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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Discrete Time Minimum Gain Lemma

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

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

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

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

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

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

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

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

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

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

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

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

 * $$\begin{bmatrix}

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

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

 * $$\begin{bmatrix}

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

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

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

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

Related LMIs
KYP Lemma State Space Stability KYP Lemma without Feedthrough


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Discrete Time Modified Minimum Gain Lemma

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

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

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

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

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

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

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

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

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

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

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

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

 * $$\begin{bmatrix}

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

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

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

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

 * $$\begin{bmatrix}

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

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

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

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

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

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


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Finsler’s Lemma

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

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

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

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

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

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

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

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

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

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

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

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

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

there exists $$\epsilon$$ such that

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

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


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Dilation

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

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

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

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

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

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

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

which is equivalent to

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

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

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

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

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

we can now rewrite the inequality (3) as

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

which is the new dilated inequality.

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

Example 1

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

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

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

Example 2

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

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

implies the inequality

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

Example 3

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

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

implies the inequality

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

Related Pages

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


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Tangential Nevanlinna Pick

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

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

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

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

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

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

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

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

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

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

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

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

Related LMIs
Nevalinna-Pick Interpolation with Scaling

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


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Nevanlinna Pick Interpolation with Scaling

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

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

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

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

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

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

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

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

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

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

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

Related LMIs
Nevalinna-Pick Interpolation

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


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Deduced LMI Conditions for Hinf Index

H-infinity Index Deduced LMI

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

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

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

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

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

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

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

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

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

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

where:

\begin{align}

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

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

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

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

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

Related LMIs
Bounded Real Lemma


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Deduced LMI Conditions for H2 Index

H2 Index Deduced LMI

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

https://github.com/yashgvd/ygovada

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


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Dissipativity of Systems

Dissipativity of Systems

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Related LMIs
Continuous_Time_Lyapunov_Inequality


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Generalized H2 Norm

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

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

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

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

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

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

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

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

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

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

Implementation
This implementation requires Yalmip and Sedumi.

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

Related LMIs
LMI for System H_{2} Norm

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LMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Passivity and Positive Realness

This section deals with passivity of a system.

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

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

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

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

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

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

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

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

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

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


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Non-Expansivity and boundedness Realness

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

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

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

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

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

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

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

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

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

and

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

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


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Variable Reduction Lemma

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

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

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

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

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

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

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

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

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https://en.wikibooks.org/wiki/LMIs_in_Control


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/D-stability Rise Poles

LMI for Rise Time Poles

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

The System
We consider the following system:

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

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

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

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

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

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

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

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

https://github.com/maxwellpeterson99/MAE509Code

Related LMIs
- D-stabilization

- D-stability Controller

- D-stability Observer

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-Matrix and LMI Properties and Tools


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/D-stability Settling time poles

LMI for Settling Time Poles

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

The System
We consider the following system:

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

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

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

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

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

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

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

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

https://github.com/maxwellpeterson99/MAE509Code

Related LMIs
- D-stabilization

- D-stability Controller

- D-stability Observer

Return to Main Page
-Matrix and LMI Properties and Tools


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/D-stability Max Percent Overshoot Poles

LMI for Max Percent Overshoot Poles

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

The System
We consider the following system:

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

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

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

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

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

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

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

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

https://github.com/maxwellpeterson99/MAE509Code

Related LMIs
- D-stabilization

- D-stability Controller

- D-stability Observer

Return to Main Page
-Matrix and LMI Properties and Tools


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Frobenius Norm

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

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

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

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

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

$$

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

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

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

$$

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


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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Dualization Lemma

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

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

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

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

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

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

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

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

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

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

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

\end{align} $$.


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LMIs in Control/Click here to continue/Software for LMIs and Optimization

Software for LMIs and Optimization

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

Gurobi :

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

Sedumi :

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

Cplex :

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

MOSEK:

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

SOSTOOLS:

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


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LMIs in Control/Click here to continue/Optimal control systems

Optimal control systems

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


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LMIs in Control/Click here to continue/Optimal control systems/H2-optimal Full-state Feedback control

Full State Feedback Optimal $$H_2$$ Control
Full State Feedback in general has the goal of positioning a system's closed loop poles in a desired location. This allows us to specify the performance of the system 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, particularly when there is information about the distribution of the noise.

The System
The system is represented using the 9-matrix notation shown below.
 * $$\begin{bmatrix} \dot{x}\\z\\y \end{bmatrix}=\begin{bmatrix} A & B_{1} & B_{2}\\ C_{1} & D_{11} & D_{12}\\ C_{2} & D_{21} & D_{22} \end{bmatrix} \begin{bmatrix} x\\w\\u \end{bmatrix}

$$ where $$x(t)\in \R^n$$ is the state, $$z(t)\in \R^p$$ is the regulated output, $$y(t)\in \R^q$$ is the sensed output, $$w(t)\in \R^r$$ is the exogenous input, and $$u(t)\in \R^m$$ is the actuator input, at any $$t\in \R$$.

The Data
$$A$$, $$B_{1}$$, $$B_{2}$$, $$C_{1}$$, $$C_{2}$$, $$D_{11}$$, $$D_{12}$$, $$D_{21}$$, $$D_{22}$$ are known.

The LMI: Optimal Output Feedback $$H_{\infty}$$ Control LMI
The following are equivalent.

1) There exists a $$K$$ such that $$||S(K,P)||_{H_{2}}<\gamma$$

2) There exists $$X>0$$, $$Z$$ and $$W$$ such that

 * $$\begin{bmatrix} A & B_2 \end{bmatrix} \begin{bmatrix}

X \\ Z \end{bmatrix}+ \begin{bmatrix} X & Z^T \end{bmatrix} \begin{bmatrix} A^T \\ B_2^T \end{bmatrix}+ B_1B_1^T <0 $$
 * $$\begin{bmatrix}

X & *^T \\ C_1X+D_{12}Z & W \end{bmatrix} > 0 $$
 * $$trace(W)<\gamma^2$$

where $$K=ZX^{-1}$$

Conclusion:
This LMI solves the $$H_2$$ optimal full state feedback problem and finds the upper bound of the $$H_2$$ norm of the system, $$\gamma$$. In addition to this the controller $$K$$ is also found in the process.

Implementation
This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/FSF_H2.m

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


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LMIs in Control/Click here to continue/Optimal control systems/Discrete-time H2-Optimal Full-state Feedback control

Discrete-Time H2-Optimal Full-State Feedback Control

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.

A full state feedback controller design for a Discrete Time system, to minimize the H2 norm of the closed loop system with exogenous input $$w_{k}$$ and performance output $$z_{k}$$.

The System
Discrete-Time LTI System with state space realization $$(A_d,B_d,C_d,D_d)$$ $$ \begin{align} x_{k+1}&=A_dx_k + B_{d1}w_k + B_{d2}u_k\\ z_{k}&=C_{d1}x_k + D_{d12}u_k\\ y_k &= x_k\\ \end{align}$$

The Data
The matrices: System $$ (A_d,B_{d1},B_{d2},C_{d1},D_{d12}), P, Z, F_d $$.

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Discrete-Time H2-Optimal Full-State Feedback Control

The LMI formulation

H2 norm < $$\mu$$

$$ \begin{align} P \in {S^{n_x}}; Z \in {S^{n_z}}; &F_d \in {R^{n_u*n_x}}; \mu \in {R_{>0}} \;\\ &P>0\\ &Z>0\\ \begin{bmatrix}P & A_dP+B_{d2}F_d & B_{d1}\\
 * & P & 0\\
 * & * & I\end{bmatrix}&>0,\\

\begin{bmatrix}Z & C_{d1}P+D_{d12}F_d\\
 * & P\end{bmatrix}&>0,\\

&trZ<\mu ^2

\end{align}$$

Conclusion:
The H2-optimal full-state feedback gain is recovered by $$K_{d}=F_{d}P^{-1}$$

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

Related LMIs
- Continuous Time H2 Optimal Full State Feedback Control


text:
LMIs in Control/Click here to continue/Optimal control systems/H2-optimal Dynamic output Feedback control

Discrete-Time H2-Optimal Dynamic Output Feedback Control

A Dynamic Output feedback controller is designed for a Continuous Time system, to minimize the H2 norm of the closed loop system with exogenous input $$w$$ and performance output $$z$$.

