9.1 Introduction

Regular polynomial matrix descriptions (PMDs) are described by

$\begin{array}{l}\hfill \begin{array}{c}{\mathrm{\Sigma }}^{\prime }:\left\{\begin{array}{l}T(\rho )\beta (t)=E(\rho )u(t)\\ y(t)=C(\rho )\beta (t)+D(\rho )u(t),t\ge 0\end{array}\right.\end{array}\end{array}$

where ρ := d/dt is the differential operator,

$\begin{array}{l}\hfill T(\rho )\in R{\left[\rho \right]}^{l\times l},{\mathrm{rank}}_{\mathfrak{R}[\rho ]}T(\rho )=l,\end{array}$

$\begin{array}{l}\hfill E\left(\rho \right)\in R{\left[\rho \right]}^{l\times m^{\prime }},C\left(\rho \right)\in R{\left[\rho \right]}^{p\times l},\end{array}$

$\begin{array}{l}\hfill D(\rho )\in R{\left[\rho \right]}^{p\times m^{\prime }},\end{array}$

$\beta \left(t\right):[0,+\infty )\to R^l$ is the pseudo-state of ${\mathrm{\Sigma }}^{\prime },u(t):\left[0\right.,\left.+\infty \right)\to R^{m^{\prime }}$ is the control input and y(t) the output of Σ′. Σ′can be written in the form:

In order to investigate the solution of the above PMD, we thus focus our interests on the following form of equation

$\begin{array}{l}\hfill \mathrm{\Sigma }:A(\rho )\xi (t)=Bu(t),t\ge 0,\end{array}$

where $A\left(\rho \right)\in R{\left[\rho \right]}^{r\times r}$, with ${\mathrm{rank}}_{\mathfrak{R}\left[\rho \right]}A\left(\rho \right)=r,\xi (t)\in R^r,B\in R^{r\times m},u(t)\in R^m$. The homogeneous case of Σ is

$\begin{array}{l}\hfill {\mathrm{\Sigma }}_0:A(\rho )\xi (t)=0,t\ge 0.\end{array}$

Both regular generalized state space systems which are described by

$\begin{array}{l}\hfill \mathrm{E}\dot{x}=Ax(t)+Bu(t),\end{array}$

where E is a singular matrix, rank(ρE − A) = r, and the (regular) state space systems which are described by

$\begin{array}{l}\hfill \dot{x}=Ax(t)+Bu(t)\end{array}$

are special cases of the regular PMDs (Eq. 9.1).

The solution of the above PMDs can be proposed on the basis of any resolvent decomposition of A(s) given by

$\begin{array}{l}\hfill A^{-1}(s)=C{(s{I}_n-J)}^{-1}Z+C_{\infty }{(s{J}_{\infty }-I_{\epsilon })}^{-1}{Z}_{\infty },\end{array}$

where (C, J) is the finite Jordan pair of A(s), and $(C_{\infty },J_{\infty })$ is the infinite Jordan pair [13, 17] of A(s). Such resolvent decompositions are not unique, but play an important role in formulating the solution of regular PMDs. The difficulties in the problem of obtaining the solution of the PMD are specific to the particular resolvent decomposition used. From a computation point of view, when the matrices in Eq. (9.3) are of minimal dimensions, it is obviously easiest to obtain the inverse matrix of A(s).

There is fundamental interest (notably [13, 17, 102]) in the solution of the regular PMD and its homogeneous case. For Eq. (9.1) Gohberg et al. [17] proposed a particular resolvent decomposition, and the solution of it was formulated according to this decomposition. However, the impulsive property of the system at t = 0 was not properly considered in this work. One advantage of this resolvent Decomposition, however, is that it is constructive, due to the fact that the matrices $Z,Z_{\infty }$ in Eq. (9.3) are formulated in terms of the finite and infinite Jordan pairs. On the other hand in this resolvent decomposition, there appears to be certain redundant information, in that the dimensions of the infinite Jordan pair are much larger than is usually necessary, this in turn brings some inconvenience in computing the inverse matrix of A(s).

In fact some of the infinite elementary divisors [18] that correspond to the infinite poles of A(s) actually contribute nothing to the solution, as will be seen subsequently. Thus a resolvent decomposition can be proposed in a simpler form that has the further advantage of giving a more precise insight into the system structure as well as bringing some convenience in actual computation. Vardulakis [13] obtained a general solution of Eq. (9.1) under the assumption that the initial conditions of ξ(t) are zero. The main idea of this approach is to find minimal realizations of the strictly proper part and the polynomial part of A⁻¹(s) and then to obtain the required resolvent decomposition. Although this procedure does give good insight into the system structure, the realization approach is not so straightforward by itself and it is consequently more difficult to apply in actual computation. Furthermore, in this procedure, no explicit formula for Z and $Z_{\infty }$ is available.

In [102] the solution and the impulsive behavior of autonomous linear multivariable systems whose pseudo-state obeys the linear matrix differential equation (9.2) are examined. The impulsive solution part of the general nonhomogeneous PMD (Eq. 9.1) is, however, not displayed in the solution of Eq. (9.2) given by Vardulakis et al. [102]. Apparently, differences arise between the solution of Gohberg et al. [17] and that of Vardulakis [13] due to the fact that these two solutions are expressed through two different resolvent decompositions. Although it is found that the redundant information contained in the solution of Gohberg et al. [17] decouples in the meaning described in this book, an overly large resolvent decomposition definitely brings some inconvenience to actual computation.

The main purpose of this chapter is to present a resolvent decomposition that is a refinement of both results obtained by Gohberg et al. [17] and Vardulakis [13]. It is formulated in terms of the notions of the finite Jordan pairs, infinite Jordan pairs, and the generalized infinite Jordan pairs that were defined by Gohberg et al. [17] and Vardulakis [13]. We clarify the issue of the infinite Jordan pair noted by Gohberg et al. [17] and Vardulakis [13]. This refined resolvent decomposition captures the essential feature of the system structure, and the redundant information that is included in the resolvent decomposition of Gohberg et al. [17] is deleted through certain transformations. The resulting resolvent decomposition thus inherits the advantages of both the results of Gohberg et al. [17] and that of Vardulakis [13].

Based on this proposed resolvent decomposition, a complete solution of a PMD follows that reflects the detailed structure of the zero state response and the zero input response of the system. The complete impulsive properties of the system are also displayed in our solution. Such impulsive properties are not properly displayed in the solution of Gohberg et al. [17]. Although for the homogeneous case it is considered in Vardulakis [13], Vardulakis et al. [102], such a complete treatment of the impulsive properties of the system for the general nonhomogeneous regular PMD is not available in Vardulakis [13] and Vardulakis et al. [102].

Compared with the complete solution of regular PMDs given in Chapter 8, where the solution is proposed based on any resolvent decomposition, the resolvent decomposition obtained in this chapter is of minimal dimension, the solution constructed subsequently is specific to this refined resolvent decomposition.

9.2 Infinite Jordan pairs

In this section, we analyze two definitions of infinite Jordan pairs. The first was given by Gohberg et al. [17] and the other was given by Vardulakis [13]. The essential differences between them are elucidated and these discussions enable us to establish a refined resolvent decomposition.

Let $A(s)=A_0+A_{1}s+\cdots +A_{q_1}{s}^{q_1}\in R^{r\times r}[s]$, ${\mathrm{rank}}_{\mathfrak{R}[s]}A(s)=r$. $S_{A(S)}^{\infty }$ denotes the Smith-McMillan form of A(s) at $s=\infty$,

$\begin{array}{l}\hfill S_{A(s)}^{\infty }(s)=\mathrm{diag}\left[\underset{k}{\underbrace{s^{q_1},s^{q_2},\dots ,s^{q_k}}},\underset{r-h-k}{\underbrace{1,\dots ,1}},\underset{h}{\underbrace{1/s^{{\widehat{q}}_{r-h+1}},\dots ,1/s^{{\widehat{q}}_r}}}\right],\end{array}$

where

$\begin{array}{l}\hfill q_1\ge q_2\ge \cdots \ge q_k>0\end{array}$

are the orders of the infinite poles at $s=\infty$ of A(s), and

$\begin{array}{l}\hfill {\hat{q}}_r\ge {\hat{q}}_{r-1}\ge \cdots \ge {\hat{q}}_{r-h+1}>0\end{array}$

are the orders of the infinite zeros at $s=\infty$ of A(s). The dual polynomial of A(s) is defined as

$\begin{array}{l}\hfill D_{A(s)}(w):=w^{q_1}A\left(\frac{1}{w}\right)=A_0{w}^{q_1}+A_1{w}^{q_1-1}+\cdots +A_{q_1}.\end{array}$

The local Smith form of D_A(s)(w) at w = 0 is

$\begin{array}{ll}\hfill S_{D_{A(s)}}^0(w)& =w^{q_1}{S}_{A(s)}^{\infty }\left(\frac{1}{w}\right)\hfill \\ \hfill & =\mathrm{diag}\left[1,w^{q_1-q_2},\dots ,w^{q_1-q_k},w^{q_1},\dots ,w^{q_1},w^{q_1+{\hat{q}}_{r-h+1}},\dots ,w^{q_1+{\hat{q}}_r}\right].\hfill \end{array}$

It is seen that the polynomial matrices have in general three kinds of IEDs, or equivalently D_A(s)(w) has three kinds of finite zero at w = 0. The first kind of IEDs with orders q₁ − q_j, j = 2, 3, …, k correspond to the poles of A(s) at $s=\infty$. The second kind of IEDs comprise those corresponding to the zeros of A(s) at $s=\infty$ with orders $q_1+{\widehat{q}}_j$, j = r − h + 1, …, r. The third kind with orders q₁ is not dynamically important [18].

Let ${Ũ}_L(w)\in R{[s]}^{r\times r}$, ${Ũ}_R(w)\in R{[s]}^{r\times r}$ be unimodular matrices reducing D_A(s)(w) to its local Smith form $S_{D_{A(s)}}^0(w)$, i.e.,

$\begin{array}{l}\hfill {\tilde{U}}_L(w)D_{A(s)}(w){\tilde{U}}_R(w)=S_{D_{A(s)}}^0(w).\end{array}$

Let u_j(w) ∈ R[w]^r×1, v_j(w) ∈ R[w]^r×1, $j\doteq 1,2,\dots ,r$ be the columns of ${Ũ}_R(w)$ and ${Ũ}_L^{-1}(w)$, then

$\begin{array}{l}\hfill D_{A(s)}(w)u_j(w)=v_j(w)w^{\tau (j)},\end{array}$

where

$\begin{array}{l}\hfill \tau (j):=\left\{\begin{array}{l}{q}_1-q_j,j=1,2,\dots ,k\\ q_1+{\hat{q}}_j,j=k+1,\dots ,r\end{array}\right.\end{array}$

Let

$\begin{array}{l}\hfill {\beta }_{jq}:=\frac{1}{q!}{u}_j^{(q)}(0),j=2,3,\dots ,r,q=0,1,2,\dots ,\tau (j)-1,\end{array}$

then for j = 2, 3, …, r, the vectors

$\begin{array}{l}\hfill {\beta }_{j0},{\beta }_{j1},\dots ,{\beta }_{j(\tau (j)-1)}\in R^r\end{array}$

form a Jordan chain corresponding to the zero w = 0 of the IEDs w^τ(j). Now we can compare the two different definitions of the infinite Jordan pair.

Definition 9.2.1

(Gohberg et al. [17])

The generalized infinite Jordan pair $(C_{\infty }^{\prime },J_{\infty }^{\prime })$ of A(s) are defined as

$\begin{array}{l}\hfill C_{\infty }^{\prime }:=\left[{C^{\prime }}_2,{C^{\prime }}_3,\dots ,{C^{\prime }}_k,{C^{\prime }}_{k+1},\dots ,{C^{\prime }}_r\right]\in R^{r\times v},\end{array}$

$\begin{array}{l}\hfill J_{\infty }^{\prime }:=\text{block diag}\left[{J^{\prime }}_2,\dots ,{J^{\prime }}_r\right]\in R^{v\times v},\end{array}$

where

$\begin{array}{l}\hfill C_j^{\prime }=\left[{\beta }_{j0},{\beta }_{j1},\dots ,{\beta }_{j(\tau (j)-1)}\right]\in R^{r\times \tau (j)},j=2,3,\dots ,r,\end{array}$

We observe that

1. $J_{\infty }^{\prime }$ is nilpotent with an index of nil-potency $q_1+{\widehat{q}}_r$, because the largest Jordan block is $(q_1+{\widehat{q}}_r)\times (q_1+{\widehat{q}}_r)$.

