Keywords

1 Introduction

Linear discrete-time periodic descriptor systems have received a lot of attention over the last twenty years. They are suitable models for several natural as well as man-made phenomena, and have applications in modeling of periodic time-varying filters and networks [16, 21], multirate sampled-data systems [16, 18], circuit simulation [3, 8, 10, 18], micro-electronics [19, 20], aerospace realm [34, 35], control of industrial processes and communication systems [1, 20].

A linear discrete-time periodic descriptor system with time-varying dimensions has the form

$$ E_{k}x_{k+1} = A_{k}x_{k}+ B_{k}u_{k},\quad y_{k}= C_{k}x_{k}, \quad k \in {\mathbb{Z}}, $$
(11.1)

where \({E_{k} \in {\mathbb{R}}^{\mu_{k+1} \times n_{k+1}}}\), \({A_{k}\in{\mathbb{R}}^{\mu_{k+1} \times n_{k}}}\), \({B_{k}\in{\mathbb{R}}^{\mu_{k+1}\times m_k}}\), \({C_{k}\in{\mathbb{R}}^{p_k\times n_{k}}}\) are periodic with a period \(K \geq 1\) and \(\sum_{k=0}^{K-1} \mu_{k}=\sum_{k=0}^{K-1} n_{k}= n\). The matrices E k are allowed to be singular for all k. For system (11.1), a reduced-order model of dimension r would be a system of the form

$$ \tilde E_{k} \tilde x_{k+1}= \tilde A_{k} \tilde x_{k}+ \tilde B_{k} u_{k},\quad \tilde y_{k}= \tilde C_{k} \tilde x_{k}, \quad k \in {\mathbb{Z}}, $$
(11.2)

where \({\tilde E_{k} \in {\mathbb{R}}^{\gamma_{k+1} \times r_{k+1}}}\), \({\tilde A_{k}\in{\mathbb{R}}^{\gamma_{k+1} \times r_{k}}}\), \({\tilde B_{k}\in{\mathbb{R}}^{\gamma_{k+1}\times m_k}}\), \({\tilde C_{k}\in{\mathbb{R}}^{p_k\times r_{k}}}\) are K-periodic matrices, \(\sum_{k=0}^{K-1} \gamma_{k}=\sum_{k=0}^{K-1} r_{k}= r\) and r ≪ n. Apart from having a much smaller state-space dimension, it is also important that the reduced-order model preserves physical properties of the original system such as regularity, stability and passivity, and that the approximation error is small.

The dynamics of the discrete-time periodic descriptor system (11.1) are often addressed by the regularity and the eigenstructure of the periodic matrix pairs \(\{E_{k},A_{k}\}_{k=0}^{K-1}\). If all E k are nonsingular, the eigenvalues (also called characteristic multipliers) of system (11.1) are given by the eigenvalues of the matrix product

$$ E_{K-1}^{-1}A_{K-1}E_{K-2}^{-1}A_{K-2} \cdots E_{0}^{-1}A_{0} $$
(11.3)

associated with the periodic matrix pairs \(\{E_{k},A_{k}\}_{k=0}^{K-1}\). This product only yields a well-defined matrix if all E k are nonsingular. Even if they are, the formulation of that matrix should be avoided for reasons of numerical stability. Note that even for some E k being singular, we use (11.3) in a formal way to denote a generalization of matrix pencils to this periodic case (see [2] for details of this formal matrix product calculus). We compute the eigenvalues of (11.3) via the generalized periodic Schur decomposition [11, 32].

There exist unitary matrices \({P_{k}\in{\mathbb{C}}^{\mu_{k+1}\times\mu_{k+1}}}\) and \({Q_{k}\in{\mathbb{C}}^{n_k\times n_k}}\), with \(Q_{k+K}=Q_{k}\) such that the transformed matrices

$$ N_{k}=P_{k}^{*}E_{k}Q_{k+1}, \quad M_{k}=P_{k}^{*}A_{k}Q_{k}, \quad k=0,\ldots, K-1, $$

are all upper triangular, where for the ease of notation we allow complex arithmetic (in practice, however, computations can be performed in real arithmetic leading to quasi-triangular structure of one of the M k ).

Then the formal matrix product

$$ N_{K-1}^{-1}M_{K-1}N_{K-2}^{-1}M_{K-2} \cdots N_{0}^{-1}M_{0} $$

has the same eigenvalues as (11.3). The blocks on the diagonals of the transformed matrices M k and N k are used to define the eigenvalues of the periodic matrix pairs \(\{E_{k},A_{k}\}_{k=0}^{K-1}\). A finite eigenvalue is given by

$$ \lambda_{l}= \prod_{k=0}^{K-1} \frac{m_{ll}^{(k)}}{n_{ll}^{(k)}}, $$

provided n (k) ll  ≠ 0 for \(k=0,\ldots, K-1\). Here \(m_{ll}^{(k)}\in \Uplambda (M_{k})\) and \(n_{ll}^{(k)}\in \Uplambda (N_{k})\), where \(\Uplambda\) denotes the eigenspectrum of the corresponding matrix. An eigenvalue is called infinite if \(\prod_{k=0}^{K-1} n_{ll}^{(k)} = 0 \), but \(\prod_{k=0}^{K-1} m_{ll}^{(k)} \neq 0 \).

In this paper, we briefly review some basic concepts of discrete-time periodic descriptor systems (Sect. 11.2). In Sect. 11.3, we study the periodic reachability and observability Gramians from [6] using a lifted representation. A balanced truncation model reduction method for periodic descriptor systems is presented in Sect. 11.4. Section 11.5 contains a numerical example that illustrates the properties of the suggested model reduction technique.

2 Periodic Descriptor Systems

Lifted representations of discrete-time periodic systems play an important role in extending many theoretical results and numerical algorithms for time-invariant systems to the periodic setting [4, 7, 33]. We consider here the cyclic lifted representation which was introduced first for standard periodic systems in [17]. The cyclic lifted representation of the periodic descriptor system (11.1) is given by

$$ {\fancyscript{E}}{\fancyscript{X}}_{k+1}={\fancyscript{A}}{\fancyscript{X}}_{k}+{\fancyscript{B}}\,{\fancyscript{U}}_{k},\quad {\fancyscript{Y}}_{k}={\fancyscript{C}}{\fancyscript{X}}_{k}, $$
(11.4)

where

$$ \begin{array}{l} {{\fancyscript{E}}}= \hbox{diag} (E_0,E_1,\ldots,E_{K-1}), \quad {{\fancyscript{B}}}= \hbox{diag} (B_0,B_1,\ldots,B_{K-1}), \\ {{\fancyscript{A}}}= \left[\begin{array}{llll} 0&\cdots&0&A_{0}\\ A_{1}&&&0\\ &{\ddots}&&{\vdots}\\ 0&&A_{K-1}&0\\ \end{array}\right],\quad {{\fancyscript{C}}}=\left[\begin{array}{llll} 0&\cdots&0&C_{0}\\ C_{1}&&&0\\ &{\ddots}&&{\vdots}\\ 0&&C_{K-1}&0\\ \end{array}\right], \end{array} $$
(11.5)

The descriptor vector, system input and output of (11.4) are related to those of (11.1) via

$$ {{\fancyscript{X}}}_{k}= [x_{1}^T, \ldots, x_{K-1}^T, x_{0}^T]^T,\quad {{\fancyscript{U}}}_{k}= [u_{0}^T, u_{1}^T, \ldots, u_{K-1}^T]^T,\quad {{\fancyscript{Y}}}_{k}= [y_{0}^T, y_{1}^T, \ldots, y_{K-1}^T]^T, $$

respectively.

A set of periodic matrix pairs \(\{{E_{k},A_{k}}\}_{k=0}^{K-1}\) is called regular if the pencil \({z {\fancyscript{E}}- {\fancyscript{A}}}\) is regular, i.e., \({\det(z {\fancyscript{E}}- {\fancyscript{A}})\not\equiv 0}\). In this case, we can define a transfer function of the lifted system (11.4) as \({{\fancyscript{H}}(z)={\fancyscript{C}}(z{\fancyscript{E}}-{\fancyscript{A}})^{-1}{\fancyscript{B}}}\).

The regular set of periodic matrix pairs \(\{E_{k},A_{k}\}_{k=0}^{K-1}\) can be transformed into a periodic Kronecker canonical form [6, 32]. For \(k=0,1,\ldots, K-1\), there exist nonsingular matrices \({W_{k}\in{\mathbb{R}}^{\mu_{k+1}\times\mu_{k+1}}}\) and \({Z_{k}\in{\mathbb{R}}^{n_k\times n_k}}\) such that

$$ W_{k}E_{k}Z_{k+1}=\left[\begin{array}{ll} I_{n_{k+1}^f}&0\\ 0 & E_{k}^{b} \end{array}\right],\quad W_{k}A_{k}Z_{k}=\left[\begin{array}{ll} A_{k}^{f}&0\\ 0 & I_{n_k^\infty} \end{array}\right], $$
(11.6)

where \(Z_{K}=Z_{0}\), \(A_{k+K-1}^{f}A_{k+K-2}^{f} \cdots A_{k}^{f}=J_k\) is an \(n_k^f\times n_k^f\) matrix corresponding to the finite eigenvalues, \(E_{k}^{b}E_{k+1}^{b} \cdots E_{k+K-1}^{b}=N_k\) is an \(n_k^\infty\times n_k^\infty\) nilpotent matrix corresponding to an eigenvalue at infinity, \(n_k=n_k^f+ n_k^\infty\) and \(\mu_{k+1}=n_{k+1}^f+ n_k^\infty\). The index ν of the periodic descriptor system (11.1) is defined as \(\nu =\max(\nu_{0},\nu_{1},\ldots, \nu_{K-1})\), where ν k is the nilpotency index of N k . Note that the finite eigenvalues of \(\{ E_{k},A_{k}\}_{k=0}^{K-1}\) coincide with the finite eigenvalues of the lifted pencil \({z {\fancyscript{E}}- {\fancyscript{A}}}\).