The System
Continuous-Time LTI System with state space realization $$(A,B,C,D)$$

\begin{align} \dot{x}&=A_x + B_{1}w + B_{2}u\\ z&=C_{1}x + D_{11}w_k+ D_{12}u\\ y&=C_{2}x + D_{21}w+ D_{22}u\\ \end{align}$$

The Data
The matrices: System $$ (A,B_{1},B_{2},C_{1},C_{1},D_{11},D_{12},D_{21},D_{22}), X_1,Y_1,Z, ,X_2,Y_2 $$

Controller $$ (A_{c},B_{c},C_{c},D_{c}) $$

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Solve for $$A_{n} \in {R^{n_x \times n_x}}, B_{n} \in {R^{n_x \times n_y}}, C_{n} \in {R^{n_u \times n_x}}, D_{n} \in {R^{n_u \times n_y}},X_{1}, Y_{1}\in {S^{n_x}}, Z \in {S^{n_z}},$$ and $$\nu \in {R_{>0}}$$ that minimize $$\mathcal{J}(\nu)<\nu$$ subject to $$X_{1}>0, Y_{1}>0,Z>0,$$

\begin{align}

\begin{bmatrix} AY_1+Y_1A^T+B_2C_n+C_n^TB_2^T &A+A_n^T+B_2D_nC_2 &B_1+B_2D_nD_{21}\\
 * &X_1A+A^TX_1+B_nC_2+C_2^TB_n^T &X_1B_1+B_nD_{21}\\
 * & * &-\mathbf{1}\end{bmatrix}&>0,\\

\begin{bmatrix} X_1 &\mathbf{1} &Y_1C_1^T+C_n^TD_{12}^T\\
 * &Y_1 &C_1^T+C_2^TD_n^TD_{12}^T \\
 * & * &Z \end{bmatrix}&>0,\\

D_{11}+D_{12}D_{n}D_{21}=0\\

\begin{bmatrix}X_{1} & \mathbf{1}\\
 * & Y_{1}\end{bmatrix}&>0,\\

trZ<\nu \end{align}$$

The controller is recovered by

$$ \begin{align} &A_{c} = A_{k}-B_{c}(1-D_{22}D_{c})^{-1}D_{22}C_{c}\\ &B_{c} = B_{k}(1-D_{c}D_{22})\\ &C_{c} = (1-D_{c}D_{22})C_{k}\\ &D_{c} = (1+D_{k}D_{22})^{-1}D_{k}\\ \end{align}$$

where, $$ \begin{align} \begin{bmatrix}A_{k} & B_{k}\\ C_{k} & D_{k}\end{bmatrix}&= \begin{bmatrix}X_{2} & X_{1}B_{2}\\ \mathbf{0} & 1\end{bmatrix}^{-1} (\begin{bmatrix}A_{n} & B_{n}\\ C_{n} & D_{n}\end{bmatrix} -\begin{bmatrix}X_{1}AY_{1} & \mathbf{0}\\ \mathbf{0} & \mathbf{0}\end{bmatrix}) \begin{bmatrix}Y_{2}^T & \mathbf{0}\\ CY_{1} & \mathbf{1}\end{bmatrix}^{-1}

\end{align}$$ and the matrices $$X_2$$ and $$Y_2$$ satisfy $$X_2Y_2^T = 1 - X_{1}Y_{1}$$. If $$D_{22}=0,$$ then $$A_c=A_k,B_c=B_k,C_c=C_k$$ and $$D_c=D_k.$$

Given $$X_1$$ and $$Y_1$$, the matrices $$X_2$$ and $$Y_2$$ can be found using a matrix decomposition, such as a LU decomposition or a Cholesky decomposition.

If $$D_{11} = 0$$, $$D_{12} \neq 0,$$ and $$D_{21} \neq 0$$, then it is often simplest to choose $$D_{n} = 0$$ in order to satisfy the equality constraint

Conclusion:
The Continuous-Time H2-Optimal Dynamic Output feedback controller is the system $$ (A_{c},B_{c},C_{c},D_{c}) $$

Related LMIs
Discrete Time H2 Optimal Dynamic Output Feedback Control

Continuous Time H∞ Optimal Dynamic Output Feedback Control


text:
LMIs in Control/Click here to continue/Optimal control systems/Discrete-time H2-optimal Dynamic output Feedback control

Discrete-Time H2-Optimal Dynamic Output Feedback Control

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.

A Dynamic Output feedback controller is designed for a Discrete Time system, to minimize the H2 norm of the closed loop system with exogenous input $$w_{k}$$ and performance output $$z_{k}$$.

The System
Discrete-Time LTI System with state space realization $$(A_d,B_d,C_d,D_d)$$ $$ \begin{align} x_{k+1}&=A_dx_k + B_{d1}w_k + B_{d2}u_k\\ z_{k}&=C_{d1}x_k + D_{d11}w_k+ D_{d12}u_k\\ y_k &=C_{d2}x_k + D_{d21}w_k+ D_{d22}u_k\\ \end{align}$$

The Data
The matrices: System $$ (A_d,B_{d1},B_{d2},C_{d1},C_{d1},D_{d11},D_{d12},D_{d21},D_{d22}), X_1,Y_1,Z, ,X_2,Y_2 $$

Controller $$ (A_{dc},B_{dc},C_{dc},D_{dc}) $$

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Discrete-Time H2-Optimal Full-State Feedback Control

The LMI formulation

H2 norm < $$\mu$$

$$ \begin{align} X_{1}, Y_{1}\in {S^{n_x}}; Z \in {S^{n_z}}; \mu \in {R_{>0}} \;\\ A_{dn} \in {R^{n_x*n_x}}; B_{dn} \in {R^{n_x*n_y}}; C_{dn} \in {R^{n_u*n_x}}; D_{dn} \in {R^{n_u*n_y}};\\ &X_{1}>0\\ &Y_{1}>0\\ &Z>0\\ \begin{bmatrix}X_{1} & 1 & X_{1}A_d+B_{dn}C_{d2} & A_{dn} & X_{1}B_{d1}+B_{dn}D_{d21}\\
 * & Y_{1} & A_d+B_{d2}D_{dn}C_{d2} & A_dY_{1}+B_{d2}C_{dn} & B_{d1}+B_{d2}D_{dn}D_{d21}\\
 * & * & X_{1} & 1 & 0\\
 * & * & * & Y_{1} & 0\\
 * & * & * & * & 1\end{bmatrix}&>0,\\

\begin{bmatrix}Z & C_{d1}+D_{d12}D_{dn}C_{d2} & C_{d1}Y_{1}^T+D_{d12}C_{dn}\\
 * & X_{1} & 1\\
 * & * & Y_{1}\end{bmatrix}&>0,\\

D_{d11}+D_{d12}D_{dn}D_{d21}=0\\

\begin{bmatrix}X_{1} & 1\\
 * & Y_{1}\end{bmatrix}&>0,\\

trZ<\mu ^2 \end{align}$$

The controller is recovered by

$$ \begin{align} &A_{dc} = A_{dk}-B_{dc}(1-D_{d22}D_{dc})^{-1}D_{d22}C_{dc}\\ &B_{dc} = B_{dk}(1-D_{dc}D_{d22})\\ &C_{dc} = (1-D_{dc}D_{d22})C_{dk}\\ &D_{dc} = 1+D_{dk}D_{d22})^{-1}D_{dk}\\ \end{align}$$

where, $$ \begin{align} \begin{bmatrix}A_{dk} & B_{dk}\\ C_{dk} & D_{dk}\end{bmatrix}&= \begin{bmatrix}X_{2} & X_{1}B_{d2}\\ 0 & 1\end{bmatrix}^{-1} (\begin{bmatrix}A_{dn} & B_{dn}\\ C_{dn} & D_{dn}\end{bmatrix} -\begin{bmatrix}X_{1}A_{d}Y_{1} & 0\\ 0 & 0\end{bmatrix}) \begin{bmatrix}Y_{2}^T & 0\\ C_{d2}Y_{1} & 1\end{bmatrix}^{-1}

\end{align}$$ and the matrixes $$X_2$$ and $$Y_2$$ satisfies $$X_2Y_2^T = 1 - X_{1}Y_{1}$$

Given $$X_1$$ and $$Y_1$$, the matrices $$X_2$$ and $$Y_2$$ can be found using a matrix decomposition, such as a LU decomposition or a Cholesky decomposition.

