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User:Margav06/sandbox/Click here to continue/Controller synthesis/LQ Regulation via H2 Control

LQ Regulation via H2 Control

Suppose people were interested in quadratic optimal regulation problem, where instead of the Ricatti equation approach (which is traditionally used in such situations), it is approached using LMI's instead (thereby making it a Linear Quadratic (LQ) problem). Such an approach would be possible by converting the LQ problem into a standard $${H2}$$ problem.

The System
Consider, for example, constant linear multi-variable system in the form of:

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

where

\begin{align} &x\in\mathbb{R}^{n}\text{ and }u\in\mathbb{R}^{r}\text{ are the state and input vectors, respectively, and }\\ &A\text{ and }B\text{ are the system coefficient matrices of appropriate dimensions,}\\ \end{align}$$ Then the LQ optimal regulation problem for the given system is stated as described below.

The Data
In order to obtain the LMI, we need the following 3 matrices: $$A$$, $$B$$, $$Q$$, and $$R$$ (the latter two of which are obtained as follows).

The Optimization Problem
Using the multi-variable system as described above, we can see that the optimal state feedback controller $$u=Kx$$ is obtained where

\begin{align} J(x,u) &=\int_0^\infty ({x^T}Qx+{u^T}Ru)\mathrm{d}t\\ \end{align}$$

is minimized where $$Q={Q^T}{\geq}0$$ and $$R={R^T}{\geq}0$$. However, it is to be noted that in order for the problem to have a solution, two assumptions are made - both of which must be held true at all times:
 * $$ 1)\quad (A,B)\text{ is stabilizable, and}$$
 * $$ 2)\quad (A,L)\text{ is observable, where }L={Q^{1/2}}$$.

Relating this to $${H_2}$$ performance, let us now consider the auxiliary system:

\begin{align} \begin{cases} \dot{x} &= Ax + Bu + {x_0}\omega\\ y &= Cx + Du\\ \end{cases} \end{align}$$,

where $$\omega$$ represents an impulse disturbance, and

\begin{align} C = \begin{bmatrix}{Q^{1/2}}\\0\end{bmatrix}&& D = \begin{bmatrix}0\\{R^{1/2}}\end{bmatrix}\\ \end{align}$$

Using the state feedback controller $$u=Kx$$ and applying it on the above auxiliary system results in the closed-loop system:

\begin{align} \begin{cases} \dot{x} &= (A+BK)x + {x_0}\omega\\ y &= (C+DK)x\\ \end{cases} \end{align}$$,

and the resulting transfer function from disturbance $$\omega$$ to output $$y$$ being:

\begin{align} {G_{y\omega}}=(C+DK)[sI-(A+BK)]^{-1}{x_0} \end{align}$$ thereby resulting in $$J(x,u)=||{G_{y\omega}}||^2_2$$.

The LMI: LQ Regulation via H2 Control
From the given pieces of information, and letting the 2 assumptions as described above hold, then there exist matrices $$X\in\mathbb{S}^n$$, $$Y\in\mathbb{S}^r$$ and $$W\in\mathbb{R}^{rxn}$$ satisfying:

\begin{align} (AX+BW)+{(AX+BW)^T}+{x_0}{x^T_0}&<0\\ trace({Q^{1/2}}X({Q^{1/2}})^T)+trace(Y)&<\gamma\\ \begin{bmatrix}-Y&&{R^{1/2}}W\\({R^{1/2}}W)^T&&-X\end{bmatrix}&<0 \end{align}$$

Conclusion:
From the LMI, it can be seen that the state feedback control in the form of $$u=Kx$$ (where $$K=WX^{-1}$$) exists such that $$J(x,u)<\gamma$$ if and only if the matrices $$X\text{, }Y\text{, and }W$$ are of the appropriate matrix sizes.

Implementation

 * Example Code - A GitHub link that contains code (titled "LQRegH2.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIs

 * ../LMI for System H2 Norm/ - LMI to determine the $$H_2$$-norm of a system.


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Controller to achieve the desired Reachable set; Polytopic uncertainty

Reachable sets with unit-energy inputs; Polytopic uncertainty
A Reachable set is a set of system States reached under the condition $$u = Kx$$. On this page we will look at the problem of finding an controller $$K$$, that  $$E$$ contains $$RS$$ - reachable set.

The System

\begin{align} \dot{x} &=A x + B_w w + B_u u \\ u &= Kx \end{align} $$

Where:

\begin{align} x & \in R^{n} \\ w & \in R^{m} \\ u & \in R^{k} \\ \end{align} $$ In case of polytopic uncertainty, we have:

\begin{align} A(t)\;\; B_w(t)\;\; B_u(t) & \in \textbf{Co} \{[A_1\;\; B_{w, 1}\;\; B_{u;1}], ... , [A_L\;\; B_{w;L}\;\; B_{u;L}] \} \\ \end{align} $$

Reachable set
The reachable set can be defined:

\begin{align} RS &= \{x(T) | u = Kx; \;\; x(0) = 0; \;\; T \geq 0 ;\;\; \int_{0}^T w^T w dt <1 \} \\ \end{align} $$

The elipsoid $$ E = \{ \varepsilon\in R^n | \varepsilon^T Q \varepsilon\leq 1\} $$ contains $$ RS $$

The Data
The matrices $$ A, A_{i} \in R^{n\times n};\; B_{w},B_{w;i}\in R^{n\times m} ;\; B_{u},B_{u;i}\in R^{n\times k}; Q \in R^{n\times n} $$. And $$$$

The Optimization Problem
The following optimization problem should be solved:

\begin{align} \text{Find} \; &Y >0:\\ & QA^T_i + A_iQ + B_{u;i}Y + Y^TB^T_{u;i} + B_{w;i}B^T_{w;i} < 0 \text{ for all } i = 1,... n\\ & K = Y Q^{-1} \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:

 * Reachable sets with unit-energy inputs; Norm-bound uncertainty
 * Reachable sets with unit-energy inputs; Diagonal Norm-bound uncertainty
 * Quadratic polytopic stabilization

Return to Main Page:

 * Main Page.

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/Controller synthesis/Controller to achieve the desired Reachable set; Norm bound uncertainty

Reachable sets with unit-energy inputs; norm bound uncertainty
A Reachable set is a set of system States reached under the condition $$u = Kx$$. On this page we will look at the problem of finding an controller $$K$$, that  $$E \supseteq RS$$ - reachable set.

The System

\begin{align} \dot{x} &=A x + B_w w + B_u u +B_q p \\ q &= C_q x + D_{qw} w + D_{qu} u + D_{qp} p\\ u &= Kx \end{align} $$

Where:

\begin{align} x & \in R^{n} \\ w & \in R^{m} \\ u & \in R^{k} \\ \end{align} $$ In case of norm-bound uncertainty, we have:

\begin{align} p & = \Delta(t) q \\ & \|\Delta\| \leq 1 \end{align} $$

The Data
The matrices $$ A \in R^{n\times n};\; B_{w} \in R^{n\times m} ;\; B_{u} \in R^{n\times k}; B_{p} \in R^{n\times N_p}; K \in R^{k\times n}  $$.

$$ C_q \in R^{N_q\times n};\; D_{qu} \in R^{N_q\times k} D_{qw} \in R^{N_q\times m} D_{qp} \in R^{N_q\times N_p} $$.

Reachable set
The reachable set can be defined:

\begin{align} RS &= \{x(T) | u = Kx; \;\; x(0) = 0; \;\; T \geq 0 ;\;\; \int_{0}^T w^T w dt <1 \} \\ \end{align} $$

The elipsoid $$ E = \{ \varepsilon\in R^n | \varepsilon^T Q \varepsilon\leq 1\} \supseteq  RS $$

The Optimization Problem
The following optimization problem should be solved:

\begin{align} \text{Find} \; &\mu >0: Y\\ & \begin{bmatrix} QA^T + A Q + B_{u }Y + Y^TB^T_{u} + B_{w}B^T_{w}+\mu B_p B_p^T & (C_q Q + D_{qu}Y)^T \\ C_q Q + D_{qu}Y & -\mu I\end{bmatrix}< 0 \\ & K = Y Q^{-1} \end{align} $$

Or

\begin{align} \text{Find} \; &\mu >0: \sigma\\ & \begin{bmatrix} QA^T + A Q + \sigma B_{u }B_u^T  + B_{w}B^T_{w}+\mu B_p B_p^T & (C_q Q -\sigma D_{qu}B_u^T + \mu D_{qp} B_p^T)^T \\ C_q Q -\sigma D_{qu}B_u^T + \mu D_{qp} B_p^T & -\mu(I - D_{qp} D_{qp}^T) \end{bmatrix}< 0 \\ & K = -\frac{\sigma}{2} B_u^T Q^{-1} \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:

 * Reachable sets with unit-energy inputs; Polytopic uncertainty
 * Reachable sets with unit-energy inputs; Diagonal Norm-bound uncertainty
 * H-inf Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

Return to Main Page

 * Main Page.


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Controller to achieve the desired Reachable set; Diagonal Norm-bound uncertainty

Reachable sets with unit-energy inputs; Diagonal Norm-bound uncertainty
A Reachable set is a set of system States reached under the condition $$u = Kx$$. On this page we will look at the problem of finding an controller $$K$$, that  $$E \supseteq RS$$ - reachable set.

The System

\begin{align} \dot{x} &=A x + B_w w + B_u u +B_q p \\ q &= C_q x + D_{qw} w + D_{qu} u + D_{qp} p\\ u &= Kx \end{align} $$

Where:

\begin{align} x & \in R^{n} \\ w & \in R^{m} \\ u & \in R^{k} \\ \end{align} $$ In case of Diagonal Norm-bound uncertainty, we have:

\begin{align} p_i & = \delta_i(t) q_i \\ & |\delta_i(t)| \leq 1;\;\;\;\; \text{for } i = 1, ... , N_q \end{align} $$

The Data
$$N_p = N_q$$

The matrices $$ A \in R^{n\times n};\; B_{w} \in R^{n\times m} ;\; B_{u} \in R^{n\times k}; B_{p} \in R^{n\times N_p}; K \in R^{k\times n}  $$.

$$ C_q \in R^{N_q\times n};\; D_{qu} \in R^{N_q\times k} D_{qw} \in R^{N_q\times m} D_{qp} \in R^{N_q\times N_p} $$.

Reachable set
The reachable set can be defined:

\begin{align} RS &= \{x(T) | u = Kx; \;\; x(0) = 0; \;\; T \geq 0 ;\;\; \int_{0}^T w^T w dt <1 \} \\ \end{align} $$

The elipsoid $$ E = \{ \varepsilon\in R^n | \varepsilon^T Q \varepsilon\leq 1\} \supseteq  RS $$

The Optimization Problem
The following optimization problem should be solved:

\begin{align} \text{Find} \; &M >0: Y\\ & \begin{bmatrix} QA^T + A Q + B_{u }Y + Y^TB^T_{u} + B_{w}B^T_{w}+  B_p M B_p^T & (C_q Q + D_{qu}Y)^T \\ C_q Q + D_{qu}Y & -M I\end{bmatrix}< 0 \\ & K = Y Q^{-1} \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:

 * - Reachable sets with unit-energy inputs; Polytopic uncertainty
 * - Reachable sets with unit-energy inputs; norm bound uncertainty
 * Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty

Return to Main Page:

 * Main Page.


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Continuous Time D-Stability Controller

Continuous-Time D-Stability Controller

In some control problems, people are still interested in designing a controller whose poles are located in specific regions of a complex plane while also simultaneously ensuring that the controller is stable. One such method with which this can be possible is called 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, and where $$A\in\mathbb{R}^{mxm}$$, $$B\in\mathbb{R}^{mxn}$$, $$C\in\mathbb{R}^{pxm}$$, and $$D\in\mathbb{R}^{qxn}$$ for any $$t\in\mathbb{R}$$. Then the 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: $$z-z^{*}+{{\pi} \over ln({M_p})}|z+z^{*}|{\leq}0$$

which not only allows us to map the relationship between complex pole locations and inequality constraints, but it also now allows us to easily formulate our LMIs for this problem.

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 $$P>0$$ 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}$$

Conclusion:
Given the resulting controller $$K=ZP^{-1}$$, 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

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

Related LMIs

 * ../Continuous Time D-Stability Observer/ - Equivalent D-stability LMI for a continuous-time observer.
 * ../D stabilization/ - LMI for $$\mathbb{D}_{(q,r)}$$-stabilization.


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

Stabilizability LMI

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

The System

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

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

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

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

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

Implementation
This implementation requires Yalmip and Sedumi.

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

Related LMIs
Hurwitz Stability LMI

Detectability LMI

Controllability Grammian LMI

Observability Grammian LMI


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User:Margav06/sandbox/Click here to continue/Controller synthesis/LMI for the Controllability Grammian

LMI to Find the Controllability Grammian

Being able to adjust a system in a desired manor using feedback and sensors is a very important part of control engineering. However, not all systems are able to be adjusted. This ability to be adjusted refers to the idea of a "controllable" system and motivates the necessity of determining the "controllability" of the system. Controllability refers to the ability to accurately and precisely manipulate the state of a system using inputs. Essentially if a system is controllable then it implies that there is a control law that will transfer a given initial state $$x(t_0)=x_0$$ and transfer it to a desired final state $$x(t_f)=x_f$$. There are multiple ways to determine if a system is controllable, one of which is to compute the rank "controllability grammian". If the grammian is full rank, the system is controllable and a state transferring control law exists.

The System

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

The Data
The matrices necessary for this LMI are $$A$$ and $$B$$. $$A$$ must be stable for the problem to be feasible.

The LMI: LMI to Determine the Controllability Grammian
$$(A,B)$$ is controllable if and only if $$W>0$$ is the unique solution to
 * $$AW+WA^T-BB^T<0$$,

where $$W$$ is the Controllability Grammian.

Conclusion:
The LMI above finds the controllability grammian $$W$$of the system $$(A,B)$$. If the problem is feasible and a unique $$W$$ can be found, then we also will be able to say the system is controllable. The controllability grammian of the system $$(A,B)$$ can also be computed as: $$W=\int_0^{\infty} e^{As}BB^Te^{A^Ts} ds$$, with control law $$u(t)=B^TW^{-1}x(t)$$ that will transfer the given initial state $$x(t_0)=x_0$$ to a desired final state $$x(t_f)=x_f$$.

Implementation
This implementation requires Yalmip and Sedumi.

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

Related LMIs
Stabilizability LMI

Hurwitz Stability LMI

Detectability LMI

Observability Grammian LMI


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User:Margav06/sandbox/Click here to continue/Controller synthesis/LMI for Decentralized Feedback Control

LMI for Decentralized Feedback Control

In large-scale systems like a multi-agent robotic system, national economies, or chemical refineries, an actuator should act based on local information, which necessitates a decentralized or distributed control strategy. In a decentralized control framework, the controllers are distributed and each controller has only access to a subset of local measurements. We describe LMI formulations for a general decentralized control framework and then provide an illustrative example of a decentralized control design.

The System
In a decentralized controller design, the state feedback controller $$ u = Kx$$ can be divided into $$n$$ sub-controllers $$u_{i} = K_{i}x_{i}, \quad i=1,2, ..., n$$.

The Data
A general state space representation of a linear time-invariant system is as follows:

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

where $$x$$ is a $$n\times n$$ vector of state variables, $$B$$ is the input matrix, $$C$$ is the output matrix, and $$D$$ is called the feedforward matrix. We assume that all the four matrices, $$A$$, $$B$$, $$C$$, and $$D$$ are given.

The Optimization Problem
We aim to solve the $$H_{\infty}$$-optimal full-state feedback control problem using a controller $$ u = Kx$$.

In a decentralized fashion, the control input $$u$$ can be divided into sub-controllers $$ u_{1}, u_{2}, ..., u_{j} $$ and can be written as $$ u = [u_{1} \quad u_{2} \quad ... \quad u_{j}]_{1\times n}^{\text{T}}$$.

For instance, let $$ j = 3$$ and $$n=6$$. Thus, there are three control inputs $$ u_{1}$$, $$u_{2}$$, and $$u_{3}$$. We also assume that u_{1} only depends on the first and the second states, while $$u_{2}$$ and $$ u_{3} $$ only depend on thrid to sixth states. For this example, the controller gain matrix can be described by:

$$ K = \begin{bmatrix} k_{1} & k_{2} & 0 & 0 & 0 & 0 \\ 0    & 0     & k_{3} & k_{4} & k_{5} & k_{6} \\ 0    & 0     & k_{7} & k_{8} & k_{9} & k_{10} \\ \end{bmatrix} $$

Thus, the decentralized controller gain consists of sub-matrices of gains.

The LMI: LMI for decentralized feedback controller
The mathematical description of the LMI formulation for a decentralised optimal full-state feedback controller can be described by:

$$ \begin{align} & \text{min} \quad \gamma\\ & \begin{bmatrix} YA^{\text{T}} + AY + Z^{\text{T}}B_{2}^{\text{T}} + B_{2}Z & *^{T} & *^{T} \\ B_{1}^{T} & -\gamma I & *^{T} \\ YC_{1}^{T} + Z^{T}D_{12} & D_{11} & -\gamma I \end{bmatrix} \end{align} $$

where $$Y > 0$$ is a positive definite matrix and $$ Z $$ such that the aforemtntioned constraints in LMIs are satisfied.

Conclusion:
The controller gain matrix is defined as:

$$ K = \begin{bmatrix} 0 & 0 \\ 0 & F \end{bmatrix} $$

where $$F$$ can be found after solving the LMIs and obtaining the variables matrices $$Y$$ and $$Z$$. Thus,

$$ F = ZY^{-1}$$.

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

https://github.com/asalimil/LMI_for_decentralized_feedback_controller/tree/master

Return to Main Page
LMIs in Control/Tools


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User:Margav06/sandbox/Click here to continue/Controller synthesis/LMI for Mixed H2 / Hinf Output Feedback Controller

LMI for Mixed $$H_{2}/H_{\infty}$$ Output Feedback Controller

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

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

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

where $$ A $$, $$ B$$, $$ C$$, and $$D $$ are the state matrix, input matrix, output matrix, and feedforward matrix, respectively.

These are the system (plant) matrices that can be shown as $$ P = (A, B, C, D) $$.

The Data
We assume that all the four matrices of the plant, $$ A, B, C, D $$, are given.

The Optimization Problem
In this problem, we use an LMI to formulate and solve the optimal output-feedback problem to minimize both the <> and <> norms. Giving equal weights to each of the norms, we will have the optimization problem in the following form:

$$ \begin{align} \text{min} \quad ||S(P,K)||_{H_{2}}^{2} + ||S(P,K)||_{H_{\infty}}^{2} \end{align} $$

The LMI: LMI for mixed $$H_{2}$$/$$H_{\infty}$$
Mathematical description of the LMI formulation for a mixed $$H_{2}$$/$$H_{\infty}$$ optimal output-feedback problem can be written as follows:

$$ \begin{align} &\text{min} \quad \gamma_{1}^{2} + \gamma_{2}^{2} \\ &\text{s.t.} \\ &\begin{bmatrix} X_{1}     & I \\ I         & Y_{1} \end{bmatrix} > 0 \\ &\begin{bmatrix} AY_{1}+Y_{1}A^{\text{T}}+B_{2}C_{n}+C_{n}B_{2}^{\text{T}} & *^{\text{T}} & *^{\text{T}} & *^{\text{T}} \\ A^{\text{T}}+A_{n}+(B_{2}D_{n}C_{2})^{\text{T}} & X_{1}A+A^{\text{T}}+B_{n}C_{2}+C_{2}^{\text{T}}B_{n}^{\text{T}} & *^{\text{T}} & *^{\text{T}} \\ (B_{1}+B_{2}D_{n}D_{21})^{\text{T}} & (X_{1}B_{1} + B_{n}D_{21})^{\text{T}} & -\gamma I & *^{\text{T}}\\ C_{1}Y_{1}+D_{12}C_{n} & C_{1}+D_{12}D_{n}C_{2} & D_{11}+D_{12}D_{n}D_{21} & -\gamma I\\ \end{bmatrix} < 0 \\ &\begin{bmatrix} Y_{1}  & I & (C_{1}Y_{1}+D_{12}C_{n})^{\text{T}} \\ I & X_{1} & (C_{1}+D_{12}D_{n}C_{2})^{\text{T}}\\ (C_{1}Y_{1}+D_{12}C_{n}) & (C_{1}+D_{12}D_{n}D_{21} & Z\\ C_{1}Y_{1}+D_{12}C_{n} & C_{1}+D_{12}D_{n}C_{2} & D_{11}+D_{12}D_{n}D_{21} & -\gamma I\\ \end{bmatrix} > 0 \\ &\begin{bmatrix} AY_{1}+Y_{1}A^{\text{T}}+B_{2}C_{n}+C_{n}\text{T}B_{2}\text{T} & *^{\text{T}} & *^{\text{T}} & *^{\text{T}} \\ (A^{\text{T}}+An+(B_{2}*D_{n}*C_{2})^{\text{T}})  & X_{1}A+A^{\text{T}}X_{1}+B_{n}C_{2}+C_{2}^{\text{T}}B_{n}^{\text{T}}  & *^{\text{T}} & *^{\text{T}} \\ (B_{1}+B_{2}D_{n}D_{21})^{\text{T}}  & (X_{1}B_{1}+B_{n}D_{21})^{\text{T}} & -\gamma_{2}^{2}I  & *^{\text{T}} \\ (C_{1}Y_{1}+D_{12}C_{n}) &  (C_{1}+D_{12}D_{n}C_{2}) & (D_{11}+D_{12}D_{n}*D_{21}) & -I \\ \end{bmatrix} < 0 \\ & \text{trace}(Z) < \gamma_{1}^{2} \\ & D_{11} + D_{12}D_{n}D_{21} = 0 \end{align} $$

where $$ \gamma_{1}^{2} $$ and $$\gamma_{1}^{2}$$ are defined as the $$ H_{2}$$ and $$ H_{\infty}$$ norm of the system:

$$ \begin{align} &||S(P,K)||_{H_{2}}^{2} = \gamma_{1}^{2} \\ &||S(P,K)||_{H_{\infty}}^{2} = \gamma_{2}^{2} \end{align} $$

Moreover, $$X_{1} $$, $$ Y_{1}$$, $$ A_{n}$$, $$ B_{n}$$, $$ C_{n}$$, and $$ D_{n}$$ are variable matrices with appropriate dimensions that are found after solving the LMIs.

Conclusion:
The calculated scalars $$ \gamma_{1}^{2} $$ and $$ \gamma_{2}^{2} $$ are the $$ H_{2}$$ and $$ H_{\infty}$$ norms of the system, respectively. Thus, the norm of mixed $$ H_{2}$$/$$ H_{\infty}$$ is defined as $$ \beta = \gamma_{1}^{2} + \gamma_{2}^{2}$$. The results for each individual $$ H_{2}$$ norm and $$H_{\infty} $$ norms of the system show that a bigger value of norms are found in comparison with the case they are solved separately.

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

https://github.com/asalimil/LMI_for_Mixed_H2_Hinf_Output_Feedback_Controller

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LMIs in Control/Tools


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

If the system is quadratically stable, then there exists some $$\mu \geq 0, P > 0,$$ and $$Z$$ such that the LMI is feasible. The $$Z$$ and $$P$$ matrices can also be used to create a quadratically stabilizing controller.

The System

\begin{align} \dot x(t)&=Ax(t)+Bu(t)+Mp(t),&&p(t) = \Delta (t)q(t),\\ q(t) &= Nx(t)+Qp(t)+D_{12}u(t),&&\Delta \in \mathbf{\Delta} \; := \{\Delta \in \mathbb{R}^{n \times n} : \| \Delta \| \leq 1 \}\\ \end{align}$$

The Data
The matrices $$ A,B,M,N,Q,D_{12} $$.

The LMI:

\begin{align} \text{Find} \; &P>0,\mu \geq 0, \text{ and } Z:\\ \begin{bmatrix} AP+BZ+PA^T+Z^TB^T & PN^T+Z^TD_{12}^T \\ NP+D_{12}Z & 0\end{bmatrix} + \mu \begin{bmatrix} MM^T & MQ^T \\ QM^T & QQ^T - I \end{bmatrix} < 0\\ \end{align}$$

Conclusion:
There exists a controller for the system with $$ u(t) = Kx(t) $$ where $$ K = ZP^{-1} $$ is the quadratically stabilizing controller, if the above LMI is feasible.

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

Related LMIs
H-infinity Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty

Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty


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User:Margav06/sandbox/Click here to continue/Controller synthesis/H-inf Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

If there exists some $$\mu \geq 0, P>0$$, and $$Z$$ such that the LMI holds, then the system satisfies $$\|y\|_{L_2} \leq \gamma \|u\|_{L_2}.$$ There also exists a controller with $$u(t) = Kx(t).$$

The System

\begin{align} \dot x(t)&=Ax(t)+Bu(t)+Mp(t)+B_2w(t),&&p(t) = \Delta (t)q(t),\\ q(t) &= Nx(t)+D_{12}u(t),&&\Delta \in \mathbf{\Delta} \; := \{\Delta \in \mathbb{R}^{n \times n} : \| \Delta \| \leq 1 \}\\ y(t) &= Cx(t)+D_{22}u(t)\\ \end{align}$$

The Data
The matrices $$ A,B,M,B_2,N,D_{12},C,D_{22} $$.

The Optimization Problem
Minimize $$ \gamma $$ subject to the LMI constraints below.

The LMI:

\begin{align} \text{Find} \; &P>0,\mu \geq 0, \text{ and } Z:\\ \begin{bmatrix} AP+BZ+PA^T+Z^TB^T+B_2B_2^T+\mu MM^T & (CP+D_{22}Z)^T & PN^T+Z^TD_{12}^T \\ CP+D_{22}Z & -\gamma ^2I & 0 \\ NP+D_{12}Z & 0 & -\mu I \end{bmatrix} < 0\\ \end{align}$$

Conclusion:
The controller gains, K, are calculated by $$ K = ZP^{-1} $$.

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

Related LMIs
Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty

Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty

$$ \begin{align} \text{The system is quadratically stable if and only if there exists some } \Theta \in P \Theta, P>0, \text{ and } Z \text{ such that the LMI is feasible.} \text{ Furthermore, there exists a quadratically stabilizing controller with } u(t) = Kx(t). \end{align} $$

The System

\begin{align} \dot x(t)&=Ax(t)+Bu(t)+Mp(t),&&p(t) = \Delta (t)q(t),\\ q(t) &= Nx(t)+Qp(t)+D_{12}u(t),&&\Delta \in \bf{\Delta} \;, || \Delta || \leq 1 \\ \end{align}$$

The Data
The matrices $$ A,B,M,N,Q,D_{12} $$.

The LMI:

\begin{align} \text{Find} \; &P>0,Z:\\ \begin{bmatrix} AP+BZ+PA^T+Z^TB^T & PN^T+Z^TD_{12}^T \\ NP+D_{12}Z & 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:
If the LMI is feasible, the controller, K, is calculated by $$ K = ZP^{-1} $$.

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

Related LMIs
Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

H-infinity Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty

$$ \begin{align} \text{If there exists some } \Theta \in P \Theta, P>0, \text{ and } Z \text{ such that the LMI is feasible, then the system satisfies } ||y||_{L_2} \leq \gamma ||u||_{L_2}. \text{ There also exists a controller with } u(t) = Kx(t). \end{align} $$

The System

\begin{align} \dot x(t)&=Ax(t)+Bu(t)+Mp(t)+B_2w(t),&&p(t) = \Delta (t)q(t),\\ q(t) &= Nx(t)+D_{12}u(t),&&\Delta \in \bf{\Delta} \;, || \Delta || \leq 1 \\ y(t) &= Cx(t)+D_{22}u(t)\\ \end{align}$$

The Data
The matrices $$ A,B,M,B_2,N,D_{12},C,D_{22} $$.

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

The LMI:

\begin{align} \text{Find} \; &P>0,Z:\\ \begin{bmatrix} AP+BZ+PA^T+Z^TB^T+B_2B_2^T+M \Theta M^T & (CP+D_{22}Z)^T & PN^T+Z^TD_{12}^T \\ CP+D_{22}Z & -\gamma ^2 I & 0 \\ NP+D_{12}Z & 0 & -\Theta\end{bmatrix} < 0\\ \end{align}$$

Conclusion:
The controller is $$ K = ZP^{-1} $$.

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

Related LMIs
Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

H-infinity Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Hinf Optimal Output Controllability for Systems With Transients

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

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

The System

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

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

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

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

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

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

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

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

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

Related LMIs
Links to other closely-related LMIs


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Quadratic Polytopic Stabilization

A Quadratic Polytopic Stabilization Controller Synthesis can be done using this LMI, requiring the information about $$A$$,$$\Delta_{A(t)} $$ ,$$B$$ and $$\Delta_{B(t)} $$ matrices.

The System

\begin{align} \dot x(t)&=Ax(t)+Bu(t),\\ x(0)&=x_0, \end{align}$$ where $$x(t)\in \R^n$$, $$u(t)\in \R^m$$, at any $$t\in \R$$. The system consist of uncertainties of the following form

\begin{align} \Delta_{A(t)} = A_1 \delta_1 (t)+....+A_k \delta_k (t)\\ \Delta_{B(t)} = B_1 \delta_1 (t)+....+B_k \delta_k (t)\\ \end{align}$$

where $$x\in\mathbb{R}^{m}$$,$$u\in\mathbb{R}^{n}$$,$$A\in\mathbb{R}^{mxm}$$ and $$B\in\mathbb{R}^{mxn}$$

The Data
The matrices necessary for this LMI are $$A$$,$$\Delta_{A(t)} \, ie \, A_i $$ ,$$B$$ and $$\Delta_{B(t)} \, ie \, B_i $$

The Optimization and LMI:LMI for Controller Synthesis using the theorem of Polytopic Quadratic Stability
There exists a K such that

\begin{align} \dot x(t)&=(A+\Delta_A+(B+\Delta_B)K)x(t)\\ \end{align}$$ is quadratically stable for $$ (\Delta_A,\Delta_B) \in C_0 ((A_1,B_2),...,(A_k,B_k)) $$ if and only if there exists some P>0 and Z such that

\begin{align} (A+A_i)P + P(A+A_i)^T + (B+B_i)Z+Z^T(B+B_i)^T < 0 \quad for \quad i = 1,...k. \end{align}$$

Conclusion:
The Controller gain matrix is extracted as $$ K = ZP^{-1} $$ Note that here the controller doesn't depend on $$ \Delta $$
 * If you want K to depend on $$ \Delta $$, the problem is harder.
 * But this would require sensing $$ \Delta $$ in real-time.