For \(k=0,1,\ldots, K-1\), the matrices

$$ P_{r}(k)=Z_{k}\left[\begin{array}{ll} {I}_{n_{k+1}^f}&0\\ 0 &0 \end{array} \right]Z_{k}^{-1}, \quad P_{l}(k)=W_{k}^{-1}\left[\begin{array}{ll} {I}_{n_{k+1}^f}&0\\ 0 &0 \end{array} \right]W_{k}, $$

are the spectral projectors onto the kth right and left deflating subspaces of the periodic matrix pairs \(\{ E_{k},A_{k}\}_{k=0}^{K-1} \) corresponding to the finite eigenvalues, and \(Q_r(k)=I-P_r(k)\) and \(Q_l(k)=I-P_l(k)\) are the complementary projectors. Let for every \(k=0,1,\ldots, K-1\), the vector \(Z_{k}^{-1}x_{k}=[\,(x_{k}^{f})^T, \; (x_{k}^{b})^T\,]^T\) and the matrices

$$ W_{k}B_{k}=\left[\begin{array}{l} B_{k}^{f}\\ B_{k}^{b} \end{array}\right], \quad C_{k}Z_{k}=\left[\begin{array}{ll} C_{k}^{f},\, & C_{k}^{b} \end{array}\right], $$

be partitioned in blocks conformally to the periodic matrix pairs \(\{ E_{k},A_{k}\}_{k=0}^{K-1} \) in (11.6). Under this transformation, system (11.1) can be decoupled into forward and backward periodic subsystems

$$ x_{k+1}^{f} = A_{k}^{f} x_{k}^{f}+B_{k}^{f} u_{k},\quad y_{k}^{f}=C_{k}^{f}x_{k}^{f}, $$
(11.7)
$$ E_{k}^{b}x_{k+1}^{b} = x_{k}^{b}+B_{k}^{b} u_{k},\quad y_{k}^{b}=C_{k}^{b}x_{k}^{b}, $$
(11.8)

respectively, with \(y_{k}=y_{k}^{f}+y_{k}^{b}\), \(k=0,1,\ldots, K-1\). The state transition matrix for the forward subsystem (11.7) is given by \(\Upphi_{f}(i,j)=A_{i-1}^{f}A_{i-2}^{f} \cdots A_{j}^{f}\) for i > j and \(\Upphi_{f}(i,i)=I_{n_i^f}\). For the backward subsystem (11.8), the state transition matrix is defined as \(\Upphi_{b}(i,j)=E_{i}^{b}E_{i+1}^{b} \cdots E_{j-1}^{b}\) for i < j and \(\Upphi_{b}(i,i)=I_{n_i^\infty}\). Using these matrices we can now define the forward and backward fundamental matrices of the periodic descriptor system (11.1) as

$$ \Uppsi_{i, j}=\left\{\begin{array}{ll} Z_{i}\left[\begin{array}{cc} \Upphi_{f}(i,j+1) & 0\\ 0 & 0 \end{array}\right] W_{j}, & \quad i>j, \\ Z_{i}\left[\begin{array}{cc} 0 & 0\\ 0 & -\Upphi_{b}(i,j) \end{array} \right]W_{j}, & \quad i\leq j. \end{array} \right. $$

These fundamental matrices play an important role in the definition of the reachability and observability Gramians of the periodic descriptor system (11.1) that we will consider in the next section.

3 Periodic Gramians and Matrix Equations<!--RH>Periodic Gramians and Matrix Equations<!--RH>

The dynamics of the periodic descriptor system (11.1) are often addressed by the eigenstructure of the periodic matrix pairs \(\{ E_{k},A_{k}\}_{k=0}^{K-1}\).

Definition 1

The periodic matrix pairs \(\{ E_{k},A_{k}\}_{k=0}^{K-1} \) are said to be periodic stable (pd-stable) if all the finite eigenvalues of \(\{ E_{k},A_{k}\}_{k=0}^{K-1} \) lie inside the unit circle.

In balanced truncation model reduction, Gramians play a fundamental role [15, 25, 28, 31]. For periodic descriptor system (11.1), the reachability and observability Gramians have been first introduced in [6].