If $$D_{d11} = 0$$, $$D_{d12}$$ ≠ 0, and $$D_{d21}$$ ≠ 0, then it is often simplest to choose $$D_{dn} = 0$$ in order to satisfy the equality constraint

Conclusion:
The Discrete-Time H2-Optimal Dynamic Output feedback controller is the system $$ (A_{dc},B_{dc},C_{dc},D_{dc}) $$

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

Related LMIs
- Continuous Time H2 Optimal Dynamic Output Feedback Control


text:
LMIs in Control/Click here to continue/Optimal control systems/Hinf-optimal Full-state Feedback Control

Full State Feedback Optimal $$H_{\infty}$$ Control
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. In a single-input single-output (SISO) system this norm represents the maximum gain on a magnitude Bode plot. In the case of multi-input multi-output (MIMO) systems it can be interpreted as maximum response to a perturbation introduced to the system. In either, by minimizing the $$H_{\infty}$$ we are minimizing the worst case effect of a disturbance to the system, whether it is noise or another perturbation.

The System
The system is represented using the 9-matrix notation shown below.
 * $$\begin{bmatrix} \dot{x}\\z\\y \end{bmatrix}=\begin{bmatrix} A & B_{1} & B_{2}\\ C_{1} & D_{11} & D_{12}\\ C_{2} & D_{21} & D_{22} \end{bmatrix} \begin{bmatrix} x\\w\\u \end{bmatrix}

$$ where $$x(t)\in \R^n$$ is the state, $$z(t)\in \R^p$$ is the regulated output, $$y(t)\in \R^q$$ is the sensed output, $$w(t)\in \R^r$$ is the exogenous input, and $$u(t)\in \R^m$$ is the actuator input, at any $$t\in \R$$.

The lower linear fractional transformation (LFT) is used to implement a controller $$K$$ into the system. The lower LFT is denoted as $$\underline{S}(P,K)$$ and is formed by $$\underline{S}(P,K)=P_{11}+P_{12}(I-KP_{22})^{-1}KP_{21}$$ with $$\begin{bmatrix} z\\y \end{bmatrix}=\begin{bmatrix} P_{11} & P_{12} \\ P_{21} & P_{22} \end{bmatrix} \begin{bmatrix} w\\u \end{bmatrix} $$. For full-state feedback we consider a controller of the form $$u(t)=Fx(t)$$. This is a special case where $$y(t)=x(t)$$ and results in a controller of the form $$ K= \begin{bmatrix} 0 & 0 \\ 0 & F \end{bmatrix} $$.

The Data
$$A$$, $$B_{1}$$, $$B_{2}$$, $$C_{1}$$, $$C_{2}$$, $$D_{11}$$, $$D_{12}$$, $$D_{21}$$, $$D_{22}$$ are known.

The LMI:Full State Feedback Optimal $$H_{\infty}$$ Control LMI
The following are equivalent.

1) There exists a $$F$$ such that $$||\underline{S}(P,K(0,0,0,F)||_{H_{\infty}}\leq \gamma$$

2) There exists $$Y>0$$ and $$Z$$ such that

 * $$\begin{bmatrix} YA^T+AY+Z^TB_2^{T}+B_2Z & B_1 & YC^{T}_1 + Z^{T} D^{T}_{12} \\ B1^{T}_1 & -\gamma I & D^{T}_{11} \\ C_1 Y + D_{12} Z & D_{11} & -\gamma I \end{bmatrix}<0

$$.

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

Conclusion:
The above LMI, if feasible, will determine the bound $$\gamma$$ on the $$H_{\infty}$$ norm of the system. In addition to this $$F$$ is also determined allowing the closed loop system to be determined using the controller $$\hat{K}(0,0,0,F)$$ found during the optimization.

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

Related LMIs
Full State Feedback Optimal H2 LMI


text:
LMIs in Control/Click here to continue/Optimal control systems/Discrete-time Hinf-Optimal Full State Feedback Control

Discrete-Time H∞-Optimal Full-State Feedback Control

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.

A full state feedback controller design for a Discrete Time system, to minimize the H∞ norm of the closed loop system with exogenous input $$w_{k}$$ and performance output $$z_{k}$$.

The System
Discrete-Time LTI System with state space realization $$(A_d,B_d,C_d,D_d)$$ $$ \begin{align} x_{k+1}&=A_dx_k + B_{d1}w_k + B_{d2}u_k\\ z_{k}&=C_{d1}x_k + D_{d12}u_k\\ y_k &= x_k\\ \end{align}$$

The Data
The matrices: System $$ (A_d,B_{d1},B_{d2},C_{d1},D_{d12}), P, F_d $$.

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Discrete-Time H∞-Optimal Full-State Feedback Control

The LMI formulation

H∞ norm < $$\gamma$$

$$ \begin{align} P \in {S^{n_x}}; &F_d \in {R^{n_u*n_x}}; \gamma \in {R_{>0}} \;\\ &P>0\\ \begin{bmatrix}P & A_dP-B_{d2}F_d & B_{d1} & 0\\
 * & P & 0 & PC_{d1}^T-F_{d}^TD_{d12}^T\\
 * & * & -\gamma I & D_{d11}^T\\
 * & * & * & -\gamma I\end{bmatrix}&>0,\\

\end{align}$$

Conclusion:
The H∞-optimal full-state feedback gain is recovered by $$K_{d}=F_{d}P^{-1}$$

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

Related LMIs
- Continuous Time H∞ Optimal Full State Feedback Control


text:
LMIs in Control/Click here to continue/Optimal control systems/Hinf-optimal Dynamic output Feedback control

Discrete-Time H∞-Optimal Dynamic Output Feedback Control

In this section, a Dynamic Output feedback controller is designed for a Continuous Time system, to minimize the H∞ norm of the closed loop system with exogenous input $$w_{k}$$ and performance output $$z_{k}$$.

The System
Continuous-Time LTI System with state space realization $$(A,B,C,D)$$

\begin{align} \dot{x}&=Ax + B_1w + B_2u\\ z&=C_1x + D_{11}w+ D_{12}u\\ y &=C_2x + D_{21}w+ D_{22}u\\ \end{align}$$

The Data
The matrices: System $$ (A,B_{1},B_{2},C_{1},C_{2},D_{11},D_{12},D_{21},D_{22}), X_1,Y_1,Z, ,X_2,Y_2 $$

Controller $$ (A_{c},B_{c},C_{c},D_{c}) $$

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Solve for $$A_{n} \in {R^{n_x*n_x}}, B_{n} \in {R^{n_x*n_y}}, C_{n} \in {R^{n_u*n_x}}, D_{n} \in {R^{n_u*n_y}}, X_{1}, Y_{1}\in {S^{n_x}}, and \gamma \in {R_{>0}};$$ that minimize $$\mathcal{J}(\gamma) = \gamma$$ subject to $$X_1>0,Y_1>0,$$ $$\begin{align}

\\ \begin{bmatrix}N_{11} &A+A_n^T+B_2D_nC_2 &B_1+B_2D_nD_{21} &Y_1^TC_1^T+C_n^TD_{12}^T \\
 * &X_1A+A^TX_1+B_nC_2+C_2^T+B_n^T &X_1B_1+B_nD_{21} &C_1^T+C_2^TD_n^TD_{12}^T \\
 * & * &-\gamma \mathbf{1} &D_{11}^T+D_{21}^TD_n^TD{12}^T \\
 * & * &* &-\gamma \mathbf{1}\end{bmatrix}&>0,\\

\begin{bmatrix}X_{1} & \mathbf{1}\\
 * & Y_{1}\end{bmatrix}&>0,\\

\end{align}$$

where $$N_{11}=AY_1+Y_1A^T+B_2C_n+C_n^TB_2^T.$$ The controller is recovered by

$$ \begin{align} &A_{c} = A_{k}-B_{c}(1-D_{22}D_{c})^{-1}D_{22}C_{c}\\ &B_{c} = B_{k}(1-D_{c}D_{22})\\ &C_{c} = (1-D_{c}D_{22})C_{k}\\ &D_{c} = 1+D_{k}D_{22})^{-1}D_{k}\\ \end{align}$$

where, $$ \begin{align} \begin{bmatrix}A_{k} & B_{k}\\ C_{k} & D_{k}\end{bmatrix}&= \begin{bmatrix}X_{2} & X_{1}B_{d2}\\ 0 & 1\end{bmatrix}^{-1} (\begin{bmatrix}A_{n} & B_{n}\\ C_{n} & D_{n}\end{bmatrix} -\begin{bmatrix}X_{1}AY_{1} & 0\\ 0 & 0\end{bmatrix}) \begin{bmatrix}Y_{2}^T & 0\\ C_{2}Y_{1} & 1\end{bmatrix}^{-1}

\end{align}$$ and the matrices $$X_2$$ and $$Y_2$$ satisfy $$X_2Y_2^T = 1 - X_{1}Y_{1}$$. If $$D_{22}=0$$, then $$A_c=A_k,B_c=B_k,C_c=C_k,$$ and $$D_c=D_k$$.

