User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Discrete-time Algebraic Riccati Inequalities

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:

$$y_t = Ay_{t-1} + Bu_{t-1},$$

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


 * $$\sum_{t=1}^T (y_t^T Qy_t + u_t^T Ru_t)$$

subject to the state equation


 * $$y_t = Ay_{t-1} + Bu_{t-1},$$

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.



\begin{align} \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^T_d PA_d - A^T_d PB_d(R + B^T_d P B_d)^{-1} B^T_d P A_d + Q - P & \geq 0,\\ \\ \text{ can be written using the Schur complement lemma as}\;\\ \begin{bmatrix} A^T_d - P + Q && A^T_d P B_d\\ \star && R+B^T_d P B_d\\ \end{bmatrix}& \geq 0 \\ \end{align}$$

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

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

$$ \begin{bmatrix} Q && 0 && A^T_d P && P\\ \star && R && B^T_d P && 0\\ \star && \star && P && 0 \\ \star && \star && \star && P \\ \end{bmatrix}$$ $$\begin{align} \geq 0 \end{align}$$,

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

$$ \begin{bmatrix} Q && 0 && A^T_d && 1\\ \star && R && B^T_d P && 0\\ \star && \star && X && 0 \\ \star && \star && \star && X \\ \end{bmatrix}$$ $$\begin{align} \geq 0 \end{align}$$,

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

Implementation:
Matlab code for this LMI in the Github repository: