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

The System Consider a Discrete-Time LTI system Consider The LMI: Discrete-Time Algebraic Riccati Inequality (DARE) 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 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 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: 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 ( 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 LMI for Continuous-Time Algebraic Riccati Inequality LMI for Schur Stabilization External Links A list of references documenting and validating the LMI. - LMI in Control Systems Analysis, Design and Applications Return to Main Page: