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User:Margav06/sandbox/Click here to continue/Applications of Linear systems/Problem of Space Rendezvous and LMI Approaches

This is a Problem of Space Rendezvous and LMI Approaches

In Section 12.4 of their book LMIs in Control Systems: Analysis, Design, and Applications, Duan and Yu discuss the problem of space rendezvous and how it can be formulated into an LMI problem. Modeling and simulating space rendezvous is of importance because it is used for any cargo or passenger spacecraft traveling to and from earth-orbiting space stations and also for satellites servicing aging in-orbit satellites, and for potential missions to mine asteroids.

The System
Though Duan and Yu first mention space rendezvous in Example 7.14 of their book. In this example, they show that the relative orbital dynamic model of spacecraft rendezvous can be described by the famous Clohessy-Wiltshire equations.

\begin{align} m\ddot{r_x}-2m\omega_0\dot{r_y}-3m\omega_0^2r_x &= T_x + d_x \\ m\ddot{r_y}+2m\omega_0\dot{r_x} &= T_y + d_y \\ m\ddot{r_z}+m\omega_0^2r_z &= T_z + d_z\\ \end{align}$$

where
 * $$r_x,r_y,r_z$$ are the components of the relative position between chaser and target
 * $$\omega_0 = \pi/12$$ [rad/h] is the orbital angular velocity of the target satellite
 * $$m$$ is the mass of the chaser
 * $$T_i, (i=x,y,z)$$ is the i-th component of the control input force acting on the relative motion dynamics
 * $$d_i, (i=x,y,z)$$ is the i-th component of the external disturbance

The C-W equations give a first-order approximation of the chaser's motion in a target-centered coordinate system and is often used in planning space rendezvous problems (ISS, Salyut, and Tiangong space stations are just some examples.)

With appropriate definitions of states and variables the dynamic equations of motion for space-rendezvous can be converted into standard state-space form for LMI optimization as follows:

\begin{align} \dot{x} &= Ax + B_1u+B_2d \\ y &= Cx \\ \end{align}$$

where the vectors in the above state-space representation are defined as follows:

\begin{align} x = \begin{bmatrix} r_x & r_y & r_z & \dot{r_x} & \dot{r_y} & \dot{r_z} \\ \end{bmatrix}^T\\ \\ y = \begin{bmatrix} r_z & r_y & r_z\end{bmatrix}^T\\ \\ u = \begin{bmatrix} T_x & T_y & T_z \end{bmatrix}^T\\ \\ d = \begin{bmatrix} d_x & d_y & d_z \end{bmatrix}^T\\ \end{align}$$

and the matrices in the above state-space representation are defined as follows:

\begin{align} A = \begin{bmatrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\               0 & 0 & 0 & 0 & 0 & 1 \\                3\omega_0^2  & 0 & 0 & 0 & 2\omega_0^2 & 0 \\ 0 & 0 & 0 & -2\omega_0 & 0 & 0 \\ 0 & 0 & -\omega_0^2 & 0 & 0 & 0 \end{bmatrix}\\ \\ B_1 = B_2 = \begin{bmatrix} 0_{3x3} \\ I_3 \\ \end{bmatrix}\\ \\ C = \begin{bmatrix} I_{3} & 0_{3x3}\\ \end{bmatrix}\\ \\ \end{align}$$

The Data
The data required are the mass properties of both the target and chaser vehicles for space rendezvous. Also required is the orbital angular velocities of the target and chasers and measurements of relative kinematics between the two.

The Optimization Problem
The optimization problem is trying to attenuate the disturbance to output transfer function using either the H-inf or H2 norm.

 * Objective: Hinf or H2 norm
 * Variables: Controller Gains
 * Constraints: Relative Dynamics/Kinematics between Chaser and Target in Orbit

The LMI: Space Rendezvous LMI Optimization
The space rendezvous problem can be approached with either H-inf or H-2 optimization formulations. Both formulations can achieve closed-loop stability which ensures that rendezvous occurs because the relative distance between target and chaser eventually approaches zero. The LMIs for the H-inf and H2 optimization problem are shown below which are easily solvable because the matrices for the space rendezvous problem are available above in standard form.

Duan and Yu approach the $$H_{\infty}$$. The minimum attenuation level from disturbance to output can be found by solving the following LMI optimization problem.

\begin{align} \\ &min \gamma_{\infty} \\ \\ &\text{s.t. } X > 0 \\ \\ &\begin{bmatrix} (AX+B_1W)^T+AX+B_1W & B_2 & (C_1X+D_1W)^T \\ B_2^T & -\gamma_{H\infty} I & D_2^T \\ C_1X+D_1W & D_2 & -\gamma_{H\infty} I \end{bmatrix} < 0 \\

\end{align}$$

which is the same as Theorem 8.1 in Duan and Yu's Book, the solution to the $$ H_{\infty} $$ problem.

Conclusion:
The LMI for Space Rendezvous is a useful and interesting method to model and simulate practical problems in spacecraft engineering. Space Rendezvous usually requires very good vision-based navigation or an exceptional human operator that can close the gap for final mating of the two docking adapters.

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

Related LMIs
Links to other closely-related LMIs

 * Optimal_Output_Feedback_Hinf_LMI
 * H2OptimalOutputFeedback


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User:Margav06/sandbox/Click here to continue/Applications of Linear systems/LMI for Attitude Control of BTT Missiles, Roll Channel

LMI for Attitude Control of BTT Missles, Roll Channel

The dynamic model of a bank-to-burn (BTT) missile can be simplified for practical application. The dynamic model for a BTT missile is given by the same model used for nonrotating missiles. However, in this case we can assume that the missile is axis-symmetrically design, and thus Jx = Jy. We assume that the roll channel is independent of the pitch and yaw channels.

The System
The state-space representation for the pitch channel can be written as follows:

$$ \begin{align} \dot{x}(t) &= A(t)x(t) + B_{1}(t)u(t) + B_{2}(t)d(t) \\ y(t) &= C(t)x(t) + D_{1}(t)u(t) + D_{2}(t)d(t) \end{align} $$

where $$ x(t) = [\omega_{x} \quad \phi]^{\text{T}}$$ is the state variable, $$ u(t) = \delta_{x} $$ is the control input, and $$ y = \phi $$ is the output. The parameters $$ \omega_{x} $$, $$ \phi $$, and $$ \delta_{x} $$ refer to the roll angular velocity, the roll angle, and the aileron deflection, respectively.

The Data
The system can be described as:

$$ \begin{align} \begin{bmatrix} \dot{\omega}_x(t) \\ \dot{\phi}(t) \end{bmatrix} = \begin{bmatrix} -c_1(t) & 0 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} \omega_x(t) \\ \phi(t) \end{bmatrix} + \begin{bmatrix} -c_3(t) \\ 0 \end{bmatrix} \delta_x(t) \end{align} $$

$$ \begin{align} y(t) = \phi(t) \end{align} $$

which can be represented in state space form as:

$$ \begin{align} A(t) = \begin{bmatrix} -c_{1}(t) & 0 \\ 1 & 0 \end{bmatrix} \end{align} $$

$$ \begin{align} B_{1}(t) = \begin{bmatrix} -c_{3}(t) \\ 0 \end{bmatrix}, \quad B_{2}(t) = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \end{align} $$

$$ \begin{align} C(t) = \begin{bmatrix} 0 & 1 \end{bmatrix} \end{align} $$

$$ \begin{align} D_{1}(t) = 0, \quad D_{2}(t) = 0 \end{align} $$

where $$ c_{1}(t) $$ and $$ c_{3}(t) $$ are the system parameters.

The Optimization Problem
The optimization problem is to find a state feedback control law $$ u=Kx$$ such that:

1. The closed-loop system:

$$ \begin{align} \dot{x} &= (A+B_{1}K)x + B_{2}d \\ z &= (C + D_{1}K)x + D_{2}d \end{align} $$

is stable.

2. The $$ H_{\infty} $$ norm of the transfer function:

$$G_{zd}(s) = (C + D_{1}K)(sI-(A+B_{1}K))^{-1}B_{2} + D_{2}$$

is less than a positive scalar value, $$ \gamma $$. Thus:

$$ ||G_{zd}(s)||_{\infty} < \gamma $$

The LMI: LMI for BTT missile attitude control
Using Theorem 8.1 in [1], the problem can be equivalently expressed in the following form:

$$ \begin{align} &\text{min} \quad \gamma \\ &\text{s.t.} \quad X > 0 \\ &\begin{bmatrix} (AX+B_{1}W)^{T} + AX+B_{1}W & B_{2} & (CX+D_{1}W)^{T} \\ B_{2}^{T} & -\gamma I & D_{2}^{T} \\ CX + D_{1}W & D_{2} & -\gamma I \end{bmatrix} < 0 \end{align}$$

Conclusion:
As mentioned, the aim is to attenuate the disturbance on the performance of the missile. The parameter $$ \gamma $$ is the disturbance attenuation level. However, it should be noted that this model for the roll channel for a BTT missile is very simple and easy to handle, there is no disturbance to attenuate. This problem is presented here for completeness when used in a full BTT missile model along with the pitch/yaw channels. When the matrices $$ W $$ and $$ X $$ are determined in the optimization problem, the controller gain matrix can be computed by:

$$ K = WX^{-1} $$

Implementation
A link to MATLAB code for the problem in the GitHub repository:

https://github.com/scarris8/LMI-for-BTT-Missile-Roll-Control

Related LMIs
LMI for Attitude Control of Nonrotating Missles, Pitch Channel

LMI for Attitude Control of Nonrotating Missles, Yaw/Roll Channel

Return to Main Page
LMIs in Control/Tools

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


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User:Margav06/sandbox/Click here to continue/Notation

Notations

In this document, matrices are denoted by boldface letters (e.g.,$$ A\in \mathbb{R}^{n\times n}$$), column matrices are denoted by lowercase boldface letters (e.g.,$$ x \in \mathbb{R} ^{n}$$), scalars are denoted by simple letters (e.g.,γ $$ \in \mathbb{R}$$), and operators are denoted by script letters (e.g., $$\mathcal{G} $$: $$ \mathcal{L _{2e}}$$ →$$\mathcal{L_{2e}}$$. The set of $$ n $$ by $$m$$ real matrices is denoted as $$\mathbb{R}^{n\times m}$$, the set of $$ n $$ by $$m$$ complex matrices is denoted as $$\mathbb{C}^{n\times m}$$, and the set of $$ n x n $$ symmetric matrices is denoted as $$ \mathbb{S} ^n$$. The identity matrix is written as 1 and a matrix filled with zeros is written as 0. The dimensions of 1 and 0 are specified when necessary. Repeated blocks within symmetric matrices are replaced by ∗ for brevity and clarity. The conjugate transpose or Hermitian transpose of the matrix $$ V \in \mathbb{C} ^{n\times m}$$ is denoted by $$V^H$$. The notation He{·} is used as a shorthand in situations with limited space, where He{·} = (·) + (·) $$^H $$. The real and imaginary parts of the complex number $$z \in \mathbb{C}$$ are denoted as Re($$z$$) and Im($$z$$), respectively. The Kroenecker product of two matrices is denoted by ⊗.