2. $v=r{q}_1-{\sum }_{i=1}^k{q}_i$ (the total number of the infinite poles of A(s))
$+{\sum }_{i=1}^r{\widehat{q}}_{k+i}$ (the total number of the infinite zeros of A(s)).

Definition 9.2.2

(Vardulakis [13])

The infinite Jordan pair $(C_{\infty },J_{\infty })$ of A(s) is defined as follows

$\begin{array}{l}\hfill C_{\infty }:=\left[C_{k+1},\dots ,C_r\right]\in R^{r\times \mu },\end{array}$

$\begin{array}{l}\hfill J_{\infty }:=\text{block diag}\left[J_{k+1},\dots ,J_r\right]\in R^{\mu \times \mu },\end{array}$

where

$\begin{array}{l}\hfill C_j=\left[{\beta }_{j0},{\beta }_{j1},\dots ,{\beta }_{j{\hat{q}}_j}\right]\in R^{r\times ({\hat{q}}_j+1)},j=k+1,\dots ,r,\end{array}$

We observe that

1. $J_{\infty }$ is nilpotent with an index of nil-potency ${\widehat{q}}_r+1$, because the largest Jordan block is $({\widehat{q}}_r+1)\times ({\widehat{q}}_r+1)$.

2. $\mu ={\sum }_{j=k+1}^r({\widehat{q}}_j+1)={\sum }_{j=k+1}^r{\widehat{q}}_j+r-k$.

The above two interpretations of infinite Jordan pairs display an essential difference. For example, the roles that the infinite poles and the infinite zeros play in the system behaviors are considered in totally different manners. It is obvious that v ≥ μ, and the dimensions of the generalized infinite Jordan pair of Definition 9.2.1 are unnecessarily larger than that of the infinite Jordan pair of Definition 9.2.2 due to some redundant information being included. This can be seen clearly from the following example.

Example 9.2.1

Consider the polynomial matrix

its Smith-McMillan form is

Thus A(s) has two infinite poles at degrees 1 and 3 and an infinite zero at degree 4. The dual polynomial matrix to A(s) is

and

is the local Smith form of D_A(S)(w) at w = 0, where

$\begin{array}{l}\hfill :=\left[u_1(w),u_2(w),u_3(w)\right].\end{array}$

So we get three kinds of infinite elementary divisor:

• w² corresponding to the infinite pole of A(s),

• w⁷ corresponding to the infinite zero of A(s) and the IED 1 = w⁰.

From

$\begin{array}{ll}\hfill {\beta }_{2q}& =\frac{1}{q!}{u}_2^{(q)}(0),q=0,1,\hfill \\ \hfill {\beta }_{3q}& =\frac{1}{q!}{u}_3^{(q)}(0),q=0,1,2,3,4,5,6,\hfill \end{array}$

we have

According to Definition 9.2.1, the generalized infinite Jordan pair in the sense of Gohberg et al. [17] is $(C_{\infty }^{\prime },J_{\infty }^{\prime })$:

While the infinite Jordan pair in the sense of Vardulakis [13] is $(C_{\infty },J_{\infty })$ with:

In the above example it is noted that the dimensions of the generalized infinite Jordan pair, which are 3 × 9 and 9 × 9, are unnecessarily larger than that of the infinite Jordan pair, which are 3 × 5 and 5 × 5. Such a tendency will become more evident as the number of the infinite poles of A(s), the dimension of A(s) and the degree of A(s) are increased. However, the following observation arising from the above two definitions of infinite Jordan pairs suggests the possibility of deleting the redundant information that is contained in the generalized infinite Jordan pair.

