Inheritance properties of the conjugate discrete-time algebraic Riccati equation

In this paper we are mainly concerned with the inheritance properties of the maximal solution to the conjugate discrete-time algebraic Riccati equation (CDARE) of the form

where $A\in\mathbb{C}^{n\times n}$, $B\in\mathbb{C}^{n\times m}$, $R\in\mathbb{H}_{m}$ is nonsingular, $H\in\mathbb{H}_{n}$, $I$ is the identity matrix of compatible size and $G:=BR^{-1}B^{H}$, respectively. Here, $\mathbb{H}_{\ell}$ denotes the set of all $\ell\times\ell$ Hermitian matrices and $\overline{M}$ is the matrix obtained by taking the complex conjugate of each entry of a matrix $M$. The solution $X\in\mathbb{H}_{n}$ of the CDARE (1a), with $R_{X}:=R+B^{H}\overline{X}B$ being positive definite, is considered in this paper.

For any $M,N\in\mathbb{H}_{n}$, the positive definite and positive semidefinite matrices are denoted by $M>0$ and $M\geq 0$, respectively. Moreover, we usually write $M\geq N$ (or $M\leq N$) if $M-N\geq 0$ (or $N-M\geq 0$) in the context. For the sake of simplicity, the spectrum and spectral radius of $A\in\mathbb{C}^{n\times n}$ are denoted by $\sigma(A)$ and $\rho(A)$, respectively. We say that the CDARE (1) has a maximal solution $X_{M}\in\mathbb{H}_{n}$ if $X_{M}\geq X$ for any solution $X\in\mathbb{H}_{n}$ of the CDARE. Thus, it follows from the definition of the maximality that $X_{M}$ is unique if it exists. A matrix operator $f:\mathbb{H}_{n}\rightarrow\mathbb{H}_{n}$ is order preserving (resp. reversing) on $\mathbb{H}_{n}$ if $f(A)\geq f(B)$ (resp. $f(A)\leq f(B)$) when $A\geq B$ and $A,B\in\mathbb{H}_{n}$.

A class of CDAREs (1) arises originally from linear quadratic regulation (LQR) optimal control problem for the discrete-time antilinear system

where $x_{k}\in\mathbb{C}^{n}$ is the state response and $u_{k}\in\mathbb{C}^{m}$ is the control input. The main goal of this control problem is to find a state feedback control $\color[rgb]{0,0,0}u_{k}=-\overline{F}x_{k}$ such that the performance index

is minimized with $H\geq 0$ and $R>0$. If the antilinear system (2) is controllable, it is shown in Theorem 12.7 of [18] that the optimal state feedback controller is $u_{k}^{*}=-\overline{F_{X_{*}}}x_{k}$ with $F_{X_{*}}:=R_{X_{*}}^{-1}B^{H}\overline{X_{*}}A$ such that the minimum value of $\mathcal{J}(u_{k}^{*},x_{0})=x_{0}^{H}X_{*}x_{0}$ is achieved and the corresponding closed-loop antilinear system

is asymptotically stable, i.e., $\lim\limits_{k\rightarrow\infty}x_{k}=0$, where $X_{*}\geq 0$ is the unique solution of the CDARE (1a) or the discrete-time algebraic anti-Riccati equation [17, 18].

Recently, some accelerated iterations have been proposed for solving the unique positive definite solution of the CDARE (1) under positive definiteness assumptions with $G>0$ and $H>0$, see, e.g., [9, 10, 12] and the references therein. In addition, this numerical technique has also been utilized to some real-life applications recently, see, e.g., [11, 14, 15].

It is worth mentioning that the authors in [9] first transformed a conjugate nonlinear matrix equation, arising from modern quantum theory, to some CDARE (1b) with $G,H>0$, and proposed theoretical results for the existence and uniqueness of the (maximal) positive definite solution to the CDARE. On the other hand, we have generalized the existence of the maximal solution for the CDARE (1) with $G,H\in\mathbb{H}_{n}$ under the framework of the fixed-point iteration (FPI) and some weaker assumptions [4], see also Theorem 2.1 below.