Consider the square matrix $$ A \in \mathbb{R} ^{n\times n}$$. The eigenvalues of $$A$$ are denoted by $$ \lambda_i (A), i = 1, 2,. . ., n$$. The matrix A is Hurwitz if all of its eigenvalues are in the open left-half complex plane (i.e., Re $$ \lambda_i (A), i = 1, 2, . . ., n$$ ). A matrix is Schur if all of its eigenvalues are strictly within a unit disk centered at the origin of the complex plane (i.e., $$|\lambda_i (A)| < 1, i = 1, . . ., n)$$. If $$A \in \mathbb{S}^n $$, then the minimum eigenvalue of A is denoted by $$\lambda(A)$$ and its maximum eigenvalue is denoted by $$ \bar{\lambda} (A) $$.

Consider the matrix B $$\in \mathbb{R} ^{n\times m}$$. The minimum singular value of B is denoted by $$ \underline{\sigma} $$ (B) and its maximum singular value is denoted by $$ \bar{\sigma} $$(B). The range and nullspace of B are denoted by $$ \mathbb{R} $$(B) and $$ \mathbb{N} $$(B), respectively. The Frobenius norm of B is ||B|| = $$ \sqrt{tr(B ^H B)}$$.

A state-space realization of the continuous-time linear time-invariant (LTI) system

$$ \dot{x}(t) = Ax(t) + Bu(t)$$,

$$ y(t) = Cx(t) + Du(t),$$. is often written compactly as (A, B,C,D) in this document. The argument of time is often omitted in continuous-time state-space realizations, unless needed to prevent ambiguity. A state-space realization of the discrete-time LTI system

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

$$y_k = C_d x_k + D_d u_k,$$

is often written compactly as $$(A_d, B_d,C_d,D_d)$$.

The $$\mathcal{H}$$∞ norm of the LTI system $$ \mathcal{G}$$ is denoted by ||$$ \mathcal{G} $$||∞ and the $$\mathcal{H}_2 $$ norm of $$\mathcal{G}$$ is denoted by
 * $$\mathcal{G}$$||$$_2$$.

The inner product spaces $$ \mathcal{L}_2 and \mathcal{L}_{2e}$$ for continuous-time signals are defined as follows.

$$ \{\mathcal{L}_2\} = \left\{x: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}^{n} | ||x||^2_2 = \int_{0}^\infty x^T(t)x(t)dt<\infty \right\}, $$

$$ \{\mathcal{L}_{2e}\} = \left\{x: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}^{n} | ||x||^2_{2T} = \int_{0}^T x^T(t)x(t)dt<\infty, \forall T \in \mathbb{R}_{\geq 0} \right\}. $$

The inner product sequence spaces ℓ2 and ℓ2e for discrete-time signals are defined as follows. $$ \{\mathcal{l}_2\} = \left\{x: \mathbb{Z}_{\geq 0} \rightarrow \mathbb{R}^{n} | ||x||^2_{2} = \sum_{k=0}^\infty x^T_k x_k <\infty, \right\}. $$ $$ \{\mathcal{l}_{2e}\} = \left\{x: \mathbb{Z}_{\geq 0} \rightarrow \mathbb{R}^{n} | ||x||^2_{2N} = \sum_{k=0}^N x^T_k x_k <\infty, \forall N \in \mathbb{Z}_{\geq 0} \right\}. $$


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Deduced LMI Conditions for Hinf Index

H-infinity Index Deduced LMI

Although the KYP Lemma, also known as the Bounded Real Lemma, is a basic condition to evaluate an upper bound on the H∞, the verification of the bound on the H∞-gain of the system can be done via the deduced condition.

The System
A state-space representation of a linear system as given below:

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

where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$ and $$w(t)\in \R^r$$ are the system state, output, and the disturbance vector respectively. The transfer function of such a system can be evaluated as:

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

The Data
Number of states n, number of outputs m and number of external noise channels r need to be known. Moreover, the system matrices A,B,C,D are also required to be known.

The Feasibility LMI
For an arbitrary $$ \gamma $$, the transfer function G(s) satisfies

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

if and only if there exists a symmetric matrix X > 0 and a matrix $$ \Omega .$$ such that:

\begin{align} \text{Find} \; X, \Omega :&\\ \Theta + \Phi^\top \Omega \Psi + \Psi^\top \Omega^\top \Phi < 0 \end{align}$$

where:

\begin{align}

\Theta &= \begin{bmatrix} 0 & X & 0 & 0 & 0 \\ X & -X & 0 & 0 & 0 \\ 0 & 0 & -\gamma I_m & 0 & 0\\ 0 & 0 & 0 & -X & 0 \\ 0 & 0 & 0 & 0 & \gamma I_r \end{bmatrix} \\ \Phi &= \begin{bmatrix} -I_n & A^\top & C^\top & I_n & 0 \\ 0 & B^\top & D^\top & 0 & -\gamma I_r\end{bmatrix} \\ \Psi &= \begin{bmatrix} I_n & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & I_r\end{bmatrix} \end{align}$$

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

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

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/Deduced_hinf_example.m

Related LMIs
Bounded Real Lemma


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Deduced LMI Conditions for H2 Index

H2 Index Deduced LMI

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

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

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

where $$x(t)\in \mathbb{R}^n$$, $$y(t)\in \mathbb{R}^m$$ and $$u(t)\in \mathbb{R}^r$$ are the system state, output, and the input vectors respectively. The transfer function of such a system can be evaluated as:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

https://github.com/yashgvd/ygovada

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


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

Dissipativity of Systems

The dissipativity of systems is associated with their supply function. In general, a linear system is dissipative if the accumulated sum (integration) of the supply function is non-negative over all the duration of $$ T \geq 0 $$.

The System
A state-space representation of a linear system as given below:

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

where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$ and $$u(t)\in \R^r$$ are the system state, output, and the input vector respectively. A, B, C and D are system coefficient matrices of appropriate dimensions. The control input u is restricted to be a piece-wise continuous vector function defined of $$[0,\infty)$$.

