# The Iterative Solution to Discrete-Time \(H_{\infty}\) Control Problems for Periodic Systems
## 2 Iterative Methods for Stabilizing Solution  
### The Iterative Procedure with Two Matrix Sequences  
We consider the iterative procedure for numerical computation of the unique stabilizing and \(\theta\)-periodic solution \(\hat{X}(. )\) of Equation (1), which is described in Section 3.1 of the paper of [PERSON] and co-authors [7]. In fact, the iterative procedure is based on the solution to \(\theta\) -periodic discrete-time Riccati equation with definite quadratic part:  
\[Z^{(k)}(t) =\mathbb{W}(Z^{(k)}):=(\hat{A}(t))^{\mathsf{T}}Z^{(k)}(t+1)\hat{ A}(t)-\left[(\hat{A}(t))^{\mathsf{T}}Z^{(k)}(t+1)B_{2}(t)\right]\] \[\times\left[R_{2}^{(k)}(t)+B_{2}(t)Z^{(k)}(t+1)B_{2}(t)\right]^{- 1}\left[(\hat{A}(t))^{\mathsf{T}}Z^{(k)}(t+1)B_{2}(t)\right]^{T}+M^{(k)}(t) \tag{6}\]  
with \(\hat{A}(t)=A(t)+B(t)F^{(k)}(t)\) where  
\[F^{(k)}(t) =-[R_{\gamma}(t)+B^{\mathsf{T}}(t)X^{(k)}(t+1)B(t)]^{-1}\left[B^ {\mathsf{T}}(t)X^{(k)}(t+1)A(t)+D^{\mathsf{T}}(t)C(t)\right]\] \[R_{2}^{(k)}(t) =D_{2}^{\mathsf{T}}(t)D_{2}(t)+B_{2}^{\mathsf{T}}(t)X^{(k)}(t+1)B _{2}(t)\] \[M^{(k)}(t) =A^{\mathsf{T}}(t)X^{(k)}(t+1)A(t)-[A^{\mathsf{T}}(t)X^{(k)}(t+1) B(t)+C^{\mathsf{T}}(t)D(t)]+C^{\mathsf{T}}(t)C(t)\] \[\times[R_{\gamma}(t)+B^{\mathsf{T}}(t)X^{(k)}(t+1)B(t)]^{-1}\left[A ^{\mathsf{T}}(t)X^{(k)}(t+1)B(t)+C^{\mathsf{T}}(t)D(t)\right]^{\mathsf{T}}-X^{ (k)}(t)\]We rewrite the above equations \((t=0,\ldots,\theta)\) using a block structure. We denote \(\check{Z}=diag(Z(0),Z(1),\ldots,Z(\theta))\) and J is a permutation matrix such that  
\[J^{-1}\check{Z}J=diag(Z(1),Z(2),\ldots,Z(\theta),Z(0))\,,\quad(J^{-1}=J^{T})\]  
Introducing the following notations  
\[\left\{ \begin{array}{rl}\check{A}&=\text{diag}(\check{A}(0),\check{A}(1), \ldots,\check{A}(\theta)),\quad\check{B}_{2}=\text{diag}(B_{2}(0),B_{2}(1), \ldots,B_{2}(\theta))\\ \check{R}_{2}^{(k)}&=\text{diag}(R_{2}^{(k)}0),R_{2}^{(k)}(1),\ldots,R_{2}^{( k)}(\theta))\\ \check{M}^{(k)}&=\text{diag}(M^{(k)}(0),M^{(k)}(1),\ldots,M^{(k)}(\theta))\\ \check{F}_{Z^{(k)}}&=-\left[\check{R}_{2}^{(k)}+\check{B}_{2}^{T}J^{-1}\check {Z}^{(k)}\,J\check{B}_{2}\right]^{-1}(\check{A}^{T}J^{-1}\check{Z}^{(k)}\,J \check{B}_{2})^{T}\end{array} \right.\]  
Equation (6) has a form:  
\[\check{Z}^{(k)}=\check{A}^{T}J^{-1}\check{Z}^{(k)}\,J\check{A}-(\check{A}^{T}J ^{-1}\check{Z}^{(k)}\,J\check{B}_{2})\times\left[\check{R}_{2}^{(k)}+\check{B }_{2}^{T}J^{-1}\check{Z}^{(k)}\,J\check{B}_{2}\right]^{-1}\,(\check{A}^{T}J^ {-1}\check{Z}^{(k)}\,J\check{B}_{2})^{T}+\check{M}^{(k)}\]  
or, in the equivalent form  
\[\check{Z}^{(k)}=(J\,\check{A}+J\,\check{B}_{2}\,\check{F}_{\check{Z}^{(k)}}) ^{T}\check{Z}^{(k)}\,(J\,\check{A}+J\,\check{B}_{2}\,\check{F}_{\check{Z}^{(k )}})+(\check{F}_{\check{Z}^{(k)}})^{T}\,\check{R}_{2}^{(k)}\,\check{F}_{ \check{Z}^{(k)}})+\check{M}^{(k)} \tag{7}\]  
A matter of practical importance is how to solve the last equation. There are several approaches for this purpose, which are briefly outlined in the text that follows. 1. **Block Successive Approximations**. The method of Block Successive Approximations solves Equation (7). It is a modification of the method derived in [12] and applied in [13]. We take \(\check{Y}^{(0)}=diag(21,\ldots,2I)\) and use the following recurrence equation:  
\[\check{Y}^{(i-1)}=\check{\mathbb{W}}(\check{Y}^{(i)}) :=(J\,\check{A}+J\check{B}_{2}\,\check{F}_{\check{Y}^{(i-1)}})^{T }\check{Y}^{(i-1)}\,(J\,\check{A}+J\check{B}_{2}\,\check{F}_{\check{Y}^{(i-1) }}) \tag{8}\] \[+(\check{F}_{\check{Y}^{(i-1)}})^{T}\,\check{R}_{2}^{(k)}\,\check {F}_{\check{Y}^{(i-1)}})+\check{M}^{(k)}\,,\quad i=1,2,...\]  
The iteration process stops with \(\|\check{Y}^{(i)}-\check{\mathbb{W}}(\check{Y}^{(i_{0})})\|\leq\textit{eps}\). 2. **A Block Stein Iteration**. The second method for solving Equation (7) is a method of A Block Stein Iteration.