Inspired by the technique using in [9], if

, where $R_{H}:=R+B^{H}\overline{H}B$, then the CDARE (1a) can be transformed into a discrete-time algebraic Riccati equation (DARE) of the form

where its coefficient matrices are given by

and the invertibility of the matrix $\widehat{R}_{X}:=\widehat{R}+\widehat{B}^{H}X\widehat{B}$ will be shown in the next section. The assumption (4) will be needed throughout the paper.

Therefore, the first question is to ask whether the CDARE (1) and its induced DARE (5) both have the same maximal solution $X_{M}\in\mathbb{H}_{n}$ under the assumptions proposed in [4]. If yes, we may further ask if there exists a suitable matrix $\widehat{F}_{X_{M}}\in\mathbb{C}^{2m\times n}$ related to $X_{M}$ so that the closed-loop matrix $\widehat{A}-\widehat{B}\widehat{F}_{X_{M}}$ inherits the stability of the closed-loop matrix $A-BF_{X_{M}}$ related to the original CDARE (1), where $F_{X_{M}}$ is defined by (3). These interesting questions will be addressed completely in this work.

Once the CDARE (1) and its transformed DARE (5) have the same maximal solution, there are dozens of numerical methods for solving the standard DAREs of small to medium sizes in the existing literatures, see, e.g., [3, 5, 6, 7, 8, 13, 16] and the references therein. Nevertheless, following the line of the FPI discussed in [4], we are mainly interested in the design of an accelerated fixed-point iteration (AFPI) for finding the maximal solution of the CDARE (1) via its transformed DARE (5) directly.

The paper is organized as follows. In Section 2, we introduce some useful notations, and auxiliary theorems and lemmas that will be used in our main results. Some inheritance properties related to the CDARE (1) and its transformed DARE (5) have been addressed in Section 3. Especially, under reasonable and mild assumptions, we have shown that these Riccati matrix equations both have the same maximal Hermitian solution in Theorem 3.1. Based on this theorem, an accelerated fixed-point iteration is proposed for computing the maximal solution of the transformed DARE (5) directly in Section 4. Numerical examples are given to illustrate the validity of the inheritance properties and the feasibility of our AFPI algorithm in Section 5. Finally, we conclude the paper in Section 6.

In this section we introduce some notations and auxiliary lemmas that will be used below. Firstly, let the conjugate Stein matrix operator $\mathcal{C}_{A}:\mathbb{H}_{n}\rightarrow\mathbb{H}_{n}$ and standard Stein matrix operator $\mathcal{S}_{A}:\mathbb{H}_{n}\rightarrow\mathbb{H}_{n}$ associated with a matrix $A\in\mathbb{C}^{n\times n}$ be defined by

for any $X\in\mathbb{H}_{n}$. In general, the operator $\mathcal{C}_{A}$ is neither order preserving nor order reversing. However, under the assumption that $\rho(\overline{A}A)<1$, its inverse operator $\mathcal{C}^{-1}_{A}$ is order preserving, since $\mathcal{C}_{A}^{-1}(X)=\sum\limits_{k=0}^{\infty}((\overline{A}A)^{k})^{H}(X+A^{H}\overline{X}A)(\overline{A}A)^{k}\geq\sum\limits_{k=0}^{\infty}((\overline{A}A)^{k})^{H}(Y+A^{H}\overline{Y}A)(\overline{A}A)^{k}=\mathcal{C}_{A}^{-1}(Y)$ for $X,Y\in\mathbb{H}_{n}$ with $X\geq Y$.

It will be clear later on that the matix operators $\mathcal{R}:\mathrm{dom}(\mathcal{R})\rightarrow\mathbb{H}_{n}$ and $\widehat{\mathcal{R}}:\mathrm{dom}(\widehat{\mathcal{R}})\rightarrow\mathbb{H}_{n}$ defined by (1a) and (5a), respectively, both play an important role in our theorems below, where $\mathrm{dom}(\mathcal{R}):=\{X\in\mathbb{H}_{n}\,|\,\det(R_{X})\neq 0\}$ and $\mathrm{dom}(\widehat{\mathcal{R}}):=\{X\in\mathbb{H}_{n}\,|\,\det(\widehat{R}_{X})\neq 0\}$. If $R_{X}$ is nonsingular for $X\in\mathrm{dom}(\mathcal{R})$, it is easily seen that the matrix $\widehat{R}_{X}$ defined by (6a) is of the form