Definition 2

Suppose that the periodic matrix pairs \(\{ E_{k},A_{k}\}_{k=0}^{K-1} \) are pd-stable. For \(k= 0,1,\ldots,K-1\), we define the causal and noncausal reachability Gramians G cr k and G ncr k of system (11.1) as

$$ G_{k}^{cr} =\sum_{j=-\infty}^{k-1} \Uppsi_{k,j}^{}B_j^{}B_j^{T}\Uppsi_{k,j}^T, \quad G_{k}^{ncr} =\sum_{j=k}^{k+\nu K-1} \Uppsi_{k,j}^{}B_j^{}B_j^T\Uppsi_{k,j}^{T}. $$

The complete reachability Gramian G r k is the sum of the causal and noncausal Gramians, i.e., \( G_{k}^{r}= G_{k}^{cr}+G_{k}^{ncr}\) for \(k=0,1,\ldots,K-1\).

Definition 3

For the pd-stable matrix pairs \(\{ E_{k},A_{k}\}_{k=0}^{K-1} \) and \(k= 0,1,\ldots,K-1\), the causal and noncausal observability Gramians G co k and G nco k of system (11.1) are defined as

$$ G_{k}^{co} =\sum_{j=k}^{\infty} \Uppsi_{j,k-1}^T C_j^TC_j^{}\Uppsi_{j,k-1}^{},\quad G_{k}^{nco} =\sum_{j=k-\nu K}^{k-1} \Uppsi_{j,k-1}^TC_j^TC_j^{}\Uppsi_{j,k-1}^{}. $$

The complete observability Gramian G o k is the sum of the causal and noncausal Gramians, i.e., \( G_{k}^{o} = G_{k}^{co}+G_{k}^{nco}\) for \(k= 0,1,\ldots,K-1\).

Note that the causal and noncausal Gramians correspond to the forward and backward subsystems (11.7) and (11.8), respectively.

3.1 Periodic Projected Lyapunov Equations

It has been shown in [26] that the Gramians of discrete-time descriptor systems satisfy projected generalized discrete-time Lyapunov equations with special right-hand sides. A similar result also holds for periodic descriptor systems.

Theorem 1

[6] Consider a periodic discrete-time descriptor system (11.1), where the periodic matrix pairs \(\{ E_{k},A_{k}\}_{k=0}^{K-1} \) are pd-stable.

  1. 1.

    For\(k=0,1, \ldots, K-1\), the causal reachability and observability Gramians\(\{G_{k}^{cr}\}_{k=0}^{K-1}\)and\(\{G_{k}^{co}\}_{k=0}^{K-1} \)are the unique symmetric, positive semidefinite solutions of the projected generalized discrete-time periodic Lyapunov equations (PGDPLEs)

    $$ \begin{aligned} A_{k}G_{k}^{cr}A_{k}^{T}-E_{k}G_{k+1}^{cr}E_{k}^{T}&= -P_{l}(k)B_{k}B_{k}^{T}P_{l}(k)^{T},\\ G_{k}^{cr}&=P_{r}(k)G_{k}^{cr}P_{r}(k)^{T}, \end{aligned} $$
    (11.9)

    and

    $$ \begin{aligned} A_{k}^{T}G_{k+1}^{co}A_{k}-E_{k-1}^{T}G_{k}^{co}E_{k-1}&=& -P_{r}(k)^{T}C_{k}^{T}C_{k}P_{r}(k),\\ G_{k}^{co}&=&P_{l}(k-1)^{T}G_{k}^{co}P_{l}(k-1), \end{aligned} $$

    respectively, where\(G_{K}^{cr}=G_{0}^{cr}\), \(G_{K}^{co}=G_{0}^{co}\), \(E_{-1}=E_{K-1}\), and\(P_{l}(-1)=P_{l}(K-1)\).

  2. 2.

    For\(k=0,1, \ldots, K-1\), the noncausal reachability and observability Gramians\(\{G_{k}^{ncr}\}_{k=0}^{K-1}\)and\(\{G_{k}^{nco}\}_{k=0}^{K-1} \)are the unique symmetric, positive semidefinite solutions of the PGDPLEs

    $$ \begin{aligned} A_{k}G_{k}^{ncr}A_{k}^{T}-E_{k}G_{k+1}^{ncr}E_{k}^{T}&= Q_{l}(k)B_{k}B_{k}^{T}Q_{l}(k)^{T},\\ G_{k}^{ncr}&=Q_{r}(k)G_{k}^{ncr}Q_{r}(k)^{T}, \end{aligned} $$

    and

    $$ \begin{aligned} A_{k}^{T}G_{k+1}^{nco}A_{k}-E_{k-1}^{T}G_{k}^{nco}E_{k-1}&= Q_{r}(k)^{T}C_{k}^{T}C_{k}Q_{r}(k)\\ G_{k}^{nco}&=Q_{l}(k-1)^{T}G_{k}^{nco}Q_{l}(k-1), \end{aligned} $$

    respectively, where \(G_{K}^{ncr}=G_{0}^{ncr}\), \(G_{K}^{nco}=G_{0}^{nco}\)and\(Q_{l}(-1)=Q_{l}(K-1)\).

