# LMIs in Control/pages/DARE

LMIs in Control/pages/DARE

**The System**[edit | edit source]

Consider a Discrete-Time LTI system

Consider A_{d} **∈** ^{nxn} ; *B _{d}*

**∈**

^{nxm}

**The Data**[edit | edit source]

The Matrices A_{d} , B_{d} , C_{d} ,
Q, R are given

Q and R should necessarily be **Hermitian** Matrices.

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

**The Optimization Problem**[edit | edit source]

Our aim is to find

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

K - State-feedback gain, returned as a matrix.

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

L - Closed-loop eigenvalues, returned as a matrix.

**The LMI: ** *Discrete-Time Algebraic Riccati Inequality (DARE)*[edit | edit source]

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

*P , Q* **∈** ^{n}
and R **∈** ^{m}
where P > 0, Q ≥ 0, R > 0

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

The above equation can be rewritten using the Schur Complement Lemma as :

**Conclusion:**[edit | edit source]

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*:

**Implementation**[edit | edit source]

( X **in the output corresponds to** P **in the LMI** )

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

https://github.com/yashgvd/LMI_wikibooks

**Related LMIs**[edit | edit source]

LMI for Continuous-Time Algebraic Riccati Inequality

LMI for Schur Stabilization

## External Links[edit | edit source]

A list of references documenting and validating the LMI.

- [1] - LMI in Control Systems Analysis, Design and Applications