Given $$X_1$$ and $$Y_1$$, the matrices $$X_2$$ and $$Y_2$$ can be found using a matrix decomposition, such as a LU decomposition or a Cholesky decomposition.

Conclusion:
The Continuous-Time H∞-Optimal Dynamic Output Feedback Controller is the system $$ (A_{dc},B_{dc},C_{dc},D_{dc}) $$

Related LMIs
Discrete Time H∞ Optimal Dynamic Output Feedback Control

Continuous Time H2 Optimal Dynamic Feedback Control


text:
LMIs in Control/Click here to continue/Optimal control systems/Discrete-time Hinf-optimal Dynamic output Feedback control

In many applications, perhaps even most, the state of the system cannot be directly known. In this case, you will need to strategically to measure key system outputs that will make the system states indirectly observable. Observers need to converge much faster than the system dynamics in order for their estimations to be accurate. Optimal observer synthesis is therefore advantageous. In this LMI, we seek to optimize both H2 and Hinf norms, to minimize both the average and the maximum error of the observer.

The System

\begin{align}

x_{k+1}&=A_{d}x_{k}+B_{d1,1}w_{1,k}+B_{d1,2}w_{2,k},\\

y_{k}&=C_{c2}x_{k} + D_{d21,1}w_{1,k}+ D_{d21,2}w_{2,k}\\ \end{align} $$

where $$ x \in R^{n}$$  and is the state vector,  $$ A \in R^{n * n}$$  and is the state matrix,  $$ B \in R^{n * r}$$  and is the input matrix,  $$ w \in R^{r}$$  and is the exogenous input,  $$ C \in R^{m * n}$$  and is the output matrix,  $$ D \in R^{m * r}$$  and is the feedthrough matrix,  $$ y \in R^{m}$$ and is the output, and it is assumed that $$(A_{d}, C_{d2})$$ is detectable.

$$ A \in R^{n * n}$$

The Data
The matrices $$ A_{d}, B_{d1}, C_{cd2}, C_{cd1}, D_{d21} $$.

The Optimization Problem
An observer of the form:

\begin{align}

\hat x_{k+1}&=A_{d}\hat x_{k}+ L_{d}(y_{k}-\hat y_{k}),\\

\hat y_{k}&=C_{d2} \hat x_{k} \\ \end{align} $$

is to be designed, where $$ L_{d} \in R^{n_{x} * n_{y}}$$ is the observer gain.

Defining the error state $$ e_{k} = x_{k} - \hat x_{k}$$, the error dynamics are found to be

$$e_{k+1} = (A_{d} - L_{d}C_{d2}) e_{k} + (B_{d1,1} - L_{d}D_{d21,1})w_{1,k} + (B_{d1,2} - L_{d}D_{d21,2})w_{2,k}$$,

and the performance output is defined as

$$ \begin{bmatrix} Z_{1,k} \\ Z_{2,k}\end{bmatrix} = \begin{bmatrix} C_{d1,1} \\ C_{d1,2} \end{bmatrix}e_{k} + \begin{bmatrix} 0 & D_{d11,12} \\ D_{d11,21} & D_{d11,22}\end{bmatrix}\begin{bmatrix} w_{1,k} \\ w_{2,k} \end{bmatrix} $$.

The observer gain $$L_{d}$$ is to be designed to minimize the $$H_{2}$$ norm of the closed loop transfer matrix $$T_{11}(z)$$ from the exogenous input $$w_{2,k}$$ to the performance output $$z_{2,k}$$ is less than $$\gamma _ {d}$$, where

$$ \begin{align} T_{11}(z) &= C_{d1,1}(z1 - (A_{d} - L_{d}C_{d2}))^{-1}(B_{d1,1} - L_{d}D_{d21,1}),\\ T_{22}(z) &= C_{d1,2}(z1 - (A_{d} - L_{d}C_{d2}))^{-1}(B_{d1,2} - L_{d}D_{d21,2}) + D_{d11,22} \end{align} $$

The LMI: Discrete-Time Mixed H2-Hinf-Optimal Observer
The discrete-time mixed-$$H_{2}H_{inf}$$-optimal observer gain is synthesized by solving for $$P \in S^{n_{x}}$$, $$Z \in S^{n_{z}}$$, $$G_{d} \in R^{n_{x} * n_{y}}$$, and $$v \in R_{>0}$$ that minimize J$$(v) = v$$ subject to $$P>0, Z>0$$,

\begin{align}

\begin{bmatrix}P & PA_{d}-G_{d}C_{d2} & PB_{d1,1}-G_{d}D_{d21,1}\\
 * & P & 0\\
 * & * & 1\end{bmatrix}&>0,\\

\begin{bmatrix}P & PA_{d}-G_{d}C_{d2} & PB_{d1,2}-G_{d}D_{d21,2} & 0\\
 * & P & 0 & C^{T}_{d1,2}\\
 * & * & \gamma_{d} 1 & D^{T}_{d11,22}\\
 * & * & * & \gamma_{d} 1\end{bmatrix}&>0,\\

\begin{bmatrix} Z & PC_{d1,1}\\
 * & P\end{bmatrix}&>0,\\

trZ<v

\end{align}$$

where $$tr$$ refers to the trace of a matrix.

Conclusion:
The mixed-$$H_{2}H_{inf}$$-optimal observer gain is recovered by $$L_{d} = P^{-1}G_{d}$$, the $$H_{2}$$ norm of $$T_{11}(z)$$ is less than $$\mu = \sqrt{v}$$, and the $$H_{inf}$$ norm of $$T_{22}(z)$$ is less than $$\gamma_{d}$$. This result gives us a matrix of observer gains $$L_{d}$$ that allow us to optimally observe the states of the system indirectly as:

\begin{align}

\hat x_{k+1}&=A_{d}\hat x_{k}+ L_{d}(y_{k}-\hat y_{k}),\\

\hat y_{k}&=C_{d2} \hat x_{k} \\ \end{align} $$

Implementation
This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/mixedh2hinfobsdiscretetime.m

Related LMIs
Discrete-Time_Hinfinity-Optimal_Observer

Discrete-Time_H2-Optimal_Observer


text:
LMIs in Control/Click here to continue/Optimal control systems/Mixed H2-Hinf-optimal Full-state Feedback Control

Discrete-Time Mixed H2-H∞-Optimal Full-State Feedback Control

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.

A full-state feedback controller $$K = K_d \in {R^{n_u*n_x}}$$ (i.e., $$uk = K_dx_k$$) is to be designed to minimize the H2 norm of the closed loop transfer matrix $$T11(z)$$ from the exogenous input $$w_{1,k}$$ to the performance output $$z_{1,k}$$ while ensuring the H∞ norm of the closed-loop transfer matrix $$T22(z)$$ from the exogenous input $$w_{2,k}$$ to the performance output $$z_{2,k}$$ is less than $$\gamma_{d}$$.

The System
Discrete-Time LTI System with state space realization

$$ \begin{align}

x_{k+1}&=A_{d}x_{k}+B_{d1,1}w_{1,k}+B_{d1,2}w_{2,k}+B_{d2}u_{k},\\

\begin{bmatrix} Z_{1,k}\\ Z_{2,k}\end{bmatrix} &= \begin{bmatrix} C_{d1,1}\\ C_{d1,2}\end{bmatrix}x_{k}+ \begin{bmatrix} 0 & D_{d11,12}\\ D_{d11,21} & D_{d11,22}\end{bmatrix} \begin{bmatrix} w_{1,k}\\ w_{2,k}\end{bmatrix}+ \begin{bmatrix} D_{d12,1}\\ D_{d12,2}\end{bmatrix}u_{k}\\

y_k &= x_k\\ \end{align} $$

The Data
The matrices: System $$ (A_d,B_{d1,1},B_{d1,2},B_{d2},C_{d1,1},C_{d1,2},D_{d11,12},D_{d11,21},D_{d11,22},,D_{d12,1},D_{d12,2}), P, F_d $$.