Numerical solution of the PGDPLEs has been considered in [6]. The method proposed there extends the periodic Schur method [5, 29, 30] and the generalized Schur-Hammarling method [23] developed for periodic standard and projected generalized Lyapunov equations, respectively. This method is based on an initial reduction of the periodic matrix pairs \(\{E_{k},A_{k}\}_{k=0}^{K-1}\) to the generalized periodic Schur form [11, 32] and solving the resulting generalized periodic Sylvester and Lyapunov equations of (quasi)-triangular structure using the recursive blocked algorithms [9]. Due to the computational complexity, the periodic generalized Schur-Hammarling method is restricted to problems of small and medium size. In [12, 27, 29], iterative methods based on Smith iterations [22] have been developed for periodic standard Lyapunov equations and also for projected generalized Lyapunov equations. These methods can also be extended to periodic projected Lyapunov equations by using the lifted representation of these equations. Therefore, we will discuss these lifted representations in the following subsection.

3.2 Lifted Representation of Periodic Lyapunov Equations

It is known that the Gramians of standard periodic systems satisfy the lifted form of the periodic Lyapunov equations and the solutions of these equations are diagonal matrices [12, 29].

The following theorem describes the block structures of the solutions of periodic Lyapunov equations in lifted form and their relations to the corresponding solutions of PGDPLEs in Theorem 1.

Theorem 2

Consider the periodic discrete-time descriptor system (11.1) and its lifted representation (11.4), where the periodic matrix pairs\(\{ E_{k},A_{k}\}_{k=0}^{K-1}\) are pd-stable. The causal and noncausal reachability Gramians\({{\fancyscript{G}}^{cr}}\) and\({{\fancyscript{G}}^{ncr}}\) satisfy the lifted projected Lyapunov equations

$$ \begin{aligned} {{\fancyscript{A}}}{{\fancyscript{G}}}^{cr}{{\fancyscript{A}}}^{T}-{{\fancyscript{E}}}{{\fancyscript{G}}}^{cr}{{\fancyscript{E}}}^{T}= -{{\fancyscript{P}}}_{l}{{\fancyscript{B}}}{{\fancyscript{B}}}^{T}{{\fancyscript{P}}}_l^T, \quad {{\fancyscript{G}}}^{cr}={{\fancyscript{P}}}_{r}{{\fancyscript{G}}}^{cr}{{\fancyscript{P}}}_r^T, \\ {{\fancyscript{A}}}{{\fancyscript{G}}}^{ncr}{{\fancyscript{A}}}^{T}-{{\fancyscript{E}}}{{\fancyscript{G}}}^{ncr}{{\fancyscript{E}}}^{T}= {{\fancyscript{Q}}}_{l}{{\fancyscript{B}}}{{\fancyscript{B}}}^{T}{{\fancyscript{Q}}}_l^T, \quad {{\fancyscript{G}}}^{ncr}={{\fancyscript{Q}}}_r{{\fancyscript{G}}}^{ncr}{{\fancyscript{Q}}}_r^T, \end{aligned} $$
(11.10)

respectively, where\({\fancyscript{E}}\), \({\fancyscript{A}}\) and\({\fancyscript{B}}\) are as in (11.5) and

$$ {{\fancyscript{G}}}^{cr}={\hbox{diag}}(G_{1}^{cr},\ldots, G_{K-1}^{cr}, G_{0}^{cr}),\quad {{\fancyscript{G}}}^{ncr}={\hbox{diag}}(G_{1}^{ncr},\ldots, G_{K-1}^{ncr}, G_{0}^{ncr}), $$
$$ \begin{aligned} {{\fancyscript{P}}}_{l}&={\text{diag}}(P_l(0), P_l(1), \ldots, P_l(K-1)), \quad {{\fancyscript{Q}}}_{l}=I-{{\fancyscript{P}}}_{l}, \\ {{\fancyscript{P}}}_{r}&={\text{diag}}(P_r(1), \ldots, P_r(K-1), P_r(0)), \quad {{\fancyscript{Q}}}_{r}=I-{{\fancyscript{P}}}_{r}. \end{aligned} $$
(11.11)

Proof

We will only sketch the proof due to space limitation of the paper. Using the block structure of matrix coefficients, a straightforward computation shows that the projected Lyapunov Equation (11.10) is equivalent to the periodic projected Lyapunov Equation (11.9). Since the periodic matrix pairs \(\{ E_{k},A_{k}\}_{k=0}^{K-1}\) are pd-stable, the pencil \({z {\fancyscript{E}}- {\fancyscript{A}}}\) is regular and all its eigenvalues lie inside the unit circle. Then (11.10) has a unique solution [24]. The proof for \({{\fancyscript{G}}^{ncr}}\) can be treated similarly.□

For the observability Gramians, the situation becomes a bit more complex. The reason is that we do not want to destroy the block diagonal structure of the lifted solutions and we would like to use the lifted solution to find a balanced realization of the original system.