Therefore, $\widehat{R}_{X}$ is nonsingular, i.e., $X\in\mathrm{dom}(\widehat{\mathcal{R}})$, since the Schur complement $R_{X}$ of $\widehat{R}_{X}$ given by

is also nonsingular. This implies $\mathrm{dom}(\mathcal{R})\subseteq\mathrm{dom}(\widehat{\mathcal{R}})$ and thus the solution set $\mathcal{R}_{=}:=\{X\in\mathrm{dom}(\mathcal{R})\,|\,X=\mathcal{R}(X)\}$ of the CDARE (1) is contained in the solution set $\widehat{\mathcal{R}}_{=}:=\{X\in\mathrm{dom}(\widehat{\mathcal{R}})\,|\,X=\widehat{\mathcal{R}}(X)\}$ of the transformed DARE (5).

Moreover, for the sake of simplicity the following sets and matrices

will be used throughout the paper for any $X\in\mathrm{dom}(\mathcal{R})$ and $\widehat{F}\in\mathbb{C}^{2m\times n}$.

The following lemma characterizes some useful identities depending on the matrix operators $\mathcal{R}(\cdot)$ and $\widehat{\mathcal{R}}(\cdot)$ with respect to the associated conjugate Stein and Stein operators, which will be quoted in the proof of Lemma 2.3 later on.

Next, a sufficient condition is provided in the following result to guarantee the equivalence between two subsets of $\mathbb{H}_{n}$, which will be described in the proof of Lemma 2.3 later.

Let $\mathbb{T}:=\{X\in\mathrm{dom}(\mathcal{R})\,|\,\rho(\widehat{T}_{X})<1\}$ and $\mathbb{P}:=\{X\in\mathrm{dom}(\mathcal{R})\,|\,R_{X}>0\}$. Starting from an initial matrix $X_{0}\in\mathcal{S}_{\geq}:=\bigcup\limits_{W\in\mathbb{T}}\{X\in\mathbb{H}_{n}\,|\,\mathcal{C}_{T_{W}}(X)\geq H_{W}\}$ if $\mathbb{T}\neq\emptyset$. It has been shown in [4] that the sequence $\{X_{k}\}_{k=0}^{\infty}$ generated by the following FPI of the form

converges at least linearly to the maximal solution of the CDARE (1) if

which are main assumptions throughout this paper. The following theorem is quoted from Theorem 3.1 of [4] and it will be applied to deduce the inheritance properties presented in Section 3.

For any $Y_{0}\in\mathrm{dom}(\mathcal{R})$, consider the sequence $\{Y_{k}\}_{k=0}^{\infty}$ generated by the FPI

where the matrix operators $\mathcal{R}(\cdot)$ and $\widehat{\mathcal{R}}(\cdot)$ are defined by (1a) and (5a), respectively. Analogously, the existence of the maximal solution $Y_{M}$ to the transformed DARE (5) can be established by modifying the proof Theorem 3.1 in [4] slightly. It is stated without proof in the following theorem, in which $\widehat{\mathbb{T}}:=\{X\in\mathbb{H}_{n}\,|\,\rho(\widehat{T}_{X}^{D})<1\}$, $\widehat{\mathbb{P}}:=\{Y\in\mathrm{dom}(\widehat{\mathcal{R}})\,|\,\widehat{R}_{Y}>0\}$ and $\widehat{\mathcal{S}}_{\geq}:=\bigcup\limits_{X_{\widehat{\mathbb{T}}}\in\widehat{\mathbb{T}}}\{X\in\mathbb{H}_{n}\,|\,\mathcal{S}_{X_{\widehat{\mathbb{T}}}}(X)\geq\widehat{H}_{X_{\mathbb{T}}}=\widehat{H}+\widehat{F}_{X_{\mathbb{T}}}^{H}\widehat{R}\widehat{F}_{X_{\mathbb{T}}}\}$ if $\widehat{\mathbb{T}}\neq\emptyset$.

It is clear that $X_{M}\leq Y_{M}$, and the interest is finding some reasonable conditions such that $X_{M}$ coincides with $Y_{M}$. Finally, the following lemma can be used to prove the inheritance of the maximal solution to CDAREs, which will be illustrated in the next section.

In this section, as mentioned previously, we shall clarify which properties will be inherited from the maximal solution of the CDARE (1). Some results of the previous section will be applied in the following properties.