View source for LMIs in Control/Discrete-Time Algebraic Riccati Inequality (DARE)

{{Discrete-Time Algebraic Riccati Inequality}}
== '''The System''' ==

'''Consider a Discrete-Time LTI system'''
:<math>

\begin{align}
x_{k+1} = A_dx_k + B_du_k 
\end{align}</math>

:<math>
\begin{align}
y_k = C_dx_k
\end{align}</math>

Consider <math>A_d \in\mathbb{R}^{n\times n}; B_d\in\mathbb{R}^{n\times m}</math>

== '''The LMI:''' ''Discrete-Time Algebraic Riccati Inequality (DARE)'' ==

An algebraic Riccati equation is a type of nonlinear equation that arises in the context of infinite-horizon optimal control problems in continuous time or discrete time

The Discrete-Time Algebraic Riccati Inequality is given by 
:<math>
\begin{align}
A_d^TPA_d - A_d^TPB_d
(R + B_d^TPB_d)^{-1}
B_d^TPA_d + Q - P 
\geq 0
\end{align}</math>

<math>P , Q\in\mathbb{S}^n</math> and <math>R\in\mathbb{S}^m</math> where <math>P > 0, Q \geq 0, R > 0</math>.

<math>P</math> is the unknown n by n symmetric matrix and <math>A, B, Q, R</math> are known real coefficient matrices.

The above equation can be rewritten using the Schur Complement Lemma as: 
:<math>
\begin{bmatrix}
A_d^TPA_d-P+Q & A_d^TPB_d \\
B_d^TPA_d & R+B_d^TPB_d 
\end{bmatrix}</math><math>\begin{align}\geq 0\end{align}</math>

== '''The Data''' ==

The Matrices <math>A_d, B_d, C_d, Q, R</math> are given

<math>Q</math> and <math>R</math> should necessarily be '''Hermitian''' matrices.

A square matrix is Hermitian if it is equal to its complex conjugate transpose.

== '''The Optimization Problem''' ==

Our aim is to find 

<math>P</math> - Unique solution to the discrete-time algebraic Riccati equation, returned as a matrix.


<math>K</math> - State-feedback gain, returned as a matrix.<br>

The algorithm used to evaluate the State-feedback gain is given by 

:<math>
\begin{align}
K = (R+B_d^TPB_d)^{-1}B_d^TPA_d
\end{align}</math>


<math>L</math> - Closed-loop eigenvalues, returned as a matrix.

== '''Conclusion:''' ==
Algebraic Riccati Inequalities play a key role in LQR/LQG control, H2- and H∞ control and Kalman filtering.
We try to find the unique stabilizing solution, if it exists. A solution is stabilizing, if controller of the system makes the closed loop system stable.

'''''Equivalently''''', this Discrete-Time algebraic Riccati Inequality is satisfied under the following ''necessary and sufficient condition'':

:<math>
\begin{bmatrix}
Q & 0 & A_d^TP & P\\
0 & R & B_d^TP & 0\\
PA_d & PB_d & P & 0\\
P & 0 & 0 & P
\end{bmatrix}</math><math>\begin{align}\geq 0\end{align}</math>

== '''Implementation''' ==
'''(<math>X</math> in the output corresponds to <math>P</math> in the LMI)'''

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

https://github.com/yashgvd/ygovada

== '''Related LMIs''' ==

LMI for Continuous-Time Algebraic Riccati Inequality

LMI for Schur Stabilization

== External Links ==

A list of references documenting and validating the LMI.

* [https://www.crcpress.com/LMIs-in-Control-Systems-Analysis-Design-and-Applications/Duan-Yu/p/book/9781466582996] - LMI in Control Systems Analysis, Design and Applications

== '''Return to Main Page:''' ==

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