Theorem 3

Consider the periodic discrete-time descriptor system (11.1) and its lifted representation (11.4). The causal and noncausal observability Gramians\({{\fancyscript{G}}^{co}}\)and\({{\fancyscript{G}}^{nco}}\)satisfy the lifted projected Lyapunov equations

$$ \begin{aligned} {\fancyscript{A}}^{T} {\fancyscript{G}}^{co} {\fancyscript{A}} - {\fancyscript{E}}^{T} {\fancyscript{G}}^{co} {\fancyscript{E}} = -{\fancyscript{P}}_r^T\hat{{\fancyscript{C}}}^{T}\hat{{\fancyscript{C}}}{\fancyscript{P}}_r, \quad {\fancyscript{G}}^{co}={\fancyscript{P}}_l^T{\fancyscript{G}}^{co}{\fancyscript{P}}_l, \\ {\fancyscript{A}}^{T} {\fancyscript{G}}^{nco} {\fancyscript{A}} - {\fancyscript{E}}^{T} {\fancyscript{G}}^{nco} {\fancyscript{E}} ={\fancyscript{Q}}_r^T\hat{{\fancyscript{C}}}^{T}\hat{{\fancyscript{C}}}{\fancyscript{Q}}_r, \quad {\fancyscript{G}}^{nco}={\fancyscript{Q}}_l^T{\fancyscript{G}}^{nco}{\fancyscript{Q}}_l, \end{aligned} $$

respectively, where \({\fancyscript{E}}\) and \({\fancyscript{A}}\) are as in (11.5), the projectors\({\fancyscript{P}_l}\), \({\fancyscript{P}_r}\), \({\fancyscript{Q}_l }\) and\({\fancyscript{Q}_r}\) are as in (11.11), \({\hat{\fancyscript{C}}={\hbox{diag}}(C_1, \ldots, C_{K-1}, C_0)}\) and

$$ {\fancyscript{G}}^{co}={\hbox{diag}}(G_{1}^{co}, \ldots, G_{K-1}^{co}, G_{0}^{co}),\quad {\fancyscript{G}}^{nco}={\hbox{diag}}( G_{1}^{nco}, \ldots, G_{K-1}^{nco}, G_{0}^{nco}). $$

Proof

The proof is analogous to the previous proof of Theorem 2.□

3.3 Hankel Singular Values

Let the periodic matrix pairs \(\{ E_{k},A_{k}\}_{k=0}^{K-1} \) be pd-stable. Then the causal and noncausal matrices \(M_{k}^{c}=G_{k}^{cr}E_{k-1}^{T}G_{k}^{co}E_{k-1}\) and \(M_{k}^{nc}=G_{k}^{ncr}A_{k}^{T}G_{k+1}^{nco}A_{k}\), \(k=0,1, \ldots, K-1\), have real and nonnegative eigenvalues. These eigenvalues are used to define the causal and noncausal Hankel singular values of system (11.1).

Definition 4

Let the set of periodic matrix pairs \(\{ E_{k},A_{k}\}_{k=0}^{K-1} \) be pd-stable. For \(k=0,1, \ldots, K-1\), the square roots of the largest n f k eigenvalues of the matrix M c k , denoted by σk,j, are called the causal Hankel singular values and the square roots of the largest n ∞ k eigenvalues of M nc k , denoted by θk,j, are called the noncausal Hankel singular values of the periodic descriptor system (11.1).

Since the causal and noncausal reachability and observability Gramians are symmetric and positive semidefinite, there exist the Cholesky factorizations

$$ G_{k}^{cr}=R_{k}R_{k}^{T},\quad G_{k}^{co}=L_{k}^{T}L_{k},\quad G_{k}^{ncr}=\check{R}_{k}\check{R}_{k}^{T},\quad G_{k}^{nco}=\check{L}_{k}^{T}\check{L}_{k}, $$
(11.12)

Simple calculations show that \(\sigma_{k,j} =\zeta_{j}(L_{k}E_{k-1}R_{k})\) and \(\theta_{k,j} =\zeta_{j}(\check{L}_{k+1}A_{k}\check{R}_{k})\), where \(\zeta_{j}(.)\) denotes the singular values of the corresponding matrices.

4 Balanced Truncation Model Reduction

In this section, we present a generalization of a balanced truncation model reduction method to periodic descriptor systems. For a balanced system, the reachability and observability Gramians are both equal to a diagonal matrix [15, 25]. Balanced realizations for periodic descriptor system have been considered in [6].