Theorem 9.2.1

If $(C_{\infty }^{\prime },J_{\infty }^{\prime })$ is a generalized infinite Jordan pair, there exists an elementary matrix P such that

$\begin{array}{l}\hfill C_{\infty }^{\prime }P=\left[C_{\infty },C_0\right],\end{array}$

where C₀ ∈ R^r×(v−μ), J′∈ R^μ×(v−μ), J₀ ∈ R^{(v−μ)×(v−μ)}, 0 ∈ R^(v−μ)×μ and $(C_{\infty },J_{\infty })$ is an infinite Jordan pair. Furthermore, J₀ is nilpotent with an index of nil-potency q₁ − 1.

Proof

Note that $C_{\infty }$ is a submatrix of $C_{\infty }^{\prime }$. By a series of interchanging of two columns in $C_{\infty }^{\prime }$, $C_{\infty }^{\prime }$ can be brought to the form $[C_{\infty },C_0]$. We denote this elementary matrix as P, then $P^{-1}=P^T,C_{\infty }^{\prime }P=[C_{\infty },C_0]$, where C₀ ∈ R^r×(v−μ). Similarly, by interchanging the corresponding columns and the corresponding rows of $J_{\infty }^{\prime },J_{\infty }^{\prime }$ can be brought to a block matrix, i.e.,

where

$\begin{array}{l}\hfill J^{\prime }={\left[a\left({\sigma }_1,{\sigma }_2\right)\right]}_{\left({\sigma }_1,{\sigma }_2\right)}\in R^{\mu \times \left(\nu -\mu \right)},\end{array}$

$\begin{array}{l}\hfill a\left({\sigma }_1,{\sigma }_2\right)=\left\{\begin{array}{ll}1,& {\sigma }_1=\sum _{\sigma =k+1}^j\left({\hat{q}}_{\sigma }+1\right),{\sigma }_2=\sum _{j=2}^k\tau \left(j\right)+\left({\hat{q}}_j+2\right),\\ & j=k+1,\dots ,r\\ 0,& \mathrm{other}\\ \end{array}\right.\end{array}$

$\begin{array}{l}\hfill J_0=\text{block diag}\left[{J_2}^{\prime },\dots ,{J_k}^{\prime },J_{k+1}^{\prime },\dots ,{J_r}^{\prime }\right],\end{array}$

$\begin{array}{l}\hfill {J_j}^{\prime }(j=2,3,\dots ,k)\text{are Jordan matrix of}\left(q_1-q_j\right)\times \left(q_1-q_j\right)\left(j=2,\dots ,k\right),\end{array}$

$\begin{array}{l}\hfill {J_i}^{\prime }(i=k+1,\dots ,r)\text{are Jordan matrix of}\left(q_1-1\right)\times \left(q_1-1\right).\end{array}$

It can easily be seen that J₀ is nilpotent with an index of nil-potency q₁ − 1.

Example 9.2.2

Recall Example 9.2.1, we have

It is noted that here J₀ is nilpotent with index of nil-potency 2.

The key point in here is to find the above elementary matrix P to delete the redundant information. Based on the information carried by the Smith-McMillan form at $s=\infty$ of A(s), P results from a series of elementary operations from an identity matrix, which is very easy to obtain. Once the redundant information is deleted from the generalized infinite Jordan pair, the generalized infinite Jordan pair and the elementary matrix P are not involved in computation in the solution procedure.

9.3 The solution of regular PMDs

We consider now the homogeneous regular PMD:

$\begin{array}{l}\hfill A(\rho )\xi (t)=0,t\ge 0.\end{array}$

Let $S_{A(s)}^{\infty }(s)$ be the Smith-McMillan form at $s=\infty$ of $A(s)=L_{-}[A(\rho )]=A_0+A_{1}s+\cdots +A_{q_1}{s}^{q_1}$:

$\begin{array}{l}\hfill S_{A(s)}^{\infty }(s)=\mathrm{diag}\left[s^{q_1},s^{q_2},\dots ,s^{q_k},\frac{1}{s^{{\hat{q}}_k+1}},\dots ,\frac{1}{s^{{\hat{q}}_r}}\right]\in R^{r\times r}(s)\end{array}$

and if A(s) has at least one zero at $s=\infty$ then the Laurent expansion of A⁻¹(s) can be written as follows

