# LMIs in Control/pages/Algebraic Riccati Inequality

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

## The System

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\\end{aligned}}}

## The Data

Following matrices are needed as Inputs:.

${\displaystyle A,B,N}$.

## The Optimization Problem

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

## The LMI: Algebraic Riccati Inequality

Title and mathematical description of the LMI formulation.

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

## Conclusion:

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

## Implementation:

Matlab code for this LMI in the Github repository:

1. REDIRECT [[1]]