Definition 5

A realization \((E_{k},A_{k},B_{k},C_{k})\) of a periodic descriptor system (11.1) is called balanced if

$$ G_{k}^{cr}=G_{k}^{co}=\left[\begin{array}{ll} \Upsigma_{k}& 0\\ 0 & 0 \end{array} \right],\quad G_{k}^{ncr}=G_{k+1}^{nco}=\left[\begin{array}{ll} 0 & 0\\ 0 & \Uptheta_{k} \end{array} \right], $$

where \(\Upsigma_{k}=\hbox{diag}(\sigma_{k,1},\ldots,\sigma_{k,n_k^f})\) and \(\Uptheta_{k}={\hbox{diag}}(\theta_{k,1},\ldots,\theta_{k,n_k^\infty})\), \(k=0, 1, \ldots, K-1\).

Consider the Cholesky factorizations (11.12) of the reachability and observability Gramians and let

$$ L_{k}E_{k-1}R_{k}=U_{k}\Upsigma_{k}V_{k}^{T},\quad \check{L}_{k+1}A_{k}\check{R}_{k}=\check{U}_{k}\Uptheta_{k}\check{V}_{k}^{T} $$
(11.13)

be the singular value decompositions of the matrices \(L_{k}E_{k-1}R_{k}\) and \(\check{L}_{k+1}A_{k}\check{R}_{k}\) for \(k=0, 1, \ldots, K-1\). Here \(U_{k}, V_{k}, \check{U}_{k}, \check{V}_{k}\) are orthogonal, and \(\Upsigma_{k}\) and \(\Uptheta_{k}\) are diagonal. If a realization \((E_{k},A_{k},B_{k},C_{k})\) with pd-stable matrix pairs \(\{ E_{k},A_{k}\}_{k=0}^{K-1}\) is minimal, i.e., \(\Upsigma_{k}\) and \(\Uptheta_{k}\) are nonsingular, then there exist nonsingular periodic matrices

$$ S_{k}=[L_{k+1}^{T}U_{k+1}\Upsigma_{k+1}^{-1/2}, \enskip \check{L}_{k+1}^{T}\check{U}_{k}\Uptheta_{k}^{-1/2}], \quad T_{k}=[R_{k}V_{k}\Upsigma_{k}^{-1/2}, \enskip\check{R}_{k}\check{V}_{k}\Uptheta_{k}^{-1/2}], $$

such that the transformed realization \((S_{k}^{T}E_{k}T_{k+1}, S_{k}^{T}A_{k}T_{k}, S_{k}^{T}B_{k}, C_{k}T_{k})\) is balanced [6]. Note that as in the case of standard state space systems, the balancing transformation matrices for periodic discrete-time descriptor system (11.1) are not unique.

Model reduction via balanced truncation is discussed very widely for standard discrete-time periodic systems [12, 31] and also for continuous-time descriptor systems [14, 25]. For a balanced system, truncation of states related to the small causal Hankel singular values does not change system properties essentially. Unfortunately, we can not do the same for the noncausal Hankel singular values. If we truncate the states that correspond to the small non-zero noncausal Hankel singular values, then the pencil for the reduced-order system may get finite eigenvalues outside the unit circle that will lead to additional errors in the system approximation.

Assume that the periodic matrix pairs \(\{ E_{k},A_{k}\}_{k=0}^{K-1} \) are pd-stable. Consider the Cholesky factorizations in (11.12). Let

$$ L_{k} E_{k-1} R_{k} = [U_{k,1}, U_{k,2}] \left[\begin{array}{ll} \Upsigma_{k,1}& \\ & \Upsigma_{k,2} \end{array}\right] [V_{k,1}, V_{k,2}]^{T},\quad \check{L}_{k+1} A_{k}\check{R}_{k} = \check{U}_{k}\Uptheta_{k}\check{V}_{k}^{T}, $$

be singular value decompositions of \(L_{k} E_{k-1} R_{k}\) and \(\check{L}_{k+1} A_{k}\check{R}_{k}\), where

$$ \Upsigma_{k,1}={\hbox{diag}}(\sigma_{k,1}, \ldots, \sigma_{k,r_k^f}), \quad \Upsigma_{k,2}={\hbox{diag}}(\sigma_{k,r_{k+1}^f},\ldots, \sigma_{n_k^{f}}), $$

with \(\sigma_{k,1}\geq \cdots \geq \sigma_{k,r_k^{f}}> \sigma_{k,r_{k+1}^f} \geq \ldots \geq \sigma_{k,n_k^{f}}>0\), and \(\Uptheta_{k}={\hbox{diag}}(\theta_{k,1}, \ldots, \theta_{k,r_k^\infty})\) is nonsingular for \(k=0, 1, \ldots, K-1\). Then the reduced-order system can be computed as

