LMIs in Control/Discrete-Time Algebraic Riccati Inequality (DARE)
Template:Discrete-Time Algebraic Riccati Inequality
The System
[edit | edit source]Consider a Discrete-Time LTI system
Consider
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
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 | edit source]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 | edit source]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 | 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]( 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 | 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