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

Related LMIs
Quadratic Polytopic $H_{\infty}$ Controller Quadratic Polytopic $H_{2}$ Controller


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Quadratic D-Stabilization

Continuous-Time D-Stability Controller

This LMI will let you place poles at a specific location based on system performance like rising time, settling time and percent overshoot, while also ensuring the stability of the system.

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, and where $$A\in\mathbb{R}^{mxm}$$, $$B\in\mathbb{R}^{mxn}$$, $$C\in\mathbb{R}^{pxm}$$, and $$D\in\mathbb{R}^{qxn}$$ for any $$t\in\mathbb{R}$$.

Adding uncertainty to the system

\begin{align} \dot x(t)&=(A+A_i)x(t)+(B+B_i)u(t)\\ \end{align}$$

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 $$A$$, $$B$$, $$A_i$$, $$B_i$$
 * 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: $$z-z^{*}+{{\pi} \over ln({M_p})}|z+z^{*}|{\leq}0$$

which not only allows us to map the relationship between complex pole locations and inequality constraints but it also now allows us to easily formulate our LMIs for this problem.

The LMI: An LMI for Quadratic D-Stabilization
Suppose there exists $$ X > 0 $$ and $$ Z $$ such that

\begin{align} \begin{bmatrix} -rP && AP+BZ \\ (AP+BZ)^T && -rP \end{bmatrix} +

\begin{bmatrix} 0 && A_iP + B_iZ \\(A_iP + B_iZ)^T&&0 \end{bmatrix} < 0 \\

AP+BZ+(AP+BZ)^T + A_iP+B_iZ+(A_iP+B_iZ)^T +2\alpha P&<0, and\\

\end{align}$$

\begin{align} \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} +

\begin{bmatrix} A_iP+B_iZ+(A_iP+B_iZ)^T && c(A_iP+B_iZ-(A_iP+B_iZ)^T) \\c((A_iP+B_iZ)^T -(A_iP+B_iZ))&& A_iP+B_iZ+(A_iP+B_iZ)^T \end{bmatrix} < 0 \\

\end{align}$$

for $$ i = 1,...,k $$

Conclusion:
Given the resulting controller $$K=ZP^{-1}$$, we can now determine that the pole locations $$z\in\mathbb{C}$$ of $$A(\Delta)+B(\Delta)K$$ satisfies the inequality constraints $$|x|{\leq}r$$, $$Re(x){\leq}-{\alpha}$$ and $$z+z^{*}{\leq}-c|z-z^{*}|$$ for all $$ \,  \Delta \, \in \, C_0(\Delta_1,...,\Delta_k) $$

Implementation
The implementation of this LMI requires Yalmip and Sedumi https://github.com/JalpeshBhadra/LMI/blob/master/quadraticDstabilization.m

Related LMIs

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


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Quadratic Polytopic Full State Feedback Optimal Hinf Control

Quadratic Polytopic Full State Feedback Optimal $$H_{\infty}$$ Control
For a system having polytopic uncertainties, Full State Feedback is a control technique that attempts to place the system's closed-loop system poles in specified locations based off of performance specifications given. $$H_{\infty}$$ methods formulate this task as an optimization problem and attempt to minimize the $$H_{\infty}$$ norm of the system.

The System
Consider System with following state-space representation.

\begin{align} \dot x(t)&=Ax(t)+B_1 q(t) + B_2 w(t)\\ p(t)& = C_1 x(t) + D_{11} q(t) + D_{12} w(t)\\ z(t)& = C_2 x(t) + D_{21} q(t) + D_{22} w(t)\\ \end{align}$$

where $$x\in\mathbb{R}^{m}$$, $$q\in\mathbb{R}^{n}$$ , $$w\in\mathbb{R}^{g}$$, $$A\in\mathbb{R}^{mxm}$$, $$B_1\in\mathbb{R}^{mxn}$$, $$B_2\in\mathbb{R}^{mxg}$$, $$p\in\mathbb{R}^{p}$$ , $$C_1\in\mathbb{R}^{pxm}$$, $$D_{11}\in\mathbb{R}^{pxn}$$,  $$D_{12}\in\mathbb{R}^{pxg}$$, $$z\in\mathbb{R}^{s}$$, $$C_2\in\mathbb{R}^{sxm}$$, $$D_{21}\in\mathbb{R}^{sxn}$$ ,  $$D_{22}\in\mathbb{R}^{sxg}$$ for any $$t\in\mathbb{R}$$.

Add uncertainty to system matrices

A, B_1, B_2, C_1, C_2, D_{11}, D_{12} $$

New state-space representation

\begin{align} \dot x(t)&=(A+A_i)x(t)+(B_1+B_i) q(t) + (B_2+B_i) w(t)\\ p(t)& = (C_1+C_i) x(t) + (D_{11}+D_i) q(t) + (D_{12}+D_i) w(t)\\ z(t)& = C_2 x(t) + D_{21} q(t) + D_{22} w(t)\\ \end{align}$$

The Optimization Problem:
Recall the closed-loop in state feedback is: $$ S(P,K) = $$

\begin{align} \begin{bmatrix} A + B_{2}F&&B_1\\C_1 + D_{12}F&& D_{11}\end{bmatrix}\\ \end{align}$$ This problem can be formulated as $$ H\infty $$ optimal state-feedback, where K is a controller gain matrix.

The LMI:
An LMI for Quadratic Polytopic $$ H\infty $$ Optimal State-Feedback Control $$ ||S(P(\Delta),K(0,0,0,F))||_{H\infty} \leq \gamma $$ $$ Y > 0 $$

\begin{align} \begin{bmatrix} Y(A+A_i)^T+(A+A_i)Y+Z^T(B_2+B_{1,i})^T+(B_2+B_{1,i})Z&&*^T&&*^T \\(B_1+B_{1,i})^T&&-\gamma I&&*^T \\(C_1+C_{1,i})Y+(D_{12}+D_{12,i})Z&&(D_{11}+D_{11,i})&&-\gamma I \end{bmatrix} < 0 \end{align}$$

Conclusion:
The $$ H\infty $$ Optimal State-Feedback Controller is recovered by $$ F = ZY^{-1} $$ Controller will determine the bound $$\gamma$$ on the $$H_{\infty}$$ norm of the system.

Implementation:
https://github.com/JalpeshBhadra/LMI/tree/master

Related LMIs
Full State Feedback Optimal $H_{\infty}$ Controller


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Quadratic Polytopic Full State Feedback Optimal H2 Control

Quadratic Polytopic Full State Feedback Optimal $$H_2$$ Control
For a system having polytopic uncertainties, Full State Feedback is a control technique that attempts to place the system's closed-loop system poles in specified locations based on performance specifications given, such as requiring stability or bounding the overshoot of the output. By minimizing the $$H_2$$ norm of this system we are minimizing the effect noise has on the system as part of the performance specifications.

The System
Consider System with following state-space representation.

\begin{align} \dot x(t)&=Ax(t)+B_1 q(t) + B_2 w(t)\\ p(t)& = C_1 x(t) + D_{11} q(t) + D_{12} w(t)\\ z(t)& = C_2 x(t) + D_{21} q(t) + D_{22} w(t)\\ \end{align}$$

where $$x\in\mathbb{R}^{m}$$, $$q\in\mathbb{R}^{n}$$ , $$w\in\mathbb{R}^{g}$$, $$A\in\mathbb{R}^{mxm}$$, $$B_1\in\mathbb{R}^{mxn}$$, $$B_2\in\mathbb{R}^{mxg}$$, $$p\in\mathbb{R}^{p}$$ , $$C_1\in\mathbb{R}^{pxm}$$, $$D_{11}\in\mathbb{R}^{pxn}$$,  $$D_{12}\in\mathbb{R}^{pxg}$$, $$z\in\mathbb{R}^{s}$$, $$C_2\in\mathbb{R}^{sxm}$$, $$D_{21}\in\mathbb{R}^{sxn}$$ ,  $$D_{22}\in\mathbb{R}^{sxg}$$ for any $$t\in\mathbb{R}$$.

Add uncertainty to system matrices

A, B_1, B_2, C_1, C_2, D_{11}, D_{12} $$

New state-space representation

\begin{align} \dot x(t)&=(A+A_i)x(t)+(B_1+B_i) q(t) + (B_2+B_i) w(t)\\ p(t)& = (C_1+C_i) x(t) + (D_{11}+D_i) q(t) + (D_{12}+D_i) w(t)\\ z(t)& = C_2 x(t) + D_{21} q(t) + D_{22} w(t)\\ \end{align}$$

The Data
The matrices necessary for this LMI are

The Optimization Problem:
Recall the closed-loop in state feedback is: $$ S(P,K) = $$

\begin{align} \begin{bmatrix} A + B_{22}F&&B_1\\C_1 + D_{12}F&& D_{11}\end{bmatrix}\\ \end{align}$$ This problem can be formulated as $$ H_2 $$ optimal state-feedback, where K is a controller gain matrix.