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

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

For such a system, a general quadratic supply function is defined as:

\begin{align} s(u,y) &= \begin{bmatrix} y & u \end{bmatrix} Q \begin{bmatrix} y \\ u \end{bmatrix} \end{align}$$

where Q is a real symmetric matrix of (m+r) dimensions. Q need not be either symmetric positive or negative definite.

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

The Feasibility LMI
The system $$ G(s) $$ defined can be evaluated to be dissipative with respect to a supply function $$ s(u,y) $$ iff there exist $$P \geq 0 $$ and a $$Q$$ (defining $$s(u,y)$$) such that the following is feasible:

\begin{align} \text{Find} \; & P, Q:\\ & P \geq 0 \\ \begin{bmatrix} A^\top P + PA & PB \\ B^\top P     & 0 \end{bmatrix} & - \begin{bmatrix} C & D \\ 0 & I \end{bmatrix}^\top Q \begin{bmatrix} C & D \\ 0 & I \end{bmatrix} \leq 0. \end{align}$$

Conclusion:
If there is a feasible solution to the aforementioned LMI, then there exists a supply function $$ s(u,y) $$ for which the system $$ G(s) $$ is dissipative. Since the assumption of the system being controllable is required for it to be dissipative, this check can be used of as a sufficient condition to check the controllability of the linear system, just like the feasibility for Lyapunov stability.

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/Dissipativity_example.m

Related LMIs
Continuous_Time_Lyapunov_Inequality


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Generalized H2 Norm

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

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

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

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

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

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

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

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

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

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

Implementation
This implementation requires Yalmip and Sedumi.

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

Related LMIs
LMI for System H_{2} Norm

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


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

This section deals with passivity of a system.

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

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

$$ x \in \mathbb{R}^{n}, y \in \mathbb{R}^{m}, u \in \mathbb{R}^{r} $$ are the state, output and input vectors respectively.

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

Definition
The linear system with the same number of input and output variables is called passive if

hold for any arbitrary $$ T\geq 0 $$, arbitrary input $$ u(t) $$, and the corresponding solution $$ y(t) $$ of the system with $$ x(0) = 0 $$. In addition, the transfer function matrix

of system is called is positive real if it is square and satisfies

LMI Condition
Let the linear system be controllable. Then, the system is passive if an only if there exists $$ P > 0 $$ such that

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

Conclusion
Thus, it is seen that passivity and positive-realness describe the same property of a linear system, one gives the time-domain feature and the other provides frequency-domain feature of this property.


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

This section deals with passivity of a system.

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

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

$$ x \in \mathbb{R}^{n}, y \in \mathbb{R}^{m}, u \in \mathbb{R}^{r} $$ are the state, output and input vectors respectively.

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

Definition
The linear system with the same number of input and output variables is called passive if

hold for any arbitrary $$ T\geq 0 $$, arbitrary input $$ u(t) $$, and the corresponding solution $$ y(t) $$ of the system with $$ x(0) = 0 $$. In addition, the transfer function matrix

of system is called is positive real if it is square and satisfies

LMI Condition
Let the linear system be controllable. Then, the system is passive if an only if there exists $$ P > 0 $$ such that

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

Conclusion
Thus, it is seen that passivity and positive-realness describe the same property of a linear system, one gives the time-domain feature and the other provides frequency-domain feature of this property.


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Non-Expansivity and boundedness Realness

This section studies the non-expansivity and bounded-realness of a system.

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

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

$$ x \in \mathbb{R}^{n}, y \in \mathbb{R}^{m}, u \in \mathbb{R}^{r} $$ are the state, output and input vectors respectively.

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

Definition
The linear system with the same number of input and output variables is called non-expansive if

hold for any arbitrary $$ T\geq 0 $$, arbitrary input $$ u(t) $$, and the corresponding solution $$ y(t) $$ of the system with $$ x(0) = 0 $$. In addition, the transfer function matrix

of system is called is positive real if it is square and satisfies

LMI Condition
Let the linear system be controllable. Then, the system is bounded-real if an only if there exists $$ P > 0 $$ such that

and

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

Conclusion:
Thus, it is seen that passivity and positive-realness describe the same property of a linear system, one gives the time-domain feature and the other provides frequency-domain feature of this property.


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

Introduction
The variable reduction lemma allows the solution of algebraic Riccati inequality that involve a matrix of unknown dimension. This often arises when finding the controller that minimizes the H∞ norm.

The Data
In order to find the unknown matrix $$M$$ we need matrices $$A$$, $$P$$ & $$Q$$.

The Optimization Problem
Given a symmetric matrix $$ A \in \mathbb{R}^{n\times n}$$ and two matrices  $$P$$ & $$Q$$ of column dimension n, consider the problem of finding matrix $$M$$ of compatible dimensions such that

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

The above equation is solvable for some $$M$$ if and only if the following two conditions hold

\begin{align} \ W_P^{T}AW_P < 0\\ \ W_Q^{T}AW_Q < 0\\ \end{align}$$

Where $$W_P$$ and $$W_Q$$ are matrices whose columns are bases for the null spaces of $$P$$ & $$Q$$, respectively.

Conclusion
Using this technique we can get the value of unknown matrix $$M$$.

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


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/D-stability Rise Poles

LMI for Rise Time Poles

The following LMI allows for the verification that poles of a system will fall within a rise time constraint. This can also be used to place poles for rise time when the system matrix includes a controller, such as in the form A+BK.

The System
We consider the following system:

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

or the matrix $$ A \in \mathbb{R}^{n\times n} $$, which is the state matrix.

The Data
The data required is the matrix A and the rise time $$t_r$$ you wish to verify.

