LMIs in Control/Matrix and LMI Properties and Tools/LMIs in Control/pages/Dicrete-time Algebraic Riccati Equation

Algebraic Riccati Equations are particularly significant in Optimal Control, filtering and estimation problems. The need to solve such equations is common in the analysis and linear quadratic Gaussian control along with general Control problems. In one form or the other, Riccati Equations play significant roles in optimal control of multivariable and large-scale systems, scattering theory, estimation, and detection processes. In addition, closed forms solution of Ricatti Equations are intractable for two reasons namely; one, they are nonlinear and two, are in matrix forms.

The System

The system is defined as:

${\displaystyle y_{t}=Ay_{t-1}+Bu_{t-1},}$

The Data

A typical specification of the discrete-time linear quadratic control problem is to minimize

${\displaystyle \sum _{t=1}^{T}(y_{t}^{T}Qy_{t}+u_{t}^{T}Ru_{t})}$

subject to the state equation

${\displaystyle y_{t}=Ay_{t-1}+Bu_{t-1},}$

where y is an n × 1 vector of state variables, u is a k × 1 vector of control variables, A is the n × n state transition matrix, B is the n × k matrix of control multipliers, Q (n × n) is a symmetric positive semi-definite state cost matrix, and R (k × k) is a symmetric positive definite control cost matrix.

The LMI: Discrete-Time Algebraic Riccati Inequality

Title and mathematical description of the LMI formulation.

{\displaystyle {\begin{aligned}{\text{ Consider}}\;A_{d}\in \mathbb {R} ^{n\times m},B_{d}\in \mathbb {R} ^{n\times m},P,Q\in \mathbb {S} ^{n},{\text{where}}\;\\P>0,\\Q\geq 0,\\R>0.\\{\text{ The Discrete-Time Algebraic Riccati inequality is given by}}\;\\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,\\\\{\text{ can be written using the Schur complement lemma as}}\;\\{\begin{bmatrix}A_{d}^{T}-P+Q&&A_{d}^{T}PB_{d}\\\star &&R+B_{d}^{T}PB_{d}\\\end{bmatrix}}&\geq 0\\\end{aligned}}}

Equivalently, this discrete-time algebraic Riccati inequality is satisfied under any of the following necessary and sufficient conditions.

1. There exist ${\displaystyle P,Q\in \mathbb {S} ^{n},{\text{and}}\;R\in \mathbb {S} ^{m},}$  where ${\displaystyle P>0,Q\geq 0,andR>0,}$  such that

${\displaystyle {\begin{bmatrix}Q&&0&&A_{d}^{T}P&&P\\\star &&R&&B_{d}^{T}P&&0\\\star &&\star &&P&&0\\\star &&\star &&\star &&P\\\end{bmatrix}}}$  {\displaystyle {\begin{aligned}\geq 0\end{aligned}}} ,

2. There exist ${\displaystyle X,Q\in \mathbb {S} ^{n},{\text{and}}\;R\in \mathbb {S} ^{m},}$  where ${\displaystyle X>0,Q\geq 0,andR>0,}$  such that

${\displaystyle {\begin{bmatrix}Q&&0&&A_{d}^{T}&&1\\\star &&R&&B_{d}^{T}P&&0\\\star &&\star &&X&&0\\\star &&\star &&\star &&X\\\end{bmatrix}}}$  {\displaystyle {\begin{aligned}\geq 0\end{aligned}}} ,

Conclusion:

If the solution exists, LMIs give a unique, stabilizing, symmetric matrix P.

Implementation:

Matlab code for this LMI in the Github repository: