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

Template:Discrete-Time Algebraic Riccati Inequality

The System edit

Consider a Discrete-Time LTI system

 
 

Consider  

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

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

 

  and   where  .

  is the unknown n by n symmetric matrix and   are known real coefficient matrices.

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

  

The Data edit

The Matrices   are given

  and   should necessarily be Hermitian matrices.

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

The Optimization Problem edit

Our aim is to find

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


  - State-feedback gain, returned as a matrix.

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

 


  - Closed-loop eigenvalues, returned as a matrix.

Conclusion: edit

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

(  in the output corresponds to   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 edit

LMI for Continuous-Time Algebraic Riccati Inequality

LMI for Schur Stabilization

External Links edit

A list of references documenting and validating the LMI.

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

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