The Optimization Problem
To begin, the constraint of the pole locations is as follows: $$z^{*} z-{1.8^2 \over {t_r}^2}{\leq}0$$, where z is a complex pole of A. We define $$r^2{\geq}z^{*} z$$. The goal of the optimization is to find a valid P > 0 such that the following LMI is satisfied.

The LMI: LMI for Rise Time Poles
The LMI problem is to find a matrix P> 0 satisfying:

$$\begin{align} \begin{bmatrix} -rP & AP\\ (AP)^{T} & -rP\end{bmatrix} < 0\\ \end{align}$$

Conclusion:
If the LMI is found to be feasible, then the pole locations of A, represented as z, will meet the rise time specification of $$z^{*} z-{1.8^2 \over {t_r}^2} {\leq} 0$$, and the poles of A satisfy the previously defined constraint.

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

https://github.com/maxwellpeterson99/MAE509Code

Related LMIs
- D-stabilization

- D-stability Controller

- D-stability Observer

Return to Main Page
-Matrix and LMI Properties and Tools


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/D-stability Settling time poles

LMI for Settling Time Poles

The following LMI allows for the verification that poles of a system will fall within a settling time constraint. This can also be used to place poles for settling time when the system matrix includes a controller, such as in the form A+BK.

The System
We consider the following system:

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

or the matrix $$ A \in \mathbb{R}^{n\times n} $$, which is the state matrix.

The Data
The data required is the matrix A and the settling time $$t_s$$ you wish to verify.

The Optimization Problem
To begin, the constraint of the pole locations is as follows: $${(z+z^{*}) \over 2}+{4.6 \over t_s}{\leq}0$$, where z is a complex pole of A. We define $$2 Re(z){\leq}-\alpha$$. The goal of the optimization is to find a valid P > 0 such that the following LMI is satisfied.

The LMI: LMI for Settling Time Poles
The LMI problem is to find a matrix P > 0 satisfying:

$$\begin{align} AP+(AP)^T +\alpha P&<0\\ \end{align}$$

Conclusion:
If the LMI is found to be feasible, then the pole locations of A, represented as z, will meet the settling time specification of $${(z+z^{*}) \over 2}+{4.6 \over t_s}{\leq}0$$, and the poles of A satisfy the previously defined constraint.

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

https://github.com/maxwellpeterson99/MAE509Code

Related LMIs
- D-stabilization

- D-stability Controller

- D-stability Observer

Return to Main Page
-Matrix and LMI Properties and Tools


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User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/D-stability Max Percent Overshoot Poles

LMI for Max Percent Overshoot Poles

The following LMI allows for the verification that poles of a system will within a maximum percent overshoot constraint. This can also be used to place poles for max percent overshoot when the system matrix includes a controller, such as in the form A+BK.

The System
We consider the following system:

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

or the matrix $$ A \in \mathbb{R}^{n\times n} $$, which is the state matrix.

The Data
The data required is the matrix A and the max percent overshoot $$M_p$$ you wish to verify.

The Optimization Problem
To begin, the constraint of the pole locations is as follows: $$z-z^{*}+{{\pi} \over ln({M_p})}|z+z^{*}|{\leq}0$$, where z is a complex pole of A. The goal of the optimization is to find a valid P > 0 such that the following LMI is satisfied.

The LMI: LMI for Max Percent Overshoot Poles
The LMI problem is to find a matrix P satisfying:

$$\begin{align}\begin{bmatrix} \pi(AP+(AP)^{T}) & ln M_p(AP-(AP)^{T})\\ ln M_p(AP-(AP)^{T})^{T} & \pi(AP+(AP)^{T})\end{bmatrix} < 0\\ \end{align}$$

Conclusion:
If the LMI is found to be feasible, then the pole locations of A, represented as z, will meet the max percent overshoot specification of $$z-z^{*}+{{\pi} \over ln({M_p})}|z+z^{*}|{\leq}0$$, and the poles of A satisfy the previously defined constraint.

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

https://github.com/maxwellpeterson99/MAE509Code

Related LMIs
- D-stabilization

- D-stability Controller

- D-stability Observer

Return to Main Page
-Matrix and LMI Properties and Tools


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

Frobenius Norm
Consider $$A\in \R^{n\times m} $$ and $$\gamma\in \R$$>0

The Frobenius norm of $$A $$ is $$||A|| $$F =$$\sqrt{tr(A^{T}A)}=\sqrt{tr(AA^{T})}$$

The Frobenius norm is less than or equal to $$\gamma$$ if and only if any of the following equivalent conditions are satisfied.

1.There exists $$S\in \R^{n} $$ such that
 * $$ \begin{bmatrix} Z & A^{T} \\

* & 1\\                           \end{bmatrix} \ge0,

$$

\begin{align} \qquad tr(Z)\le\gamma^{\text{2}}. \qquad   \\ \end{align}                              $$

2.There exists $$S\in \R^{m} $$ such that
 * $$ \begin{bmatrix} Z & A \\

* & 1\\                           \end{bmatrix} \ge0,

$$

\begin{align} \qquad tr(Z)\le\gamma^{\text{2}}. \qquad   \\ \end{align}                              $$


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

Dualization Lemma
Consider $$P_{i}\in \text{S}^{n} $$ and the subspaces $$U, V$$, where $$P$$ is invertible and $$U +V = \text{R}^{n} $$. The following are equivalent.

$$ X^{T}PX<0$$ for all $$X\in \text{U}$$ \ $$\left\{0\right\}$$ and  $$ X^{T}PX \ge 0$$ for all $$X\in \text{V} $$.

$$ X^{T}P^{-1}X>0$$ for all $$X\in \text{U}^{\bot}$$ \ $$\left\{0\right\}$$ and  $$ X^{T}P^{-1}X \le 0$$ for all $$X\in \text{V}^{\bot} $$.