The Optimization Problem
The following feasibility problem should be optimized:

Minimize the H2 norm of the closed loop transfer matrix $$T11(z)$$, while ensuring the H∞ norm of the closed-loop transfer matrix $$T22(z)$$ is less than $$\gamma_{d}$$, while obeying the LMI constraints.

The LMI:
Discrete-Time Mixed H2-H∞-Optimal Full-State Feedback Controller is synthesized by solving for $$P \in S^{n_{x}}, Z \in S^{n_{w}}, F_{d} \in R^{n_{u} * n_{x}}$$, and $$\mu \in R_{>0}$$ that minimize $$\mu$$ subject to $$P>0, Z>0$$

The LMI formulation

H∞ norm < $$\gamma_{d}$$

H2 norm < $$\mu$$

$$ \begin{align}

\begin{bmatrix}P & A_{d}P-B_{d2}F_{d} & B_{d1,1}\\
 * & P & 0\\
 * & * & 1\end{bmatrix}&>0\\

\begin{bmatrix}P & A_{d}P-B_{d2}F_{d} & B_{d1,2} & 0\\
 * & P & 0 & PC_{d1,2}^T-F_{d}^TD_{d12,2}^T\\
 * & * & \gamma_{d} I & D_{d11,22}^T\\
 * & * & * & \gamma_{d} I\end{bmatrix}&>0\\

\begin{bmatrix} Z & C_{d1,1}P-D_{d12,1}\\
 * & P\end{bmatrix}&>0\\

trace(Z) &< \mu^2

\end{align}$$

Conclusion:
The H2-optimal full-state feedback controller gain is recovered by $$K_{d}=F_{d}P^{-1}$$

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

Related LMIs
- Continuous Time Mixed H2-H∞ Optimal Full State Feedback Control


text:
LMIs in Control/Click here to continue/Optimal control systems/Discrete-time H2-Hinf-optimal Full state Feedback control

Discrete-Time Mixed H2-H∞-Optimal Full-State Feedback Control

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.

A full-state feedback controller $$K = K_d \in {R^{n_u*n_x}}$$ (i.e., $$uk = K_dx_k$$) is to be designed to minimize the H2 norm of the closed loop transfer matrix $$T11(z)$$ from the exogenous input $$w_{1,k}$$ to the performance output $$z_{1,k}$$ while ensuring the H∞ norm of the closed-loop transfer matrix $$T22(z)$$ from the exogenous input $$w_{2,k}$$ to the performance output $$z_{2,k}$$ is less than $$\gamma_{d}$$.

The System
Discrete-Time LTI System with state space realization

$$ \begin{align}

x_{k+1}&=A_{d}x_{k}+B_{d1,1}w_{1,k}+B_{d1,2}w_{2,k}+B_{d2}u_{k},\\

\begin{bmatrix} Z_{1,k}\\ Z_{2,k}\end{bmatrix} &= \begin{bmatrix} C_{d1,1}\\ C_{d1,2}\end{bmatrix}x_{k}+ \begin{bmatrix} 0 & D_{d11,12}\\ D_{d11,21} & D_{d11,22}\end{bmatrix} \begin{bmatrix} w_{1,k}\\ w_{2,k}\end{bmatrix}+ \begin{bmatrix} D_{d12,1}\\ D_{d12,2}\end{bmatrix}u_{k}\\

y_k &= x_k\\ \end{align} $$

The Data
The matrices: System $$ (A_d,B_{d1,1},B_{d1,2},B_{d2},C_{d1,1},C_{d1,2},D_{d11,12},D_{d11,21},D_{d11,22},,D_{d12,1},D_{d12,2}), P, F_d $$.

The Optimization Problem
The following feasibility problem should be optimized:

Minimize the H2 norm of the closed loop transfer matrix $$T11(z)$$, while ensuring the H∞ norm of the closed-loop transfer matrix $$T22(z)$$ is less than $$\gamma_{d}$$, while obeying the LMI constraints.

The LMI:
Discrete-Time Mixed H2-H∞-Optimal Full-State Feedback Controller is synthesized by solving for $$P \in S^{n_{x}}, Z \in S^{n_{w}}, F_{d} \in R^{n_{u} * n_{x}}$$, and $$\mu \in R_{>0}$$ that minimize $$\mu$$ subject to $$P>0, Z>0$$

The LMI formulation

H∞ norm < $$\gamma_{d}$$

H2 norm < $$\mu$$

$$ \begin{align}

\begin{bmatrix}P & A_{d}P-B_{d2}F_{d} & B_{d1,1}\\
 * & P & 0\\
 * & * & 1\end{bmatrix}&>0\\

\begin{bmatrix}P & A_{d}P-B_{d2}F_{d} & B_{d1,2} & 0\\
 * & P & 0 & PC_{d1,2}^T-F_{d}^TD_{d12,2}^T\\
 * & * & \gamma_{d} I & D_{d11,22}^T\\
 * & * & * & \gamma_{d} I\end{bmatrix}&>0\\

\begin{bmatrix} Z & C_{d1,1}P-D_{d12,1}\\
 * & P\end{bmatrix}&>0\\

trace(Z) &< \mu^2

\end{align}$$

Conclusion:
The H2-optimal full-state feedback controller gain is recovered by $$K_{d}=F_{d}P^{-1}$$

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

Related LMIs
- Continuous Time Mixed H2-H∞ Optimal Full State Feedback Control


text:
LMIs in Control/Click here to continue/Optimal control systems/Mixed H2-Hinf-optimal Dynamic output Feedback control

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

Return to Main Page
LMIs in Control/Tools


text:
LMIs in Control/Click here to continue/Optimal control systems/Discrete-time H2-Hinf-optimal Dynamic output Feedback control

In many applications, perhaps even most, the state of the system cannot be directly known. In this case, you will need to strategically to measure key system outputs that will make the system states indirectly observable. Observers need to converge much faster than the system dynamics in order for their estimations to be accurate. Optimal observer synthesis is therefore advantageous. In this LMI, we seek to optimize both H2 and Hinf norms, to minimize both the average and the maximum error of the observer.

The System

\begin{align}

x_{k+1}&=A_{d}x_{k}+B_{d1,1}w_{1,k}+B_{d1,2}w_{2,k},\\

y_{k}&=C_{c2}x_{k} + D_{d21,1}w_{1,k}+ D_{d21,2}w_{2,k}\\ \end{align} $$

where $$ x \in R^{n}$$  and is the state vector,  $$ A \in R^{n * n}$$  and is the state matrix,  $$ B \in R^{n * r}$$  and is the input matrix,  $$ w \in R^{r}$$  and is the exogenous input,  $$ C \in R^{m * n}$$  and is the output matrix,  $$ D \in R^{m * r}$$  and is the feedthrough matrix,  $$ y \in R^{m}$$ and is the output, and it is assumed that $$(A_{d}, C_{d2})$$ is detectable.

$$ A \in R^{n * n}$$

The Data
The matrices $$ A_{d}, B_{d1}, C_{cd2}, C_{cd1}, D_{d21} $$.

The Optimization Problem
An observer of the form:

\begin{align}

\hat x_{k+1}&=A_{d}\hat x_{k}+ L_{d}(y_{k}-\hat y_{k}),\\

\hat y_{k}&=C_{d2} \hat x_{k} \\ \end{align} $$

is to be designed, where $$ L_{d} \in R^{n_{x} * n_{y}}$$ is the observer gain.

Defining the error state $$ e_{k} = x_{k} - \hat x_{k}$$, the error dynamics are found to be

$$e_{k+1} = (A_{d} - L_{d}C_{d2}) e_{k} + (B_{d1,1} - L_{d}D_{d21,1})w_{1,k} + (B_{d1,2} - L_{d}D_{d21,2})w_{2,k}$$,

and the performance output is defined as

$$ \begin{bmatrix} Z_{1,k} \\ Z_{2,k}\end{bmatrix} = \begin{bmatrix} C_{d1,1} \\ C_{d1,2} \end{bmatrix}e_{k} + \begin{bmatrix} 0 & D_{d11,12} \\ D_{d11,21} & D_{d11,22}\end{bmatrix}\begin{bmatrix} w_{1,k} \\ w_{2,k} \end{bmatrix} $$.