$\begin{array}{l}\hfill \begin{array}{cc}{A}^{-1}(s)& =H_{{\hat{q}}_r}{s}^{{\hat{q}}_r}+H_{{\hat{q}}_r-1}{s}^{{\hat{q}}_r-1}+\cdots +H_{1}s+H_0+H_{-1}{s}^{-1}+H_{-2}{s}^{-2}+\cdots \\ & =H_{pol}(s)+H_{sp}(s),\end{array}\end{array}$

where $H_{pol}(s)\in R^{r\times r}\left[s\right]$ is the polynomial part of A⁻¹(s) and H_sp(s) ∈ R^r×r(s) is the strictly proper part of A⁻¹(s). Let a resolvent decomposition of A(s) be given by

$\begin{array}{l}\hfill A^{-1}(s)=C_{\infty }{(s{J}_{\infty }-I_{\mu })}^{-1}{Z}_2+C{(s{I}_n-J)}^{-1}{Z}_1,\end{array}$

where $n:=\mathrm{deg}det(A(s))$. It should be noted that (C, J, Z₁) is a realization of H_sp(s), and $(C_{\infty },J_{\infty },Z_2)$ is a realization of H_pol(s). Considering the Laplace transformation of Eq. (9.4), we obtain

$\begin{array}{l}\hfill \widehat{\xi }(s)=A^{-1}(s)\widehat{\alpha }(s)\in R^{r\times 1},\end{array}$

where $\widehat{\alpha }(s)\in R^{r\times 1}[s]$ is the initial condition vector associated with the initial values of ξ(t) and its (q₁ − 1)-derivatives at t = 0 −, i.e., $\xi (0-),{\xi }^{(1)}(0-),\dots ,{\xi }^{(q_1-1)}(0-)$ given by Fossard [108], Pugh [107]

We obtain

where the vector

Now by taking the inverse L₋ Laplace transformation of Eq. (9.7), we have the following result.

The next result constructs a resolvent decomposition for a regular polynomial matrix, which follows from Gohberg et al. [17].

Lemma 9.3.2

Let $A(s)=A_0+A_{1}s+\cdots +A_{q_1}{s}^{q_1}\in R^{r\times r}[s]$, ${\mathrm{rank}}_{\mathfrak{R}[s]}A(s)=r$. If (C, J) is the finite Jordan pair of A(s), and $(C_{\infty }^{\prime },J_{\infty }^{\prime })$ is the generalized infinite Jordan pair of A(s) in the sense of Gohberg et al. [17], $n:=\mathrm{deg}det(A(s))$. Put

and

(9.9)

then

(9.10)

An interesting observation from Eq. (9.9) is the following.

Proposition 9.3.1

If we partition

$\begin{array}{l}\hfill Z^{\prime }:=\left[\begin{array}{l}\hfill Z_1\hfill \\ \hfill Z_2^{\prime }\hfill \end{array}\right]:=\left[\begin{array}{l}\hfill Z_1\hfill \\ \hfill Z_{21}^{\prime }\hfill \\ \hfill Z_{22}^{\prime }\hfill \end{array}\right],\end{array}$

where $Z_1\in R^{n\times r},Z_{21}^{\prime }\in R^{\mu \times r},Z_{22}^{\prime }\in R^{(v-\mu )\times r}$, then

where Z₂ ∈ R^μ×r.

Proof

Let

$\begin{array}{l}\hfill \left[\begin{array}{l}\hfill {\mathrm{\Pi }}_1\hfill \\ \hfill {\mathrm{\Pi }}_2\hfill \\ \hfill {\mathrm{\Pi }}_3\hfill \end{array}\right]:={\left[\begin{array}{l}\hfill S\hfill \\ \hfill V\hfill \end{array}\right]}^{-1}{[0,\dots ,0,I]}^T,\end{array}$

where Π₁ ∈ R^n×1, Π₂ ∈ R^μ×1, Π₃ ∈ R^(v−μ)×1. From Theorem 9.2.1, there exists an elementary matrix P ∈ R^v×v, such that

so

where $Q=J_{\infty }^{q_1-2}{J}^{\prime }+J_{\infty }^{q_1-3}{J}^{\prime }{J}_0+\cdots +J^{\prime }{J}_0^{q_1-2}$.