Example
Consider the matrices $$Q\in \text{S}^{n},S\in \text{R}^{n \times m},R\in \text{S}^{m},M\in \text{R}^{m \times n} $$ where $$ R\ge 0 ,$$ which define the quadratic matrix inequality

$$ \begin{align} \qquad \begin{bmatrix} 1 & M\\ \end{bmatrix} \begin{bmatrix} Q & S\\ S^{T} & R\\ \end{bmatrix} \begin{bmatrix} 1\\ M\\ \end{bmatrix} <0. \qquad (1) \end{align} $$

Define $$ \begin{align} P= \begin{bmatrix} Q & S\\ S^{T} & R\\ \end{bmatrix} ,U=R( \begin{bmatrix} 0\\ 1\\ \end{bmatrix} ) \end{align} $$ where $$U+V=R^{n+m}$$. Notice that $$(1)$$ is equivalent to $$X^{T}PX<0$$ for all $$X\in \text{U}$$ \ $$\left\{0\right\}$$.Additionally, $$X^{T}PX<0 \ge$$ for all $$X\in \text{V} $$ is euaivalent to

$$ \begin{align} \qquad \begin{bmatrix} 0 & 1\\ \end{bmatrix} \begin{bmatrix} Q & S\\ S^{T} & R\\ \end{bmatrix} \begin{bmatrix} 0\\ 1\\ \end{bmatrix} =R \ge 0, \end{align} $$

which is satisfied based on the definition of $$R$$. By the dualization lemma, $$(1)$$ is satisfied with $$R \ge 0$$ if and only if

$$ \begin{align} \qquad \begin{bmatrix} -M^{T} & 1\\ \end{bmatrix} \begin{bmatrix} \tilde{Q} & \tilde{S}\\ \tilde{S}^{T} & \tilde{R}\\ \end{bmatrix} \begin{bmatrix} -M^{T}\\ 1\\ \end{bmatrix} >0,\qquad \tilde{Q}\le 0, \end{align} $$

where $$ \begin{align} \qquad \begin{bmatrix} \tilde{Q} & \tilde{S}\\ \tilde{S}^{T} & \tilde{R}\\ \end{bmatrix} = \begin{bmatrix} Q & S\\ S^{T} & R\\ \end{bmatrix}^{-1}, U^{\bot}=N([1\quad M^{T}])=R( \begin{bmatrix} -M^{T}\\ 1\\ \end{bmatrix} )

\end{align} $$, and $$ \begin{align}V^{\bot}=N([0\quad 1])=R( \begin{bmatrix} 1\\ 0\\ \end{bmatrix} )

\end{align} $$.


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

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

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

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

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

$$


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


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User:Margav06/sandbox/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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User:Margav06/sandbox/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


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User:Margav06/sandbox/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.


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User:Margav06/sandbox/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.


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User:Margav06/sandbox/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.


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User:Margav06/sandbox/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.


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User:Margav06/sandbox/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.


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

Discrete-Time Lyapunov Stability

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

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

The System
Discrete-Time System

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

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

The Optimization Problem
The following feasibility problem should be optimized:

Find P obeying the LMI constraints.

The LMI:
Discrete-Time Bounded Real Lemma

The LMI formulation

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

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

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

Related LMIs
Continuous_Time_Lyapunov_Inequality - Lyapunov_Inequality


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User:Margav06/sandbox/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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User:Margav06/sandbox/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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User:Margav06/sandbox/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

Return to Main Page:

 * Main Page.


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User:Margav06/sandbox/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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User:Margav06/sandbox/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


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Stability Of Lure's System

The System

\begin{align} \dot x(t)&=Ax(t)+B_pp(t)+B_ww(t),\\ z(t)&=C_zx(t)\\ p_i(t)&=\phi_i(q_i(t)), i=1,\dots, n_p\\ q&=C_qx,\\ 0 &\leq \sigma \phi_i(\sigma) \leq \sigma^2 \ \forall \sigma \in \mathbb{R} \end{align}$$

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

The LMI: The Lure's System's Stability
The following feasibility problem should be solved as sufficient condition for the stability of the above Lur'e system.

\begin{align} \text{Find} \; &P>0,\Lambda=diag(\lambda_1,\dots,\lambda_{n_p})\succeq 0, T=diag(\tau_1,\dots,\tau_{n_p}) \succeq 0:\\ &\begin{bmatrix} A^\top P+PA & PB_p+A^\top C^\top_q \Lambda +C^\top_q T \\ B^\top_p P + \Lambda C_qA+TC_q & \Lambda C_q B_p+B^\top_p C^\top_q \Lambda-2T \end{bmatrix} \prec 0\\ \end{align}$$

Implementation
https://codeocean.com/capsule/0232754/tree

Conclusion
If the feasibility problem with LMI constraints has solution, then the Lure's system is stable.

Remark
The LMI is only a sufficient condition for the existence of a Lur’e Lyapunov function that proves stability of Lur'e system. It is also necessary when there is only one nonlinearity, i.e., when $$ n_p = 1 $$.


