LMIs in Control/pages/DARE

LMIs in Control/pages/DARE

The SystemEdit

Consider a Discrete-Time LTI system

 
 

Consider Ad  nxn ; Bd  nxm

The DataEdit

The Matrices Ad , Bd , Cd , 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 ProblemEdit

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

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

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:

  

ImplementationEdit

( 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 LMIsEdit

LMI for Continuous-Time Algebraic Riccati Inequality

LMI for Schur Stabilization

External LinksEdit

A list of references documenting and validating the LMI.

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


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