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

The System

Consider a Discrete-Time LTI system

${\begin{aligned}x_{k+1}=A_{d}x_{k}+B_{d}u_{k}\end{aligned}}$

${\begin{aligned}y_{k}=C_{d}x_{k}\end{aligned}}$

Consider $A_{d}\in \mathbb {R} ^{n\times n};B_{d}\in \mathbb {R} ^{n\times m}$

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

${\begin{aligned}A_{d}^{T}PA_{d}-A_{d}^{T}PB_{d}(R+B_{d}^{T}PB_{d})^{-1}B_{d}^{T}PA_{d}+Q-P\geq 0\end{aligned}}$

$P,Q\in \mathbb {S} ^{n}$ and $R\in \mathbb {S} ^{m}$ where $P>0,Q\geq 0,R>0$.

$P$ 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:

${\begin{bmatrix}A_{d}^{T}PA_{d}-P+Q&A_{d}^{T}PB_{d}\\B_{d}^{T}PA_{d}&R+B_{d}^{T}PB_{d}\end{bmatrix}}$${\begin{aligned}\geq 0\end{aligned}}$

The Data

The Matrices $A_{d},B_{d},C_{d},Q,R$ 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

${\begin{aligned}K=(R+B_{d}^{T}PB_{d})^{-1}B_{d}^{T}PA_{d}\end{aligned}}$

$L$ - 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:

${\begin{bmatrix}Q&0&A_{d}^{T}P&P\\0&R&B_{d}^{T}P&0\\PA_{d}&PB_{d}&P&0\\P&0&0&P\end{bmatrix}}$${\begin{aligned}\geq 0\end{aligned}}$

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/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.

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