LMIs in Control/pages/DARE
Consider a Discrete-Time LTI system
Consider Ad ∈ nxn ; Bd ∈ nxm
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 :
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:
( 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:
LMI for Continuous-Time Algebraic Riccati Inequality
LMI for Schur Stabilization
A list of references documenting and validating the LMI.
-  - LMI in Control Systems Analysis, Design and Applications