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/L2 Gain of Lure's Systems

The System

\begin{align} \dot x(t)&=Ax(t)+B_pp(t)+B_ww(t),\\ z(t)&=C_zx(t)\\ p_i(t)&=\phi_i(q_i(t)), i=1,\dots, n_p\\ q&=C_qx,\\ 0 &\leq \sigma \phi_i(\sigma) \leq \sigma^2 \ \forall \sigma \in \mathbb{R} \end{align}$$

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

The Optimization Problem:
The following semi-definite problem should be solved.

\begin{align} &\min_{\{P \succ 0, \Lambda=diag(\lambda_1,\dots,\lambda_{n_p})\succeq 0,T=diag(\tau_1,\dots,\tau_{n_p}) \succeq 0\}} \gamma^2 \; \\ &\quad \quad \quad \quad \quad \quad \quad \quad  s.t. \quad \begin{bmatrix} A^\top P+PA+C^\top_zC_z & PB_p+A^\top C^\top_q \Lambda +C^\top_q T & PB_w \\ B^\top_p P + \Lambda C_qA+TC_q & \Lambda C_q B_p+B^\top_p C^\top_q \Lambda-2T & \Lambda C_q B_w \\ B^\top_wP & B^\top_w C^\top_q\Lambda & -\gamma^2I \end{bmatrix} \preceq 0\\ \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/12a7039f9e3d966e24b43fd58a3cce3725282ed2

Conclusion
The value function returns the square of the smallest provable upper bound on the $$ \mathcal{L}_2 $$ gain of the Lure's system.

Remark
The Lyapunov function which is used to proof is similar to the one for the systems with unknown parameters.


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Output Energy Bound for Lure's Systems

The System

\begin{align} \dot x(t)&=Ax(t)+B_pp(t)+B_ww(t),\\ z(t)&=C_zx(t)\\ p_i(t)&=\phi_i(q_i(t)), i=1,\dots, n_p\\ q(t)&=C_qx(t),\\ 0 &\leq \sigma \phi_i(\sigma) \leq \sigma^2 \ \forall \sigma \in \mathbb{R} \end{align}$$

The Data
The matrices $$ A,B_p,B_w,C_q,C_z,x(0) $$.

The Optimization Problem:
The following optimization problem should be to find the tightest upper bound for the output energy of the above Lur'e system.

\begin{align} &\min_{P \succ 0,\Lambda=diag(\lambda_1,\dots,\lambda_{n_p})\succeq 0,T=diag(\tau_1,\dots,\tau_{n_p}) \succeq 0} x^\top(0)(P+C^\top_q \Lambda C_q)x(0)\\ & \quad \quad \quad \quad \quad \quad \quad \begin{bmatrix} A^\top P+PA & PB_p+A^\top C^\top_q \Lambda +C^\top_q T \\ B^\top_p P + \Lambda C_qA+TC_q & \Lambda C_q B_p+B^\top_p C^\top_q \Lambda-2T \end{bmatrix} \preceq 0\\ \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/blob/master/LMIs%20for%20Output%20Energy%20Bounds%20of%20Lure's%20Systems

Conclusion
The value function returns the the lowest bound for the energy function of the Lure's systems, i.e., $$ J=\int_{0}^{\infty} z^\top z \ dt $$ with initial conditions $$ x(0) $$.

Remark
The key step in the proof is to satisfy $$ \frac{d}{dt}V(x)+z^\top z \leq 0 $$, where $$ V(.)$$ is Lyapunov function in a special form.


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User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Stability of Quadratic Constrained Systems

The System

\begin{align} \dot x(t)&=Ax(t)+B_pp(t)+B_u u(t)+B_ww(t),\\ q(t)&=C_qx(t)+D_{qp}p(t)+D_{qu}u(t)+D_{qw}w(t),\\ z(t)&=C_zx(t)+D_{zp}p(t)+D_{zu}u(t)+D_{zw}w(t)\\ \int_0^t &p^\top(\tau)p(\tau) \ d\tau \leq \int_{0}^t q^\top(\tau)q(\tau) \ d\tau. \end{align}

$$

The Data
The matrices $$ A,B_p,B_w,C_q,C_z,D_{qp},D_{zw} $$.

The LMI:
The following feasibility problem should be solved.

\begin{align} \text{Find} \; &\{P\succ0,\lambda \geq 0\}:\\ &  s.t. \quad \begin{bmatrix} A^\top P+PA+\lambda C^\top_q C_q & PB_p+ \lambda C^\top_q D_{qp}   \\ (PB_p+ \lambda C^\top_q D_{qp})^\top &  \lambda (I-D^\top_{qp}D_{qp})  \end{bmatrix} \prec 0. \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/38f3b55ca7060a1260384a96e9dc31142af07a9a

Conclusion
The integral quadratic constrained system is stable if the provided LMI is feasible

Remark
The key point of the proof is to satisfy $$ \dot{V}<0 $$ whenever $$ p^\top p \leq q^\top q $$, using $$ \mathcal{S} $$-procedure.


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User:Margav06/sandbox/Click here to continue/Controller synthesis

Controller synthesis

 * 1) Continuous Time
 * 2) State Feedback
 * 3) /Stabilization of Second-Order System/
 * 4) /LQ Regulation via H2 Control/
 * 5) /Controller to achieve the desired Reachable set; Polytopic uncertainty/
 * 6) /Controller to achieve the desired Reachable set; Norm bound uncertainty/
 * 7) /Controller to achieve the desired Reachable set; Diagonal Norm-bound uncertainty/
 * 8) D-Stability
 * 9) /Continuous Time D-Stability Controller/
 * 10) /Stabilizability LMI/
 * 11) /LMI for the Controllability Grammian/
 * 12) /LMI for Decentralized Feedback Control/
 * 13) /LMI for Mixed H2 / Hinf Output Feedback Controller/
 * 14) /Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty/
 * 15) /H-inf Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty/
 * 16) /Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty/
 * 17) /Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty/
 * 18) /Hinf Optimal Output Controllability for Systems With Transients/
 * 19) /Quadratic Polytopic Stabilization/
 * 20) /Quadratic D-Stabilization/
 * 21) /Quadratic Polytopic Full State Feedback Optimal Hinf Control/
 * 22) /Quadratic Polytopic Full State Feedback Optimal H2 Control/
 * 23) /Continuous-Time Static Output Feedback Stabilizability/
 * 24) /Multi-Criterion LQG/
 * 25) /Inverse Problem of Optimal Control/
 * 26) /Nonconvex Multi-Criterion Quadratic Problems/
 * 27) /Static-State Feedback Problem/
 * 28) /Mixed H2 Hinf with desired pole location control/
 * 29) /Mixed H2 Hinf with desired pole location control for perturbed systems/
 * 30) /Robust H2 State Feedback Control/
 * 31) /LQ Regulation via H2 control/
 * 32) Discrete Time
 * 33) /Discrete Time Stabilizability/
 * 34) /Quadratic Schur Satbilization/
 * 35) Design For Insensitive Region
 * 36) /Generic Insensitive Strip Region Design/
 * 37) /Generic Insensitive Disk Region Design/
 * 38) /Design for Insensitive Strip Region/
 * 39) /Design for Insensitive Disk Region/
 * 40) Takagi Sugeno Models for Polytopic Controller Design
 * 41) /Quadratic Stability/
 * 42) /Apkarian Filter and State Feedback/
 * 43) /Minimum Decay Rate in State Feedback/
 * 44) /Maximum Natural Frequency in State Feedback/