The LMI: An LMI for Quadratic Polytopic $$ H_2 $$ Optimal
State-Feedback Control $$ ||S(P(\Delta),K(0,0,0,F))||_{H_2} \leq \gamma $$ $$ X > 0 $$

\begin{align} \begin{bmatrix} AX + B_2Z + XA^T + Z^TB^T_{2} &&B_1 \\B_1^T&&-I \end{bmatrix} +

\begin{bmatrix} A_iX + B_{2,i}Z + XA^T_i + Z^TB^T_{2,I} &&B_{1,i} \\B_{1,i}^T&&0 \end{bmatrix} < 0 \quad i = 1,......,k

\end{align}$$

\begin{align} \begin{bmatrix} X && (C_1X+D_{12}Z)^T \\C_1X+D_{12}Z && W \end{bmatrix} +

\begin{bmatrix} 0 && (C_{1,i}X+D_{12,i}Z)^T \\C_{1,i}X+D_{12,i}Z&&0 \end{bmatrix} > 0 \quad i = 1,......,k

\end{align}$$

\begin{align} \\TraceW < \gamma \end{align}$$

Conclusion:
The $$ H_2 $$ Optimal State-Feedback Controller is recovered by $$ F = ZX^{-1} $$

Implementation:
https://github.com/JalpeshBhadra/LMI/blob/master/H2_optimal_statefeedback_controller.m

Related LMIs
$ H_2 $ Optimal State-Feedback Controller


text:
User:Margav06/sandbox/Click here to continue/Controller synthesis/Continuous-Time Static Output Feedback Stabilizability

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

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

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

\end{align}$$

The Data

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

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

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

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

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

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

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

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

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

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

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

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

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

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

https://github.com/yashgvd/LMI_wikibooks

Related LMIs
Discrete time Static Output Feedback Stabilizability Static Feedback Stabilizability


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Multi-Criterion LQG

The Multi-Criterion Linear Quadratic Gaussian (LQG) linear matrix inequality will allow one to form an optimized controller, similar to that in an LQR framework, for a state space system with gaussian noise based on several different criterions defined in the Q and R matrices, that are optimized as a part of the arbitrary cost function. Just like traditional LQR, the cost matrices must be tuned in much a similar fashion as traditional gains in classical control. In the LQR and LQG framework however, the gains are more intuitive as each correlates directly to a state or an input.

The System
The system is a linear time-invariant system, that can be represented in state space as shown below:

\begin{align} \dot x &= Ax + Bu + w,\\ y &= Cx + v,\\ z &=  \begin{bmatrix} Q^{1/2} & 0 \\ 0 & R^{1/2}\end{bmatrix}\begin{bmatrix} x \\ u \end{bmatrix} \end{align}$$

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

$$ Q\geq 0$$ and $$ R> 0 $$, and the system is controllable and observable.

The Data
The matrices $$ A, B, C, Q, R, W, V $$ and the noise signals $$ w, v$$.

The Optimization Problem
In the Linear Quadratic Gaussian (LQG) control problem, the goal is to minimize a quadratic cost function while the plant has random initial conditions and suffers white noise disturbance on the input and measurement.

There are multiple outputs of interest for this problem. They are defined by

\begin{align} z &=  \begin{bmatrix} Q^{1/2} & 0 \\ 0 & R^{1/2}\end{bmatrix}\begin{bmatrix} x \\ u \end{bmatrix}, Q_{i} \geq 0, R_{i}>0, i = 0,...,p. \end{align}$$

For each of these outputs of interest, we associate a cost function:

\begin{align} J^{i}_{LQG} = \lim_{t\to\infty} Ez_{i}(t)^{T}z_{i}(t), i = 0,...,p. \end{align}$$

Additionally, the matrices $$X_{LQG}$$ and $$Y_{LQG}$$ must be found as the solutions to the following Riccati equations:

\begin{align} A^{T}X_{LQG} + X_{LQG}A = X_{LQG}BR^{-1}B^{T}X_{LQG} + Q &= 0\\ AY_{LQG} + Y_{LQG}A^{T} - Y_{LQG}C^{T}V^{-1}CY_{LQG}+W &= 0\\ \end{align}$$

The optimization problem is to minimize $$J^{0}_{LQG}$$ over u subject to the measurability condition and the constraints $$J^{i}_{LQG}<\gamma_{i}, i = 0,...,p.$$. This optimization problem can be formulated as:

\begin{align} \max trace(X_{LQG}U + QY_{LQG}) - \sum_{i=1}^{p} \gamma_{i}\tau_{i}, \end{align}$$ over $$\tau_{1},...,\tau_{p}$$, with:

\begin{align} Q &= Q_{0} + \sum_{i=1}^{p} \tau_{i}Q_{i},\\ R &= R_{0} + \sum_{i=1}^{p} \tau_{i}R_{i}. \end{align}$$

The LMI: Multi-Criterion LQG

\begin{align} \max:     trace(XU + (Q_{0} + \sum_{i=1}^{p} \tau_{i} Q_{i} )Y_{LQG}) - \sum_{i=1}^{p} \gamma_{i}\tau_{i}, \end{align}$$ over $$X, \tau_1, ... \tau_{p}$$, subject to the following constraints:

\begin{align} X&>0,\\ \tau_{1} \geq 0, ..., \tau_{p} &\geq 0,\\ A^{T}X + XA - XB(R_{0} + \sum_{i=1}^{p} \tau_{i} R_{i})^{-1}B^{T}X + Q_{0} + \sum_{i=1}^{p} \tau_{i} Q_{i} &\geq 0.\\ \end{align}$$

Conclusion:
The result of this LMI is the solution to the aforementioned Ricatti equations:

\begin{align} A^{T}X_{LQG} + X_{LQG}A = X_{LQG}BR^{-1}B^{T}X_{LQG} + Q &= 0\\ AY_{LQG} + Y_{LQG}A^{T} - Y_{LQG}C^{T}V^{-1}CY_{LQG}+W &= 0\\ \end{align}$$

Implementation
This implementation requires Yalmip and Sedumi.

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

Related LMIs

 * Inverse Problem of Optimal Control


text:
User:Margav06/sandbox/Click here to continue/Controller synthesis/Inverse Problem of Optimal Control

In some cases, it is needed to solve the inverse problem of optimal control within an LQR framework. In this inverse problem, a given controller matrix needs to be verified for the system by assuring that it is the optimal solution to some LQR optimization problem that is controllable and detectable. In other words: in this inverse problem, the controller is known and the LQR gain matrices are to be calculated such that the controller is the optimal solution.

The System
The system is a linear time-invariant system, that can be represented in state space as shown below:

\begin{align} \dot x &= Ax + Bu,\\ z &=  \begin{bmatrix} Q^{1/2} & 0 \\ 0 & R^{1/2}\end{bmatrix}\begin{bmatrix} x \\ u \end{bmatrix} \end{align}$$ where $$x \in R^{n}, y \in R^{l}, z \in R^{m}$$ represent the state vector, the measured output vector, and the output vector of interest, respectively, $$w \in R^{p}$$ is the disturbance vector, and $$A, B, C, Q, R$$ are the system matrices of appropriate dimension. To further define: $$x$$ is $$\in R^{n}$$ and is the state vector, $$A $$ is $$\in R^{n*n}$$ and is the state matrix, $$B$$ is $$\in R^{n*r}$$ and is the input matrix, $$w$$ is $$\in R^{r}$$ and is the exogenous input, $$C, Q, R $$ is $$\in R^{m*n}$$ and are the output matrices, and $$y$$ and $$z$$ are $$\in R^{m}$$ and are the output and the output of interest, respectively.

The Data
The matrices $$ A, B, C $$ that define the system, and a given controller $$ K $$ for which the inverse problem is to be solved.

The Optimization Problem
In this LMI, the following cost function is to be minimized for a given controller K by finding an optimal input:

\begin{align} \int_{0}^{\infty} z^{T}z dt \end{align}$$

the solution of the problem can be formulated as a state feedback controller given as:

\begin{align} K &= -R^{-1}B^{T}P,\\ A^{T}P + PA - PBR^{-1}B^{T}P+Q &=0 \end{align}$$

The LMI: Inverse Problem of Optimal Control
the inverse problem of optimal control is the following: Given a matrix $$ K $$, determine if there exist $$ Q \geq 0 $$ and $$ R > 0 $$, such that $$ (Q, A) $$ is detectable and $$ u = Kx $$ is the optimal control for the corresponding LQR problem. Equivalently, we seek $$ R>0 $$ and $$ Q \geq 0 $$ such that there exist $$ P $$ nonnegative and $$ P_{1} $$ positive-definite satisfying

\begin{align} (A+BK)^{T}P + P(A+BK) + K^{T}RK + Q&=0\\ B^{T}P + RK &=0\\ A^{T}P_{1} + P_{1}A &< Q \end{align}$$

Conclusion:
If the solution exists, then $$ K $$ is the optimal controller for the LQR optimization on the matrices $$ Q $$ and $$ R $$

Implementation
This implementation requires Yalmip and Sedumi.

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

Related LMIs

 * Multi-Criterion LQG]


text:
User:Margav06/sandbox/Click here to continue/Controller synthesis/Nonconvex Multi-Criterion Quadratic Problems

The Non-Concex Multi-Criterion Quadratic linear matrix inequality will allow one to form an optimized controller, similar to that in an LQR framework, for a non-convex state space system based on several different criterions defined in the Q and R matrices, that are optimized as a part of the arbitrary cost function. Just like traditional LQR, the cost matrices must be tuned in much a similar fashion as traditional gains in classical control. In the LQR and LQG framework however, the gains are more intuitive as each correlates directly to a state or an input.

The System
The system for this LMI is a linear time invariant system that can be represented in state space as shown below:

\begin{align} \dot x &=Ax+Bw, x(0) = x_{0}\\ \end{align}$$ where the system is assumed to be controllable.

where $$x \in R^{n}$$ represents the state vector, respectively, $$w \in R^{p}$$ is the disturbance vector, and $$A, B$$ are the system matrices of appropriate dimension. To further define: $$x$$ is $$\in R^{n}$$ and is the state vector, $$A $$ is $$\in R^{n*n}$$ and is the state matrix, $$B$$ is $$\in R^{n*r}$$ and is the input matrix, $$w$$ is $$\in R^{r}$$ and is the exogenous input.

for any input, we define a set $$p+1$$ cost indices $$J_{0},...,J_{P}$$ by

\begin{align} J_{i}(u) = \int_{0}^{\infty}\begin{bmatrix} x^{T}&u^{T} \end{bmatrix}\begin{bmatrix} Q_i & C_i\\ C_i^T & R_i \end{bmatrix}\begin{bmatrix} x\\ u \end{bmatrix} dt,\\ i = 0,...,p \end{align}$$

Here the symmetric matrices,

\begin{align} \begin{bmatrix} Q_i & C_i\\ C_i^T & R_i \end{bmatrix}, i=0,...,p \end{align}$$,

are not necessarily positive-definite.

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

The Optimization Problem
The constrained optimal control problem is:

\begin{align} \max:     J_0,\\ \end{align}$$ subject to

\begin{align} J_i \leq \gamma_i, i=1,...,p, x \rightarrow 0, t \rightarrow \infty \end{align}$$

The LMI: Nonconvex Multi-Criterion Quadratic Problems
The solution to this problem proceeds as follows: We first define

\begin{align} Q = Q_{0} + \sum_{i=1}^{p} \tau_i Q_i ,\\ R = R_{0} + \sum_{i=1}^{p} \tau_i R_i ,\\ C = C_{0} + \sum_{i=1}^{p} \tau_i C_i ,\\ \end{align}$$

where $$\tau_i \geq 0$$ and for every $$\tau_i$$, we define

\begin{align} S = J_0 + \sum_{i=1}^{p} \tau_i J_i - \sum_{i=1}^{p} \tau_i \gamma_i \end{align}$$

then, the solution can be found by:

\begin{align} \max: x(0)^{T}Px(0) - \sum_{i=1}^{p} \tau_i \gamma_i \end{align}$$

subject to

\begin{align} \begin{bmatrix} A^TP + PA + Q & PQ+C^T \\ B^TP+C &R \end{bmatrix} &\geq 0\\ \tau_i &\geq 0 \end{align}$$

Conclusion:
If the solution exists, then $$ K $$ is the optimal controller and can be solved for via an EVP in P.

Implementation
This implementation requires Yalmip and Sedumi.

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

Related LMIs

 * Multi-Criterion LQG
 * Inverse Problem of Optimal Control
 * Nonconvex Multi-Criterion Quadratic Problems
 * Static-State Feedback Problem


text:
User:Margav06/sandbox/Click here to continue/Controller synthesis/Static-State Feedback Problem

We are attempting to stabilizing The Static State-Feedback Problem

The System
Consider a continuous time Linear Time invariant system

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

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

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

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

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

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

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

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

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

The Problem 1 is equivalent to Problem 2

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

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

https://github.com/yashgvd/ygovada

Related LMIs
Hurwitz Stability


text:
User:Margav06/sandbox/Click here to continue/Controller synthesis/Mixed H2 Hinf with desired pole location control

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

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

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

$$

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

where

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Return to Main Page
LMIs in Control/Tools


text:
User:Margav06/sandbox/Click here to continue/Controller synthesis/Mixed H2 Hinf with desired pole location control for perturbed systems

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

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

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

$$

\begin{align} \dot{x} &= (A + \Delta A) x + (B_1 + \Delta B_1) u + B_2w \\ z_{\infty} &= C_{\infty} + D_{\infty1}u + D_{\infty2}w\\ z_2 &= C_2x+D_{21}u \end{align} $$

where

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

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

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

 * $$ \Delta A $$ and $$ \Delta B_1 $$ are real valued matrix functions which represent the time varying parameters uncertainities.

Furthermore, the parameter uncertainties $$ \Delta A $$ and $$ \Delta B_1 $$ are in the form of $$ [\Delta A \,\,\,\,\, \Delta B_1] = HF[E_1 \,\,\,\,\, E_2] $$ where $$ F^T F < I $$
 * $$ H $$, $$ E_1 $$ and $$ E_2 $$ are known matrices of appropriate dimensions.
 * $$ F $$ is a matrix containing the uncertainty, which satisfies

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

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

\begin{align} \dot{x} &= ((A+\Delta A) + (B_1 + \Delta B_1)K)x + B_2w \\ z_{\infty} &= (C_{\infty} + D_{\infty1}K)x + D_{\infty2}w\\ z_2 &= (C_2+D_{21}K)u \end{align} $$

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

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

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

&\begin{bmatrix} \Psi (X,W) & B_2 & (C_{\infty}X+D_{\infty 1}W)^T & (E_1X + E_2W)^T \\ B_2^T & -\gamma_{\infty}I & D_{\infty 2}^T & 0\\ C_{\infty}X+D_{\infty 1}W & D_{\infty 2} &-\gamma_{\infty}I & 0\\ (E_1X + E_2W) & 0 & 0 & -\alpha I \\ \end{bmatrix} < 0 \\

&\begin{bmatrix} \langle AX + B_1W \rangle + B_2B_2^T + \beta HH^T & (E_1X + E_2W)^T \\ E_1X + E_2W & -\beta I \\ \end{bmatrix} < 0 \\

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

where $$ \Psi (X,W) = \langle AX + B_1W \rangle + \alpha HH^T $$

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

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

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

Related LMIs
Mixed H2 Hinf  with desired poles controller

Return to Main Page
LMIs in Control/Tools


text:
User:Margav06/sandbox/Click here to continue/Controller synthesis/Robust H2 State Feedback Control

Robust $$H_{2}$$ State Feedback Control
For the uncertain linear system given below, and a scalar $$ \gamma > 0 $$. The goal is to design a state feedback control $$ u(t) $$ in the form of $$ u(t) =Kx(t) $$ such that the closed-loop system is asymptotically stable and satisfies.

\begin{align} \end{align}$$
 * G_{zw}(s)||_{2} < \gamma

The System
Consider System with following state-space representation.

\begin{align} \dot x(t)&=(A + \Delta{A})x(t)+ (B_{1} + \Delta{B_{1}}) u(t) + B_2 w(t)\\ z(t)& = C x(t) + D_{1} u(t) + D_{2} w(t)\\ \end{align}$$

where $$x\in\mathbb{R}^{n}$$, $$u\in\mathbb{R}^{r}$$ , $$w\in\mathbb{R}^{p}$$, $$z\in\mathbb{R}^{m}$$. For $$ H_{2} $$ state feedback control $$ D_{2}=0 $$

$$ \Delta{A} $$ and $$ \Delta{B_{1}} $$ are real valued matrix functions that represent the time varying parameter uncertainties and of the form

\begin{align} \begin{bmatrix} \Delta{A} & \Delta{B_{1}} \end{bmatrix} = HF \begin{bmatrix} E_{1} & E_{2} \end{bmatrix} \end{align}$$ where matrices $$ E_{1},E_{2}$$ and $$ H $$ are some known matrices of appropriate dimensions, while $$ F $$ is a matrix which contains the uncertain parameters and satisfies.

\begin{align} F^{T}F \leq I \end{align}$$

For the perturbation, we obviously have

\begin{align} \begin{bmatrix} \Delta{A} & \Delta{B_{1}} \end{bmatrix} =

\begin{bmatrix} 0 & 0 \end{bmatrix} \end{align}$$, for $$ F=0 $$

\begin{align} \begin{bmatrix} \Delta{A} & \Delta{B_{1}} \end{bmatrix} = H \begin{bmatrix} E_{1} & E_{2} \end{bmatrix} \end{align}$$, for $$ F=0 $$

The Problem Formulation:
The $$ H_{2} $$ state feedback control problem has a solution if and only if there exist a scalar $$ \beta $$, a matrix $$ W $$, two symmetric matrices $$ Z $$ and $$ X $$ satisfying the following LMI's problem.

The LMI:
$$ \min \gamma^{2}

\begin{align} \begin{bmatrix} \langle AX+B_{1}W \rangle_{s} + B_{2}B_{2}^{T} + \beta{H}H^(T) & (E_{1}X + E_{2}W)^{T} \\ E_{1}X + E_{2}W & -\beta{I} \end{bmatrix} < 0 \end{align}$$

\begin{align} \begin{bmatrix} -Z & CX + D_{1}W \\ (CX + D_{1}W)^{T} & -X \end{bmatrix} < 0 \end{align}$$

\begin{align} trace(Z) < \gamma^2 \end{align}$$ where $$ \langle M \rangle_{s} = (M+M^T) $$ is the definition that is need for the above LMI.

Conclusion:
In this case, an $$ H_{2} $$ state feedback control law is given by $$ u(t)=WX^{-1}x(t) $$.


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User:Margav06/sandbox/Click here to continue/Controller synthesis/LQ Regulation via H2 control

LQ Regulation via $$H_{2}$$ Control
The LQR design problem is to build an optimal state feedback controller $$ u=Kx $$ for the system $$ \dot x = Ax + Bu, x(0)=x_0 $$ such that the following quadratic performance index.

\begin{align} J(x,u) = \int_{0}^{\infty}(x^TQx+u^TRu)dt \end{align}$$ is minimized, where

\begin{align} Q = Q^T \geq 0, R = R^T > 0 \end{align}$$ The following assumptions should hold for a traditional solution.

$$ \boldsymbol{A1}. (A,B) $$ is stabilizable. $$ \boldsymbol{A2}. (A,L) $$ is observable, with $$ L = Q^{1/2} $$.

Relation to $$ H_{2} $$ performance
For the system given above an auxiliary system is constructed

\begin{align} \dot x =Ax + Bu + x_0\omega, y= Cx +Du \end{align}$$ where

\begin{align} C = \begin{bmatrix} Q^{1/2}\\ 0 \end{bmatrix}, D = \begin{bmatrix} 0\\ R^{1/2} \end{bmatrix} \end{align}$$ Where $$ \omega $$ represents an impulse disturbance. Then with state feedback controller $$ u=Kx $$ the closed loop transfer function from disturbance $$ \omega $$ to output $$ y $$ is

\begin{align} G_{y\omega}(s)=(C+DK)[sI - (A+BK)]^{-1}x_0 \end{align}$$ Then the LQ problem and the $$ H_{2} $$ norm of $$ G_{y\omega} $$ are related as

\begin{align} J(x,u) = ||G_{y\omega}(s)||_{2}^2 \end{align}$$ Then $$ H_{2} $$ norm minimization leads minimization of $$ J $$.

Data
The state-representation of the system is given and matrices $$ Q,R $$ are chosen for the optimal LQ problem.

The Problem Formulation:
Let assumptions $$ A1 $$ and $$ A2 $$ hold, then the state feedback control of the form $$ u=Kx $$ exists such that $$ J(x,u) < \gamma $$ if and only if there exist $$ X \in \mathbb{S}^n $$, $$ Y \in \mathbb{S}^{r} $$ and $$ W \in \mathbb{R}^{rxn} $$. Then $$ K $$ can be obtained by the following LMI.

The LMI:
$$ \min \gamma

\begin{align} (AX + BW) + (AX + BW)^T +x_0x_0^T < 0 \end{align}$$

\begin{align} trace(Q^{1/2}X(Q^{1/2})) + trace(Y) < \gamma \end{align}$$

\begin{align} \begin{bmatrix} -Y & R^{1/2}W \\ (R^{1/2}W)^T & -X \end{bmatrix} < 0 \end{align}$$

Conclusion:
In this case, a feedback control law is given as $$ K = WX^{-1} $$.


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Discrete Time Stabilizability

Discrete-Time Stabilizability

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

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

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

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

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Discrete-Time Stabilizability

The LMI formulation

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

K_d = WP^{-1}

\end{align}$$

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

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

Related LMIs
- Continuous Time Stabilizability


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Quadratic Schur Satbilization

LMI for Quadratic Schur Stabilization 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 with polytopic uncertainties and a linear time-invariant system with this property is called a Schur stable system.

The System
Consider discrete time system

\begin{align} x_{k+1}=Ax_k+Bu_k,\\ \end{align}$$ where $$x_k\in \R^n$$, $$u_k\in \R^m$$, at any $$t\in \R$$. The system consist of uncertainties of the following form

\begin{align} \Delta_{A(t)} = A_1 \delta_1 (t)+....+A_k \delta_k (t)\\ \Delta_{B(t)} = B_1 \delta_1 (t)+....+B_k \delta_k (t)\\ \end{align}$$

where $$x\in\mathbb{R}^{m}$$,$$u\in\mathbb{R}^{n}$$,$$A\in\mathbb{R}^{mxm}$$ and $$B\in\mathbb{R}^{mxn}$$

The Data
The matrices necessary for this LMI are $$A$$,$$\Delta_{A(t)} \, ie \, A_i $$ ,$$B$$ and $$\Delta_{B(t)} \, ie \, B_i $$

The LMI:
There exists some X > 0 and Z such that

\begin{align} \begin{bmatrix} X && AX+BZ \\ (AX+BZ)^T && X \end{bmatrix} +

\begin{bmatrix} 0 && A_iX + B_iZ \\(A_iX + B_iZ)^T&&0 \end{bmatrix} > 0 \quad i = 1,......,k

\end{align}$$

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:

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

Conclusion:
The Controller gain matrix is extracted as $$ F = ZX^{-1} $$ $$ u_k = Fx_k $$

\begin{align} x_{k+1}=Ax_k+Bu_k,\\ \quad\quad\quad = Ax_k + BFx_k\\ \quad\quad = (A+BF)x_k \end{align}$$

It follows that the trajectories of the closed-loop system (A+BK) are stable for any $$ \, \Delta \, \in \, C_0(\Delta_1,...,\Delta_k) $$

Implementation
https://github.com/JalpeshBhadra/LMI/blob/master/quadratic_schur_stabilization.m

Related LMIs
Schur Complement Schur Stabilization


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Generic Insensitive Strip Region Design

Insensitive Strip Region Design

Suppose if one were interested in robust stabilization where closed-loop eigenvalues are placed in particular regions of the complex plane where the said regions has an inner boundary that is insensitive to perturbations of the system parameter matrices. This would be accomplished with the help of 2 design problems: the insensitive strip region design and insensitive disk region design (see link below for the latter).

The System
Suppose we consider the following continuous-time linear system that needs to be controlled:

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

where $$x\in\mathbb{R}^{n}$$, $$y\in\mathbb{R}^{m}$$, and $$u\in\mathbb{R}^{r}$$ are the state, output and input vectors respectively. Then the steps to obtain the LMI for insensitive strip region design would be obtained as follows.

The Data
Prior to obtaining the LMI, we need the following matrices: $$A$$, $$B$$, and $$C$$.

The Optimization Problem
Consider the above linear system as well as 2 scalars $$\gamma_1$$ and $$\gamma_2$$. Then the output feedback control law $$u=Ky$$ would be such that $${\gamma_1} < {\lambda_i}(A^s_c) < {\gamma_1}$$, where:

\begin{align} {A^s_c}&\triangleq\frac{1}{2}{\langle{A_c}\rangle_s}=\frac{{(A+BKC)^T}+(A+BKC)}{2}\\ \end{align}$$

Letting $$K$$ being the solution to the above problem, then

\begin{align} {\gamma_1}&<{\alpha_1}\leq{Re({\lambda_i}(A+BKC))\leq{\alpha_2}<{\gamma_2}}, &i=1,2,...,n\\ \end{align}$$ where

\begin{align} \begin{cases} {\alpha_1}&={\lambda_{min}}({A^s_c})\\ {\alpha_2}&={\lambda_{max}}({A^s_c})\\ \end{cases} \end{align}$$

The LMI: Insensitive Strip Region Design
Using the above info, we can simplify the problem by setting $$\gamma_1$$ to $$-\infty$$ for all practical applications. This then simplifies our problem and results in the following LMI:

\begin{align} \begin{cases} \text{min }\gamma\\ \text{s.t. }{(A+BKC)^T}+(A+BKC)<{\gamma}I\\ \end{cases} \end{align}$$

Conclusion:
If the resulting solution from the LMI above produces a negative $$\gamma$$, then the output feedback controller $$K$$ is Hurwitz-stable. Hoewever, if $$\gamma$$ is a really small positive number, then $${\alpha_2}=\lambda_{max}(A^s_c)$$ must be negative for the controller to be Hurwitz-stable.

Implementation

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

Related LMIs

 * Insensitive Disk Region Design - Equivalent LMI for Insensitive Disk Region Design
 * $H_2$ Strip Region Design - LMI for $$H_2$$ Strip Region Design


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Generic Insensitive Disk Region Design

Insensitive Disk Region Design

Similar to the insensitive strip region design problem, insensitive disk region design is another way with which robust stabilization can be achieved where closed-loop eigenvalues are placed in particular regions of the complex plane where the said regions has an inner boundary that is insensitive to perturbations of the system parameter matrices.

The System
Suppose we consider the following linear system that needs to be controlled:

\begin{align} \begin{cases} \rho x&=Ax+Bu,\\ y&=Cx\\ \end{cases} \end{align}$$

where $$x\in\mathbb{R}^{n}$$, $$y\in\mathbb{R}^{m}$$, and $$u\in\mathbb{R}^{r}$$ are the state, output and input vectors respectively, and $$\rho$$ represents the differential operator (in the continuous-time case) or one-step shift forward operator (i.e., $$\rho x(k)=x(k+1)$$) (in the discrete-time case). Then the steps to obtain the LMI for insensitive strip region design would be obtained as follows.

The Data
Prior to obtaining the LMI, we need the following matrices: $$A$$, $$B$$, and $$C$$.

The Optimization Problem
Consider the above linear system as well as 2 positive scalars $$\gamma$$ and $$q$$. Then the output feedback control law $$u=Ky$$ would be designed such that:

\begin{align} {\eta}&=||A+BKC+qI||<{\gamma}\\ \end{align}$$

Recalling the definition, we have:

\begin{align} {\mathbb{D}_{q,\eta}}&=\{ {s|s\in\mathbb{C},|s+q|<{\eta}}\}\\ &=\{ x+jy|x,y\in\mathbb{R},{(x+q)^2}+{y^2}<{\eta^2}\}\\ \end{align}$$

and

\begin{align} {\mathbb{D}_{q,r}}&=\{ {s|s\in\mathbb{C},|s+q|<{\gamma}}\}\\ &=\{ x+jy|x,y\in\mathbb{R},{(x+q)^2}+{y^2}<{\gamma^2}\}\\ \end{align}$$

Letting $$K$$ being the solution to the above problem, then

\begin{align} {\lambda_i}(A+BKC)&\in{\mathbb{D}_{q,\eta}}\subset{\mathbb{D}_{q,r}}, &i=1,2,...,n\\ \end{align}$$

The LMI: Insensitive Strip Region Design
Using the above info, we can convert the given problem into an LMI, which - after using Schur compliment Lemma - results in the following:

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

Conclusion:
For Schur stabilization, we can choose to solve the problem with $$q=0$$. Schur stability is achieved when $${\gamma}{\leq}1$$. Alternately, if $${\gamma}$$ is greater than (but very close to) 1, then Schur stability is also achieved when $${\eta}=||A+BKC+qI||_2{\leq}1$$.

Implementation

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

Related LMIs

 * $H_2$ Disk Region Design - LMI for Disk Region Design with minimal $$H_2$$ gain
 * Insensitive Strip Region Design - Equivalent LMI for Insensitive Strip Region Design


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Design for Insensitive Strip Region

Insensitive Strip Region Design with Minimum $$ H_2 $$ Gain

When designing controllers with insensitive region conditions, the aim is to place the closed-loop poles of the system in a particular region defined by its inner boundary. These regions are specified based on their insensitivity to perturbations to the system parameter matrices.

One type of such design is the Insensitive Strip Region Design. In this section, building upon that, optimization problems will be provided that ensure that the conditions for insensitive strip region design are satisfied with some bounds on the $$ H_2 $$ gain of the closed-loop system.

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) \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.

The Data
To solve the design optimization problem, the linear system matrices A,B,C are required. Furthermore, to define the strip region on the eigenvalue-space, two parameters $$ \gamma_1 $$ and $$ \gamma_2 $$ are required.

The Optimization Problem
The problem of designing an $$ H_2 $$ optimal controller that results in the closed loop system insensitive to a certain strip region involves two sub-problems:

 * Finding a control gain $$ K $$ such that: $$ \left\| K \right\|_2 < \gamma $$.
 * The conditions for insensitive strip region design for the closed-loop system, as provided in the section Insensitive Strip Region Design are fulfilled.
 * The optimization goal is to minimize $$ \gamma $$ such that above two hold.

The LMI: $$ H_2 $$ Optimal Control Design for Insensitive Strip Region
The problem above has a solution if and only if the following optimization problem has a solution $$ (K, \gamma) $$:

$$ \begin{align} \text{min   } & \gamma \\ \text{s.t.  } & \begin{bmatrix} - \gamma I & K \\ K^\top & - \gamma I \end{bmatrix} < 0 \\ & 2 \gamma_1 I < (A+BKC)^\top + (A+BKC) < 2 \gamma_2 I \end{align} $$

Conclusion:
By using the design problem provided here, an optimal $$ H_2 $$ controller is designed to make the closed-loop system robust to perturbations in the system matrices.