The observer gain $$L_{d}$$ is to be designed to minimize the $$H_{2}$$ norm of the closed loop transfer matrix $$T_{11}(z)$$ from the exogenous input $$w_{2,k}$$ to the performance output $$z_{2,k}$$ is less than $$\gamma _ {d}$$, where

$$ \begin{align} T_{11}(z) &= C_{d1,1}(z1 - (A_{d} - L_{d}C_{d2}))^{-1}(B_{d1,1} - L_{d}D_{d21,1}),\\ T_{22}(z) &= C_{d1,2}(z1 - (A_{d} - L_{d}C_{d2}))^{-1}(B_{d1,2} - L_{d}D_{d21,2}) + D_{d11,22} \end{align} $$

The LMI: Discrete-Time Mixed H2-Hinf-Optimal Observer
The discrete-time mixed-$$H_{2}H_{inf}$$-optimal observer gain is synthesized by solving for $$P \in S^{n_{x}}$$, $$Z \in S^{n_{z}}$$, $$G_{d} \in R^{n_{x} * n_{y}}$$, and $$v \in R_{>0}$$ that minimize J$$(v) = v$$ subject to $$P>0, Z>0$$,

\begin{align}

\begin{bmatrix}P & PA_{d}-G_{d}C_{d2} & PB_{d1,1}-G_{d}D_{d21,1}\\
 * & P & 0\\
 * & * & 1\end{bmatrix}&>0,\\

\begin{bmatrix}P & PA_{d}-G_{d}C_{d2} & PB_{d1,2}-G_{d}D_{d21,2} & 0\\
 * & P & 0 & C^{T}_{d1,2}\\
 * & * & \gamma_{d} 1 & D^{T}_{d11,22}\\
 * & * & * & \gamma_{d} 1\end{bmatrix}&>0,\\

\begin{bmatrix} Z & PC_{d1,1}\\
 * & P\end{bmatrix}&>0,\\

trZ<v

\end{align}$$

where $$tr$$ refers to the trace of a matrix.

Conclusion:
The mixed-$$H_{2}H_{inf}$$-optimal observer gain is recovered by $$L_{d} = P^{-1}G_{d}$$, the $$H_{2}$$ norm of $$T_{11}(z)$$ is less than $$\mu = \sqrt{v}$$, and the $$H_{inf}$$ norm of $$T_{22}(z)$$ is less than $$\gamma_{d}$$. This result gives us a matrix of observer gains $$L_{d}$$ that allow us to optimally observe the states of the system indirectly as:

\begin{align}

\hat x_{k+1}&=A_{d}\hat x_{k}+ L_{d}(y_{k}-\hat y_{k}),\\

\hat y_{k}&=C_{d2} \hat x_{k} \\ \end{align} $$

Implementation
This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/mixedh2hinfobsdiscretetime.m

Related LMIs
Discrete-Time_Hinfinity-Optimal_Observer

Discrete-Time_H2-Optimal_Observer


text:
LMIs in Control/Click here to continue/Optimal control systems/H2-Optimal Observer

State observer is a system that provides estimates of internal states of a given real system, from measurements of the inputs and outputs of the real system.The goal of $$ H_2$$ -optimal state estimation is to design an observer that minimizes the $$ H_2 $$ norm of the closed-loop transfer matrix from w to z. Kalman filter is a form of Optimal Observer.

The System
Consider the continuous-time generalized plant $$ P $$ with state-space realization

\begin{align} \dot x&=Ax+B_1w(t),\\ y&=C_2x+D_{21}w\\ \end{align}$$

The Data
The matrices needed as input are $$ A,B,C,D $$.

The Optimization Problem
The task is to design an observer of the following form:

\begin{align} \dot{\hat{x}}=A\hat{x} + L(y - \hat{y}),\\ \hat{y} = C_2\hat{x}\\ \end{align}$$

The LMI: $$ H_2 $$ Optimal Observer
LMIs in the variables $$ P, G, Z, \nu $$ are given by:

\begin{align}

\begin{bmatrix} PA+ A^TP-GC_2-{C_2}^TG^T && PB_1-GD_{21}\\ \star && -1 \end{bmatrix} <0\\ trZ < \nu \end{align}$$

Conclusion:
The $$ H_2$$ -optimal observer gain is recovered by $$ L = P^{-1}G $$ and the $$ H_2 $$ norm of T(s) is $$ \mu = \sqrt {\nu} $$

Implementation
https://github.com/Ricky-10/coding107/blob/master/H2%20Optimal%20Observer

= External Links =

A list of references documenting and validating the LMI.

 * LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
 * LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
 * LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
 * https://onlinelibrary.wiley.com/doi/abs/10.1002/rnc.1310


text:
LMIs in Control/Click here to continue/Optimal control systems/Discrete-Time H2-Optimal observer

In many applications, perhaps even most, the state of the system cannot be directly known. In this case, you will need to strategically to measure key system outputs that will make the system states indirectly observable. Observers need to converge much faster than the system dynamics in order for their estimations to be accurate. Optimal observer synthesis is therefore advantageous. In this LMI, we seek to optimize the H2 norm, which conceptually is minimizing the average magnitude of error in the observer.

The System

\begin{align}

x_{k+1}&=A_{d}x_{k}+B_{d1}w_{k},\\

y_{k}&=C_{c2}x_{k} + D_{d21}w_{k}\\ \end{align} $$

where $$ x \in R^{n}$$  and is the state vector,  $$ A \in R^{n\times n}$$  and is the state matrix,  $$ B \in R^{n\times r}$$  and is the input matrix,  $$ w \in R^{r}$$  and is the exogenous input,  $$ C \in R^{m\times n}$$  and is the output matrix,  $$ D \in R^{m\times r}$$  and is the feedthrough matrix,  $$ y \in R^{m}$$ and is the output, and it is assumed that $$(A_{d}, C_{d2})$$ is detectable.

The Data
The matrices $$ A_{d}, B_{d1}, C_{cd2}, C_{cd1}, D_{d21} $$.

The Optimization Problem
An observer of the form:

\begin{align}

\hat x_{k+1}&=A_{d}\hat x_{k}+ L_{d}(y_{k}-\hat y_{k}),\\

\hat y_{k}&=C_{d2} \hat x_{k} \\ \end{align} $$

is to be designed, where $$ L_{d} \in R^{n_{x}\times n_{y}}$$ is the observer gain.

Defining the error state $$ e_{k} = x_{k} - \hat x_{k}$$, the error dynamics are found to be

$$e_{k+1} = (A_{d} - L_{d}C_{d2} e_{k} + (B_{d1} - L_{d}D_{d21})w_{k}$$,

and the performance output is defined as

$$z_{k} = C_{d1}e_{k}$$.

The observer gain $$L_{d}$$ is to be designed such that the $$H_{2}$$ of the transfer matrix from $$w_{k}$$ to $$z_{k}$$, given by

$$L_{d} \begin{align} T(z) &= C_{d1}(z1 - (A_{d} - L_{d}C_{d2}))^{-1}(B_{d1} - L_{d}D_{d21}),\\ \end{align} $$

is minimized.

The LMI: Discrete-Time H2-Optimal Observer
The discrete-time $$H_{2}$$-optimal observer gain is synthesized by solving for $$P \in S^{n_{x}}$$, $$Z \in S^{n_{z}}$$, $$G_{d} \in R^{n_{x}\times n_{y}}$$, and $$v \in R_{>0}$$ that minimize $$J(v) = v$$ subject to $$P>0, Z>0$$,

\begin{align}

\begin{bmatrix}P & PA_{d}-G_{d}C_{d2} & PB_{d1}-G_{d}D_{d21}\\
 * & P & 0\\
 * & * & 1\end{bmatrix}&>0,\\

\begin{bmatrix} Z & PC_{d1}\\
 * & P\end{bmatrix}&>0,\\

\operatorname{tr}Z<v

\end{align}$$ where $$\operatorname{tr}$$ refers to the trace of a matrix.

