI solver such as MOSEK.

Related LMIs

 * Output Energy Bound for Discrete-Time Autonomous LTI Systems