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Stabilization of Second-Order System

Stabilization is a vastly important concept in controls, and is no less important for second order systems. A second order system can be conceptualized most simply by the model of a mass-spring-damper. Velocity and position are of course chosen as the states for this system, and the state space model can be written as it is below. The goal of stabilization in this context is to design a control law that is made up of two controller gain matrices $$ K_{p} \in R^{r*m_{p}} $$, $$ K_{d} \in R^{r*m_{d}}  $$. These allow the construction of a stabilized closed loop controller.

The System
Here, we want to stabilize a second order system of the following form:

\begin{align} M \ddot x + D \dot x + K x&=B u, \end{align}$$ where $$x \in R^{n}$$ and $$u \in R^{r}$$ are the state vector and the control vector, respectively, and M (called the "mass matrix"), D (called the "structural damping matrix"), K (called the "stiffness matrix"), and B are the system coefficient matrices of appropriate dimensions.

To make the system follow standard convention, we reformulate the system as:

\begin{align} A_{2} \ddot x + A_{1} \dot x + A_{0} x&=B u\\ y_{d} &= C_{d}\dot x\\ y_{p} &= C_{p}x, \end{align}$$ where: $$x \in R^{n}$$ and $$u \in R^{r}$$ are the state vector and the control vector, respectively; $$y_{d} \in R^{m_{d}}$$ and $$y_{p} \in R^{m_{p}}$$ are the derivative output vector and the proportional output vector, respectively; and $$A_{2}, A_{1}, A_{0}, B, C_{d},$$ and $$C_{p}$$ are the system coefficient matrices of appropriate dimension. Note that $$A_{2}$$ must be $$ > 0 $$, and $$A_{0}, A_{2}$$ must be $$\in S^{n}$$.

To further define: $$x$$ is$$\in R^{n}$$ and is the state vector, $$A_{0} $$ is $$\in R^{n*n}$$ and is the state matrix on $$ x $$, $$A_{1} $$ is $$\in R^{n*n}$$ and is the state matrix on $$ \dot x $$ , $$A_{2} $$ is $$\in R^{n*n}$$ and is the state matrix on $$ \ddot x $$, $$B$$ is $$\in R^{n*r}$$ and is the input matrix, $$u$$ is $$\in R^{r}$$ and is the input, $$C_{d} $$ and $$C_{p} $$ are $$\in R^{m*n}$$ and are the output matrices, $$y_{d}$$ is $$\in R^{m}$$ and is the output from $$C_{d} $$, and $$y_{p}$$ is $$\in R^{m}$$ and is the output from $$C_{p} $$.

$$\in R^{n}$$

The Data
The matrices $$ A_{2}, A_{1}, A_{0}, B, C_{d}, C_{p}$$.

The Optimization Problem
For the system described, we choose the following control law

\begin{align} u &= K_{p}y_{p} + K_{d}y_{d}\\ &= K_{p}C_{p}\hat x + K_{d}C_{d}x, \end{align} $$ with $$ K_{p} \in R^{r*m_{p}} $$, $$ K_{d} \in R^{r*m_{d}}  $$, we obtain the closed-loop system as follows:

\begin{align} A_{2}\ddot x + (A_{1} - BK_{p}C_{p})\dot x + (A_{0} - BK_{d}C_{d})x &=0.\\ \end{align} $$ We are tasked to design a state feedback control law such that the above system is hurwitz stable.

First, in order to solve this problem, we need to introduce a Lemma. This Lemma comes from Appendix A.6 in "LMI's in Control systems" by Guang-Ren Duan and Hai-Hua Yu. This Lemma states the following:

\begin{align} A_{2}> 0, A_{1} + A^{T}_{1} > 0, A_{0} > 0\\ \end{align} $$

The LMI: Stabilization of Second Order Systems
There is a solution if there exists matrices $$K_{p} \in R^{r*m_{p}}$$ and $$K_{d} \in R^{r*m_{d}}$$ that satisfy the following LMIs:

\begin{align} A_{0} - BK_{d}C_{d} > 0, \end{align}$$ and

\begin{align} (A_{1} - BK_{p}C_{p}) + (A_{1} - BK_{p}C_{p})^{T} > 0. \end{align}$$

Conclusion:
Finally, having solved the LMI the optimization will produce two matrices, $$ K_{p}$$ and $$ K_{d}$$ that can be substituted into the system as

\begin{align} u &= K_{p}C_{p} \dot x + K_{d}C_{d}x \end{align}$$

to obtain a stabilized second order system.

Implementation
This implementation requires Yalmip and Sedumi. https://github.com/rezajamesahmed/LMImatlabcode/blob/master/stab2ndorder.m

Related LMIs
Robust Stabilization of Second-Order Systems