Conclusion:
The $$H_{2}$$-optimal observer gain is recovered by $$L_{d} = P^{-1}G_{d}$$ and the $$H_{2}$$ norm of $$T(z)$$ is $$\mu = \sqrt{v}$$. The $$L_{d} $$ matrix is the observer gains that can be used to form the optimal observer:

\begin{align}

\hat x_{k+1}&=A_{d}\hat x_{k}+ L_{d}(y_{k}-\hat y_{k}),\\

\hat y_{k}&=C_{d2} \hat x_{k} \\ \end{align} $$

Implementation
This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/Discrete_Time_H2_Optimal_Observer_LMIs_Wikibook_Example.m

Related LMIs
Mixed H2-Hinfinity discrete time observer

Discrete-Time_Hinfinity-Optimal_Observer


text:
LMIs in Control/Click here to continue/Optimal control systems/Hinf-Optimal Observer

$$ H\infin $$-Optimal observers yield  robust  estimates  of  some  or  all  internal  plant states  by  processing  measurement  data. Robust observers are increasingly  demanded in industry  as they  may provide state  and  parameter  estimates  for  monitoring  and  diagnosis purposes even in the presence of large disturbances such as noise etc. It is there where Kalman filters may tend to fail. State observer is a system that provides estimates of internal states of a given real system, from measurements of the inputs and outputs of the real system. The goal of $$ H_\infin$$ -optimal state estimation is to design an observer that minimizes the $$ H_\infin $$ norm of the closed-loop transfer matrix from w to z.

The System
Consider the continuous-time generalized plant $$ P $$ with state-space realization

\begin{align} \dot x&=Ax+B_1w,\\ y&=C_2x+D_{21}w\\ \end{align}$$

The Data
The matrices needed as input are $$ A,B_1,B_2,C_2,D_{21},D_{11} $$.

The Optimization Problem
The observer gain $$ L $$ is to be designed such that the $$ H\infin $$ of the transfer matrix from w to z, given by

\begin{align} T(s) = C_1(s1-(A-LC_2))^{-1}(B_1-LD_{21})+D_{11}\\ \end{align} $$ is minimized. The form of the observer would be:

\begin{align} \dot{\hat{x}}=A\hat{x} + L(y - \hat{y}),\\ \hat{y} = C_2\hat{x}\\ \end{align}$$

The LMI: $$ H_\infin $$ Optimal Observer
The $$H\infin $$-optimal observer gain is synthesized by solving for $$ P \in \mathbb{S} ^{n_x}, G \in \mathbb{R} ^{n_x \times n_y} $$, and $$ \gamma \in \mathbb{R}_{>0} $$ that minimize $$ \zeta(\gamma) = \gamma $$ subject to $$ P>0 $$ and

\begin{align}

\begin{bmatrix} PA+ A^TP-GC_2-{C_2}^TG^T && PB_1-GD_{21} && C_1^T\\ \star && -\gamma1 && {D_{11}}^T\\ \star && \star && -\gamma1 \end{bmatrix} <0\\ \end{align}$$

Conclusion:
The $$ H_\infin$$ -optimal observer gain is recovered by $$ L = P^{-1}G $$ and the $$ H_\infin $$ norm of T(s) is $$ \gamma$$.

Implementation
Link to the MATLAB code designing $$ H\infin $$- Optimal Observer

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

= External Links =

 * LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
 * LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
 * LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
 *
 *


text:
LMIs in Control/Click here to continue/Optimal control systems/Discrete-time Hinf-Optimal Observer

In many applications, perhaps even most, the state of the system cannot be directly known. In this case, you will need to strategically to measure key system outputs that will make the system states indirectly observable. Observers need to converge much faster than the system dynamics in order for their estimations to be accurate. Optimal observer synthesis is therefore advantageous. In this LMI, we seek to optimize the H-infinity norm, which conceptually is minimizing the maximum magnitude of error in the observer.

The System
The system needed for this LMI is a discrete-time LTI plant $$G$$, which has the state space realization:

\begin{align}

x_{k+1}&=A_{d}x_{k}+B_{d1}w_{k},\\

y_{k}&=C_{d2}x_{k} + D_{d21}w_{k}\\ \end{align} $$

where $$ x \in R^{n}$$  and is the state vector,  $$ A \in R^{n * n}$$  and is the state matrix,  $$ B \in R^{n * r}$$  and is the input matrix,  $$ w \in R^{r}$$  and is the exogenous input,  $$ C \in R^{m * n}$$  and is the output matrix,  $$ D \in R^{m * r}$$  and is the feedthrough matrix,  $$ y \in R^{m}$$ and is the output, and it is assumed that $$(A_{d}, C_{d2})$$ is detectable.

The Data
The matrices $$ A_{d}, B_{d1}, C_{d2}, D_{d21}, D_{d11} $$.

The Optimization Problem
An observer of the form:

\begin{align}

\hat x_{k+1}&=A_{d}\hat x_{k}+ L_{d}(y_{k}-\hat y_{k}),\\

\hat y_{k}&=C_{d2} \hat x_{k} \\ \end{align} $$

is to be designed, where $$ L_{d} \in R^{n_{x} * n_{y}}$$ is the observer gain.

Defining the error state $$ e_{k} = x_{k} - \hat x_{k}$$, the error dynamics are found to be

$$e_{k+1} = (A_{d} - L_{d}C_{d2})e_{k} + (B_{d1} - L_{d}D_{d21})w_{k}$$,

and the performance output is defined as

$$z_{k} = C_{d1}e_{k} + D_{d11}w_{k}$$.

The observer gain $$L_{d}$$ is to be designed such that the $$H_{inf}$$ of the transfer matrix from $$w_{k}$$ to $$z_{k}$$, given by

$$ \begin{align} T(z) &= C_{d1}(z1 - (A_{d} - L_{d}C_{d2}))^{-1}(B_{d1} - L_{d}D_{d21}) + D_{d11},\\ \end{align} $$

is minimized.

The LMI: Discrete-Time Hinf-Optimal Observer
The discrete-time $$H_{inf}$$-optimal observer gain is synthesized by solving for $$P \in S^{n_{x}}$$, $$G_{d} \in R^{n_{x} * n_{y}}$$, and $$\gamma \in R_{>0}$$ that minimize J$$(\gamma) = \gamma$$ subject to $$P>0$$, and

\begin{align}

\begin{bmatrix}P & PA_{d}-G_{d}C_{d2} & PB_{d1}-G_{d}D_{d21} & 0\\
 * & P & 0 & C^{T}_{d1}\\
 * & * & \gamma * 1 & D^{T}_{d11} \\
 * & * & * & \gamma * 1\end{bmatrix}&>0.\\

\end{align}$$

Conclusion:
The $$H_{inf}$$-optimal observer gain is recovered by $$L_{d} = P^{-1}G_{d}$$ and the $$H_{inf}$$ norm of $$T(z)$$ is $$\gamma$$. This matrix of observer gains can then be used to form the optimal observer formulated by:

\begin{align}

\hat x_{k+1}&=A_{d}\hat x_{k}+ L_{d}(y_{k}-\hat y_{k}),\\

\hat y_{k}&=C_{d2} \hat x_{k} \\ \end{align} $$

Implementation
This implementation requires Yalmip and Sedumi.

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

Related LMIs
Mixed H2-Hinfinity discrete time observer

Discrete-Time_H2-Optimal_Observer


text:
LMIs in Control/Click here to continue/Optimal control systems/Mixed H2-Hinf-Optimal Observer

The goal of mixed $$ H_2-H\infin $$-optimal state estimation is to design an observer that minimizes the $$ H_2 $$ norm of the closed-loop transfer matrix from $$ w_1 $$ to $$ z_1 $$, while ensuring that the $$ H\infin $$ norm of the closed-loop transfer matrix from $$ w_2 $$ to $$ z_2 $$ is below a specified bound.

The System
Consider the continuous-time generalized plant $$ P $$ with state-space realization

\begin{align} \dot x&=Ax+B_{1,1}w_1 +B_{1,2}w_2,\\ y&=C_2x+D_{21,1}w_1+D_{21,1}w_2\\ \end{align}$$

where it is assumed that $$ (A, C_2) $$ is detectable.

The Data
The matrices needed as input are $$ A,B_1,B_2,C_2,D_{21},D_{11} $$.

