# The Iterative Solution to Discrete-Time \(H_{\infty}\) Control Problems for Periodic Systems
## 2 Iterative Methods for Stabilizing Solution
### The Iterative Procedure with One Matrix Sequences  
An alternative globally convergent iterative method for the computation of the unique stabilizing and \(\theta\)-periodic solution \(\tilde{X}(. )\) of Equation (1) is derived in [8]. The method preserves similar desirable characteristics as global convergence and local quadratic rate of convergence. The main advantage of this method is that the matrix sequence of approximations of the stabilizing solution are computed directly without exploiting an additional sequence as it is done in the Algorithm TMS. The procedure is described in Section III in [8]. Compute \(X^{(k)}(\cdot)\) as the stabilizing solution of the discrete-time Riccati equations with a defined sign [8]:  
\[X^{(k)}(t)=\mathbb{Q}(X^{(k)}) :=(\tilde{A}(t))^{T}X^{(k)}(t+1)\tilde{A}(t)-\left[(\tilde{A}(t) )^{T}X^{(k)}(t+1)B_{2}(t)+L^{(k-1)}(t)\right]\] \[\times\left[D_{2}^{T}(t)\,D_{2}(t)+B_{2}^{T}(t)X^{(k)}(t+1)B_{2} (t)\right]^{-1}\] \[\times\left[(\tilde{A}(t))^{T}X^{(k)}(t+1)B_{2}(t)+L^{(k-1)}(t) \right]^{T}+M^{(k-1)}(t) \tag{11}\]  
where  
\[F_{1}^{(k-1)}(t) =(I_{m_{1}}\ \ 0)\ F^{(k)}(t)\,,\] \[=-\left(I_{m_{1}}\ \ 0\right)\left[R_{\gamma}(t)+B^{T}(t)X^{(k-1)}(t+ 1)B(t)\right]^{-1}\left[B^{T}(t)X^{(k-1)}(t+1)A(t)+D^{T}(t)C(t)\right]\] \[\tilde{A}(t) =A(t)+B_{1}(t)\,F_{1}^{(k-1)}(t)\] \[L^{(k-1)}(t) =(C(t)+D_{1}(t)F_{1}^{(k-1)}(t))^{T}\,D_{2}(t)\] \[M^{(k-1)} =(C(t)+D_{1}(t)F_{1}^{(k-1)}(t))^{T}\,(C(t)+D_{1}(t)F_{1}^{(k-1)} (t))-\gamma^{2}(F_{1}^{(k-1)}(t))^{T}F_{1}^{(k-1)}(t)\]We present the above set of Equations (11) as one equation with a block structure. We denote \(\tilde{X}=diag(X(0),X(1),\ldots,X(\theta))\) and J is a permutation matrix such that  
\[J^{-1}\tilde{X}J=diag(X(1),X(2),\ldots,X(\theta),X(0))\]  
and  
\[\left\{ \begin{array}{ll}\tilde{A}&=\text{diag}(\tilde{A}(0),\tilde{A}(1), \ldots,\tilde{A}(\theta))\,,\quad\tilde{B}_{2}=\text{diag}(B_{2}(0),B_{2}(1), \ldots,B_{2}(\theta))\\ \tilde{D}&=\text{diag}(D_{2}^{T}(0)\,D_{2}(0),D_{2}^{T}(1)\,D_{2}(1),\ldots,D_ {2}^{T}(\theta)\,D_{2}(\theta))\\ \tilde{L}&=\text{diag}(L^{(k-1)}(0),L^{(k-1)}(1),\ldots,L^{(k-1)}(\theta))\\ \tilde{M}&=\text{diag}(M^{(k-1)}(0),M^{(k-1)}(1),\ldots,M^{(k-1)}(\theta))\\ \tilde{F}_{\tilde{X}}&=-\left[\tilde{D}+\tilde{B}_{2}^{T}J^{-1}\tilde{X}\,j \tilde{B}_{2}\right]^{-1}(\tilde{A}^{T}J^{-1}\tilde{X}\,j\tilde{B}_{2}+L)^{T} \end{array} \right.\]  
The set of equations \(X^{(k)}(t)=\mathbb{Q}(X^{(k)})\) is transformed:  
\[\tilde{X}=\tilde{A}^{T}J^{-1}\tilde{X}\,J\tilde{A}-(\tilde{A}^{T}J^{-1}\tilde {X}\,J\tilde{B}_{2}+\tilde{L})\,\left[\tilde{D}+\tilde{B}_{2}^{T}J^{-1}\tilde {X}\,j\tilde{B}_{2}\right]^{-1}\,(\tilde{A}^{T}J^{-1}\tilde{X}\,J\tilde{B}_{2 }+\tilde{L})^{T}+\tilde{M}\]  
or  
\[\tilde{X}=(J\,\tilde{A}+J\,\tilde{B}_{2}\,\tilde{F}_{\tilde{X}})^{T}\tilde{X} \,(J\,\tilde{A}+J\,\tilde{B}_{2}\,\tilde{F}_{\tilde{X}})+(\tilde{F}_{\tilde{ X}})^{T}\,\tilde{D}\,\tilde{F}_{\tilde{X}}+\tilde{M} \tag{12}\]  
We present two iterative approaches to implement iteration Equation (12). 1. **A Block Stein Iteration. ** The first method for solving Equation (12) is a method of A Block Stein Iteration. We take \(\tilde{Y}_{0}=diag(2I,\ldots,2I)\) and use the following recurrence equation:  
\[\tilde{Y}^{(i)}=(J\,\tilde{A}+J\tilde{B}_{2}\,\tilde{F}_{\tilde{Y}^{(i-1)}}) ^{T}\tilde{Y}^{(i)}\,(J\,\tilde{A}+J\tilde{B}_{2}\,\tilde{F}_{\tilde{Y}^{(i-1 )}})+(\tilde{F}_{\tilde{Y}^{(i-1)}})^{T}\,\tilde{D}\,\tilde{F}_{\tilde{Y}^{(i -1)}}+\tilde{M}^{(k)} \tag{13}\]  
\(i=1,2,\)... The iteration process stops with \(\|\tilde{Y}^{(i_{0})}-\mathbb{Q}(\tilde{Y}^{(i_{0})})\|\leq\textit{eps}\). We use the MATLAB command dlyap for solving Equation (13). 2. **Successive Approximations: Equation by Equation**. The second method for the solution of Equation (11) is a method of Successive Approximations: Equation by Equation. We apply this approach to Equation (11) with \(Y^{(0)}(0)=I\).