LMIs in Control/pages/DARE

The System Consider a Discrete-Time LTI system Consider Ad ∈ nxn ; Bd ∈ nxm The Data 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 Problem 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) 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: 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 ( 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 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: