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

The System

Consider a Discrete-Time LTI system

$\begin{array}{c}{x}_{k+1}=A_d{x}_k+B_d{u}_k\end{array}$

$\begin{array}{c}{y}_k=C_d{x}_k\end{array}$

Consider $A_d\in {ℝ}^{n\times n};B_d\in {ℝ}^{n\times m}$

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

$\begin{array}{c}{A}_d^{T}P{A}_d-A_d^{T}P{B}_d(R+B_d^{T}P{B}_d{)}^{-1}{B}_d^{T}P{A}_d+Q-P\ge 0\end{array}$

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

$P$ 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:

$\left[\begin{array}{cc}{A}_d^{T}P{A}_d-P+Q& A_d^{T}P{B}_d\\ B_d^{T}P{A}_d& R+B_d^{T}P{B}_d\end{array}\right]$$\begin{array}{c}\ge 0\end{array}$

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

$\begin{array}{c}K=(R+B_d^{T}P{B}_d{)}^{-1}{B}_d^{T}P{A}_d\end{array}$

$L$ - 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:

$\left[\begin{array}{cccc}Q& 0& A_d^{T}P& P\\ 0& R& B_d^{T}P& 0\\ P{A}_d& P{B}_d& P& 0\\ P& 0& 0& P\end{array}\right]$$\begin{array}{c}\ge 0\end{array}$

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