The Optimization Problem
The observer gain L is to be designed to minimize the $$ H_2 $$ norm of the closed-loop transfer matrix $$ T_{11}(s) $$ from the exogenous input $$ w_1 $$ to the performance output $$ z_1 $$ while ensuring the $$ H\infin $$ norm of the closed-loop transfer matrix $$ T_{22}(s) $$ from the exogenous input $$ w_2 $$ to the performance output $$ z_2 $$ is less than $$ \gamma_d $$, where

\begin{align} T_{11}(s) = C_{1,1}(s1-(A-LC_2))^{-1}(B_{1,1}-LD_{21,1})\\ T_{22}(s) = C_{1,2}(s1-(A-LC_2))^{-1}(B_{1,2}-LD_{21,2})+D_{11,22} \end{align} $$

is minimized. The form of the observer would be:

\begin{align} \dot{\hat{x}}=A\hat{x} + L(y - \hat{y}),\\ \hat{y} = C_2\hat{x}\\ \end{align}$$

is to be designed, where $$ L \in \mathbb{R}^{n_x \times n_y} $$ is the observer gain.

The LMI: $$ H_\infin $$ Optimal Observer
The mixed $$H_2 -H\infin $$-optimal observer gain is synthesized by solving for $$ P \in \mathbb{S} ^{n_x}, G \in \mathbb{R} ^{n_x \times n_y} $$, and $$ \nu \in \mathbb{R}_{>0} $$ that minimize $$ \zeta(\nu) = \nu $$ subject to $$ P>0, Z>0 $$,

\begin{align} \begin{bmatrix} PA+ A^TP-GC_2-{C_2}^TG^T && PB_1-GD_{21}\\ \star && -1 \end{bmatrix} <0\\

\begin{bmatrix} PA+ A^TP-GC_2-{C_2}^TG^T && PB_1-GD_{21} && C_1\\ \star && -\gamma1 && {D_{11}}^T\\ \star && \star && -\gamma1 \end{bmatrix} <0\\ \begin{bmatrix} P && C{_{1,1}}^T \\ \star && Z\end{bmatrix} >0\\ trZ < \nu \end{align}$$

Conclusion:
The mixed $$ H_2- H_\infin$$ -optimal observer gain is recovered by $$ L = P^{-1}G $$, the $$ H_2 $$ norm of $$ T_{11}(s) $$ is less than $$ \mu = \sqrt {\nu} $$ and the $$ H_\infin $$ norm of T(s) is less than $$ \gamma_d$$.

Implementation
Link to the MATLAB code designing $$ H_2- H\infin $$- Optimal Observer

Code $ H_2-H\infin $ Optimal Observer

Related LMIs

 * $ H_2 $ Optimal observer
 * $ H\infin $ Optimal observer


text:
LMIs in Control/Click here to continue/Optimal control systems/Discrete-Time Mixed H2-Hinf Optimal Observer

In many applications, perhaps even most, the state of the system cannot be directly known. In this case, you will need to strategically to measure key system outputs that will make the system states indirectly observable. Observers need to converge much faster than the system dynamics in order for their estimations to be accurate. Optimal observer synthesis is therefore advantageous. In this LMI, we seek to optimize both H2 and Hinf norms, to minimize both the average and the maximum error of the observer.

The System

\begin{align}

x_{k+1}&=A_{d}x_{k}+B_{d1,1}w_{1,k}+B_{d1,2}w_{2,k},\\

y_{k}&=C_{c2}x_{k} + D_{d21,1}w_{1,k}+ D_{d21,2}w_{2,k}\\ \end{align} $$

where $$ x \in R^{n}$$  and is the state vector,  $$ A \in R^{n * n}$$  and is the state matrix,  $$ B \in R^{n * r}$$  and is the input matrix,  $$ w \in R^{r}$$  and is the exogenous input,  $$ C \in R^{m * n}$$  and is the output matrix,  $$ D \in R^{m * r}$$  and is the feedthrough matrix,  $$ y \in R^{m}$$ and is the output, and it is assumed that $$(A_{d}, C_{d2})$$ is detectable.

$$ A \in R^{n * n}$$

The Data
The matrices $$ A_{d}, B_{d1}, C_{cd2}, C_{cd1}, D_{d21} $$.

The Optimization Problem
An observer of the form:

\begin{align}

\hat x_{k+1}&=A_{d}\hat x_{k}+ L_{d}(y_{k}-\hat y_{k}),\\

\hat y_{k}&=C_{d2} \hat x_{k} \\ \end{align} $$

is to be designed, where $$ L_{d} \in R^{n_{x} * n_{y}}$$ is the observer gain.

Defining the error state $$ e_{k} = x_{k} - \hat x_{k}$$, the error dynamics are found to be

$$e_{k+1} = (A_{d} - L_{d}C_{d2}) e_{k} + (B_{d1,1} - L_{d}D_{d21,1})w_{1,k} + (B_{d1,2} - L_{d}D_{d21,2})w_{2,k}$$,

and the performance output is defined as

$$ \begin{bmatrix} Z_{1,k} \\ Z_{2,k}\end{bmatrix} = \begin{bmatrix} C_{d1,1} \\ C_{d1,2} \end{bmatrix}e_{k} + \begin{bmatrix} 0 & D_{d11,12} \\ D_{d11,21} & D_{d11,22}\end{bmatrix}\begin{bmatrix} w_{1,k} \\ w_{2,k} \end{bmatrix} $$.

The observer gain $$L_{d}$$ is to be designed to minimize the $$H_{2}$$ norm of the closed loop transfer matrix $$T_{11}(z)$$ from the exogenous input $$w_{2,k}$$ to the performance output $$z_{2,k}$$ is less than $$\gamma _ {d}$$, where

$$ \begin{align} T_{11}(z) &= C_{d1,1}(z1 - (A_{d} - L_{d}C_{d2}))^{-1}(B_{d1,1} - L_{d}D_{d21,1}),\\ T_{22}(z) &= C_{d1,2}(z1 - (A_{d} - L_{d}C_{d2}))^{-1}(B_{d1,2} - L_{d}D_{d21,2}) + D_{d11,22} \end{align} $$

The LMI: Discrete-Time Mixed H2-Hinf-Optimal Observer
The discrete-time mixed-$$H_{2}H_{inf}$$-optimal observer gain is synthesized by solving for $$P \in S^{n_{x}}$$, $$Z \in S^{n_{z}}$$, $$G_{d} \in R^{n_{x} * n_{y}}$$, and $$v \in R_{>0}$$ that minimize J$$(v) = v$$ subject to $$P>0, Z>0$$,

\begin{align}

\begin{bmatrix}P & PA_{d}-G_{d}C_{d2} & PB_{d1,1}-G_{d}D_{d21,1}\\
 * & P & 0\\
 * & * & 1\end{bmatrix}&>0,\\

\begin{bmatrix}P & PA_{d}-G_{d}C_{d2} & PB_{d1,2}-G_{d}D_{d21,2} & 0\\
 * & P & 0 & C^{T}_{d1,2}\\
 * & * & \gamma_{d} 1 & D^{T}_{d11,22}\\
 * & * & * & \gamma_{d} 1\end{bmatrix}&>0,\\

\begin{bmatrix} Z & PC_{d1,1}\\
 * & P\end{bmatrix}&>0,\\

trZ<v

\end{align}$$

where $$tr$$ refers to the trace of a matrix.

Conclusion:
The mixed-$$H_{2}H_{inf}$$-optimal observer gain is recovered by $$L_{d} = P^{-1}G_{d}$$, the $$H_{2}$$ norm of $$T_{11}(z)$$ is less than $$\mu = \sqrt{v}$$, and the $$H_{inf}$$ norm of $$T_{22}(z)$$ is less than $$\gamma_{d}$$. This result gives us a matrix of observer gains $$L_{d}$$ that allow us to optimally observe the states of the system indirectly as:

\begin{align}

\hat x_{k+1}&=A_{d}\hat x_{k}+ L_{d}(y_{k}-\hat y_{k}),\\

\hat y_{k}&=C_{d2} \hat x_{k} \\ \end{align} $$

Implementation
This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/mixedh2hinfobsdiscretetime.m

Related LMIs
Discrete-Time_Hinfinity-Optimal_Observer

Discrete-Time_H2-Optimal_Observer