Implementation
To solve the optimization problem with LMI presented here, 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/H2_Strip_example.m

Related LMIs
Insensitive Strip Region Design


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Design for Insensitive Disk Region

Insensitive Disk Region Design with Minimum $$ H_2 $$ Gain

Apart from the design for the insensitive strip region with minimum $ H_2 $ gain, another type of such design is the Insensitive Disk Region Design. In this section, optimization problems will be provided that ensure that the conditions for insensitive disk region design are satisfied with some bounds on the $$ H_2 $$ gain of the closed-loop system.

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

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

where $$x\in \R^n$$, $$y\in \R^m$$ and $$u\in \R^r$$ are the system state, output, and the input vector respectively. $$ \rho $$ represents the differential operation for continuous time systems, or the one-step shift forward operator for discrete time case.

The Data
To solve the design optimization problem, the linear system matrices A,B,C are required. Furthermore, to define the disk region on the eigenvalue-space, its radius $$ \gamma_0 $$ is required.

The Optimization Problem
The problem of designing an $$ H_2 $$ optimal controller that results in the closed loop system insensitive to a certain disk region involves two sub-problems:

 * Finding a control gain $$ K $$ such that: $$ \left\| K \right\|_2 < \gamma $$.
 * The conditions for insensitive disk region design for the closed-loop system, as provided in the section Insensitive Disk Region Design are fulfilled.
 * The optimization goal is to minimize $$ \gamma $$ such that above two hold.

The LMI: $$ H_2 $$ Optimal Control Design for Insensitive Disk Region
The problem above has a solution if and only if the following optimization problem has a solution $$ (K, \gamma) $$:

$$ \begin{align} \text{min   } & \gamma \\ \text{s.t.  } & \begin{bmatrix} - \gamma I & K \\ K^\top & - \gamma I \end{bmatrix} < 0 \\ & \begin{bmatrix} -\gamma_0 I & A+ BKC + qI \\ (A+ BKC + qI)^\top & -\gamma_0 I \end{bmatrix} < 0 \end{align} $$

Conclusion:
By using the design problem provided here, an optimal $$ H_2 $$ controller is designed to make the closed-loop system robust to perturbations in the system matrices.

Implementation
To solve the optimization problem with LMI presented here, 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/H2_Disk_example.m

Related LMIs
Insensitive Disk Region Design


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$Q > 0$$

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

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

Conclusion:
The LMI is feasible.


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Apkarian Filter and State Feedback

The System:
The number of LMI constraints needed to check quadratic stability is reduced if all the subsystems in the polytopic model has the same matrix $$B$$. This can be achieved by adding an Apkarian filter in the input of the system.

The Optimization Problem:
Apkarian Filter

Let consider our TS-LIA model. This can be re written in linear form as:

\dot{x} = A(z(t)) x + B(z(t)) u

$$

The filter should be such that the equilibrium of the states are the input values and the dynamics should be fast, so we could assume the dynamics of the filter negligible (i.e. the input of the filter is equivalent to the input of the quadrotor). One possible filter is shown, where $$ A_F$$ = −100$$ I_4$$, $$B_F$$ = 100$$I_4$$ and $$I_4$$ ∈ $$R^{4\times4}$$ is the identity matrix.

 * $$ \dot{x_F} = A_F x_F + B_F u_F; y_f = x_f$$.

When applying the filter, we are imposing that the output of the filter is the new input of the TS-LIA model (i.e. $$u$$ = $$y_F$$ ). Then, the extended model is:

\dot{x_c}= \begin{bmatrix} A(z(t))&B(z(t))\\ 0&A_F\\ \end{bmatrix} x_c + \begin{bmatrix}0\\B_F\end{bmatrix} u_F = A_e(z(t)) x_e + B_e u_f;

x_e = \begin{bmatrix}x\\x_F \end{bmatrix} $$

This prefiltering does not affect the procedure followed to obtain the TS-LIA model, so the premise variables, membership functions and activations functions remains the same.

State Feedback Controller Design

Let consider the state feedback control law for the extended TS-LIA model:$$ \dot{x_e} = \sum_{i=1}^{32} h_i(z(t)) [ A_{ei} x_e + B_{ei} u_F ]$$, where the state feedback control laws are :$$ u_F= \sum_{i=1}^{32} h_i(z(t)) K_i x(t) $$, we get the closed loop system  :$$ \dot{x_e} = \sum_{i=1}^{32} \sum_{j=1}^{32} h_j(z(t)) [ A_{ei} x_e + B_{ei} K_j ]x_e$$

The LMI:
The design of the controller is done by solving an LMI problem involving the quadratic stability constraints. In case we want D- stabilization, the following set of LMI constraints are needed:

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

Conclusion:
The LMI is feasible.

Related LMIs

 * LMI for Natural Frequency in State Feedback. https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Maximum_Natural_Frequency_in_State_Feedback#The_LMI%3A
 * LMI for Minimum Decay Rate in State Feedback. https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Minimum_Decay_Rate_in_State_Feedback#The_LMI%3A


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Minimum Decay Rate in State Feedback

The System:
The number of LMI constraints needed to check quadratic stability is reduced if all the subsystems in the polytopic model has the same matrix $$B$$. This can be achieved by adding an Apkarian filter in the input of the system.

The Optimization Problem:
Apkarian Filter

Let consider our TS-LIA model. This can be re written in linear form as:

\dot{x} = A(z(t)) x + B(z(t)) u

$$

The filter should be such that the equilibrium of the states are the input values and the dynamics should be fast, so we could assume the dynamics of the filter negligible (i.e. the input of the filter is equivalent to the input of the quadrotor). One possible filter is shown, where $$ A_F$$ = −100$$ I_4$$, $$B_F$$ = 100$$I_4$$ and $$I_4$$ ∈ $$R^{4\times4}$$ is the identity matrix.

 * $$ \dot{x_F} = A_F x_F + B_F u_F; y_f = x_f$$.

When applying the filter, we are imposing that the output of the filter is the new input of the TS-LIA model (i.e. $$u$$ = $$y_F$$ ). Then, the extended model is:

\dot{x_c}= \begin{bmatrix} A(z(t))&B(z(t))\\ 0&A_F\\ \end{bmatrix} x_c + \begin{bmatrix}0\\B_F\end{bmatrix} u_F = A_e(z(t)) x_e + B_e u_f;

x_e = \begin{bmatrix}x\\x_F \end{bmatrix} $$

This prefiltering does not affect the procedure followed to obtain the TS-LIA model, so the premise variables, membership functions and activations functions remains the same.

State Feedback Controller Design

Let consider the state feedback control law for the extended TS-LIA model:$$ \dot{x_e} = \sum_{i=1}^{32} h_i(z(t)) [ A_{ei} x_e + B_{ei} u_F ]$$, where the state feedback control laws are :$$ u_F= \sum_{i=1}^{32} h_i(z(t)) K_i x(t) $$, we get the closed loop system  :$$ \dot{x_e} = \sum_{i=1}^{32} \sum_{j=1}^{32} h_j(z(t)) [ A_{ei} x_e + B_{ei} K_j ]x_e$$

The LMI:
The design of the controller is done by solving an LMI problem involving the quadratic stability constraints. In case we want D- stabilization, the following set of LMI constraints are needed:

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

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

Minimum Decay Rate:

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

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

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

Conclusion:
The LMI is feasible.

Related LMIs

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


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User:Margav06/sandbox/Click here to continue/Controller synthesis/Maximum Natural Frequency in State Feedback

The System:
The number of LMI constraints needed to check quadratic stability is reduced if all the subsystems in the polytopic model has the same matrix $$B$$. This can be achieved by adding an Apkarian filter in the input of the system.

The Optimization Problem:
Apkarian Filter

Let consider our TS-LIA model. This can be re written in linear form as:

\dot{x} = A(z(t)) x + B(z(t)) u

$$

The filter should be such that the equilibrium of the states are the input values and the dynamics should be fast, so we could assume the dynamics of the filter negligible (i.e. the input of the filter is equivalent to the input of the quadrotor). One possible filter is shown, where $$ A_F$$ = −100$$ I_4$$, $$B_F$$ = 100$$I_4$$ and $$I_4$$ ∈ $$R^{4\times4}$$ is the identity matrix.

 * $$ \dot{x_F} = A_F x_F + B_F u_F; y_f = x_f$$.

When applying the filter, we are imposing that the output of the filter is the new input of the TS-LIA model (i.e. $$u$$ = $$y_F$$ ). Then, the extended model is:

\dot{x_c}= \begin{bmatrix} A(z(t))&B(z(t))\\ 0&A_F\\ \end{bmatrix} x_c + \begin{bmatrix}0\\B_F\end{bmatrix} u_F = A_e(z(t)) x_e + B_e u_f;

x_e = \begin{bmatrix}x\\x_F \end{bmatrix} $$

This prefiltering does not affect the procedure followed to obtain the TS-LIA model, so the premise variables, membership functions and activations functions remains the same.

State Feedback Controller Design

Let consider the state feedback control law for the extended TS-LIA model:$$ \dot{x_e} = \sum_{i=1}^{32} h_i(z(t)) [ A_{ei} x_e + B_{ei} u_F ]$$, where the state feedback control laws are :$$ u_F= \sum_{i=1}^{32} h_i(z(t)) K_i x(t) $$, we get the closed loop system  :$$ \dot{x_e} = \sum_{i=1}^{32} \sum_{j=1}^{32} h_j(z(t)) [ A_{ei} x_e + B_{ei} K_j ]x_e$$

The LMI:
The design of the controller is done by solving an LMI problem involving the quadratic stability constraints. In case we want D- stabilization, the following set of LMI constraints are needed:

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

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

Maximizing Natural Frequency:

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

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

Conclusion:
The LMI is feasible.

Related LMIs

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


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

Observer synthesis

 * 1) Continuous Time
 * 2) /FDI Filter Design For Systems With Sensor Faults/
 * 3) /Hurwitz Detectability/
 * 4) /Full-order state Observer/
 * 5) /Full-order Hinf state Observer/
 * 6) /Reduced-Order State Observer/
 * 7) Discrete-Time
 * 8) /Discrete Time Detectability/
 * 9) /Schur Detectability/