$$ \tilde E_{k}= S_{k,r}^{T}E_{k}T_{k+1,r},\quad \tilde A_{k}= S_{k,r}^{T}A_{k}T_{k,r},\quad \tilde B_{k}= S_{k,r}^{T}B_{k},\quad \tilde C_{k}=C_{k}T_{k,r}, $$
(11.14)

where

$$ \begin{aligned} S_{k,r}& = & [L_{k+1}^{T}U_{k+1,1}\Upsigma_{k+1,1}^{-1/2},\check{L}_{k+1}^{T}\check{U}_{k}\Uptheta_{k}^{-1/2}]\in{\mathbb{R}}^{\mu_{k+1},r_{k+1}},\\ T_{k,r} & = & [R_{k}V_{k,1}\Upsigma_{k,1}^{-1/2},\quad \check{R}_{k}\check{V}_{k}\Uptheta_{k}^{-1/2}]\in {\mathbb{R}}^{n_k,r_k}, \end{aligned} $$

with \(r_k = r_k^{f}+r_k^{\infty}\). Let \({\tilde{\fancyscript{H}}(z)}\) be the transfer function of the reduced-order lifted system formed from the reduced-order subsystems in (11.14). Then we have the following \({\mathbb{H}_\infty}\)-norm error bound

$$ \Vert {\fancyscript{H}}-\tilde{{\fancyscript{H}}}\Vert _{{\mathbb{H}}_\infty} = \sup_{\omega\in[0,2\pi]}\Vert{\fancyscript{H}}(e^{i\omega})-\tilde{{\fancyscript{H}}}(e^{i\omega})\Vert_2 \leq 2\, \sum_{k=0}^{K-1}{\hbox{trace}}(\Upsigma_{k,2}), $$
(11.15)

where \(\Vert.\Vert_2\) denotes the matrix spectral norm and \(\Upsigma_{k,2}\) contains the truncated causal Hankel singular values. This error bound can be obtained similarly to the standard state space case [13, 31].

5 Example

We consider a periodic discrete-time descriptor system with \(\mu_k=n_k=10,\) n = 10, m = 2, \(p_k=3\), and period K = 3 as presented in [6, Example 1]. The periodic matrix pairs \(\{ E_{k},A_{k}\}_{k=0}^{K-1} \) are pd-stable with n f k  = 8 and n ∞ k  = 2 for k = 0, 1, 2. The norms of the computed solutions of the periodic Lyapunov equations and the corresponding residuals, e.g.,

$$ \rho_{k}^{cr}=\Vert A_{k}G_{k}^{cr}A_{k}^{T}-E_{k}G_{k+1}^{cr}E_{k}^{T}+P_{l}(k)B_{k}B_{k}^{T}P_{l}(k)^{T}\Vert_{2}, $$

are shown in Tables 11.1 and 11.2.

The original lifted system has order n = 30. Figure 11.1a shows the causal Hankel singular values of the different subsystems for k = 0, 1, 2. We see that they decay fast, and, hence system (11.1) can be well approximated by a reduced-order model. We have 24 causal Hankel singular values for the original lifted system and the remaining 6 are noncausal Hankel singular values which are positive. We approximate system (11.1) to the tolerance 10−2 by truncating the states corresponding to the smallest 7 causal Hankel singular values.

Table 11.1 Norms and relative residuals for the reachability Gramians
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Table 11.2 Norms and relative residuals for the observability Gramians
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Fig. 11.1
figure 1

a Causal Hankel singular values of different subsystems, b finite eigenvalues of the original and the reduced-order lifted systems

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Figure 11.1b shows the finite eigenvalues of the original and reduced-order lifted systems. We observe that stability is preserved for the reduced-order system. In Fig. 11.2a, we present the norms of the frequency responses \({\fancyscript{H}(e^{i\omega})}\) and \({\tilde{\fancyscript{H}}(e^{i\omega})}\) of the original and reduced-order lifted systems for a frequency range [0, 2π]. We observe nice match of the system norms.

Fig. 11.2
figure 2

a The frequency responses of the original and the reduced-order lifted systems; b absolute error and error bound

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In Fig. 11.2b, we display the absolute error \({\Vert \fancyscript{H}(e^{i\omega})-\tilde{\fancyscript{H}}(e^{i\omega})\Vert _{2}}\) and the error bound (11.15). One can see that the absolute error is smaller than the error bound.

6 Conclusion

In this paper, we have considered the reachability and observability Gramians as well as Hankel singular values for periodic discrete-time descriptor systems. For such systems, a balanced truncation model reduction method has been presented. The proposed method delivers a reduced-order model that preserves the regularity and stability properties of the original system. A computable global error bound for the approximate system is also available.