Noticing J₀ is nilpotent with an index of nil-potency q₁ − 1, i.e., $J_0^{q_1-1}=0,$ it follows that:

The above observation is interesting, not least for the way in which the redundant information contained in $Z^{\prime }$ is deleted. More importantly than this, however, is that such a mechanism of decoupling makes the computation of P attractive and facilitates the proposed refined resolvent decomposition.

The following result proposes a new approach to constructing a resolvent decomposition in terms of the finite Jordan pair, the infinite Jordan pair and the generalized infinite Jordan pair, which is fundamental for us to formulate the solution of the PMD. As one of our main results, we are ready to state it as follows.

Proof

This follows readily from Eq. (9.10) and Theorem 9.2.1 on noting that

For the nonhomogeneous regular PMD

$\begin{array}{l}\hfill A(\rho )\xi (t)=Bu(t),t\ge 0\end{array}$

the following theorem gives the complete solution.

Proof

From Lemma 9.3.1, one knows that

$\begin{array}{l}\hfill \xi (t)=C{e}^{Jt}{x}_s(0-)-C_{\infty }\sum _{i=1}^{\widehat{q_r}}\delta {(t)}^{(i-1)}{J}_{\infty }^{i-1}(J_{\infty }{x}_f(0-))\end{array}$

generates the general solution of the homogeneous PMD, so we only need to check that the formula (9.13) does indeed produce a solution of the nonhomogeneous PMD (Eq. 9.12). To this end, when t > 0, we have

$\begin{array}{ll}\hfill {\xi }^{(j)}(t)& =C{J}^j{e}^{Jt}{x}_s(0-)+\sum _{k=0}^{j-1}C{J}^{j-1-k}{Z}_{1}B{u}^{(k)}(t)\hfill \\ \hfill & +C{\int }_{0-}^t{J}^j{e}^{J(t-\tau )}{Z}_{1}Bu(\tau )d\tau -C_{\infty }\sum _{i=0}^{\widehat{q_r}}{J}_{\infty }^i{Z}_{2}B{u}^{(i+j)}(t),\hfill \end{array}$

so

$\begin{array}{ll}\hfill A(\rho )\xi (t)& =\sum _{j=0}^{q_1}{A}_j{\xi }^{(j)}(t)\hfill \\ \hfill & =\sum _{j=0}^{q_1}{A}_{j}C{J}^j{e}^{Jt}{x}_s(0-)+\sum _{j=0}^{q_1}\sum _{k=0}^{j-1}{A}_{j}C{J}^{j-1-k}{Z}_{1}B{u}^{(k)}(t)\hfill \\ \hfill & +{\int }_{0-}^t\left[\sum _{j=0}^{q_1}{A}_{j}C{J}^j\right]e^{J(t-\tau )}{Z}_{1}Bu(\tau )d\tau -\sum _{j=0}^{\widehat{q_r}}\sum _{j=0}^{q_1}{A}_j{C}_{\infty }{J}_{\infty }^i{Z}_{2}B{u}^{(i+j)}(t).\hfill \end{array}$

Using the property ${\sum }_{j=0}^{q_1}{A}_{j}C{J}^j=0$ and rearranging the terms, we have

$\begin{array}{ll}\hfill A(\rho )\xi (t)& =\sum _{j=0}^{q_1}{A}_j{\xi }^{(j)}(t)\hfill \\ \hfill & =\sum _{k=0}^{q_1+\widehat{q_r}}\left\{\sum _{j=k+1}^{q_1}{A}_{j}C{J}^{j-1-k}{Z}_1-\sum _{j=0}^k{A}_j{C}_{\infty }{J}_{\infty }^{k-j}{Z}_2\right\}B{u}^{(k)}(t).\hfill \end{array}$

Noticing

from

$\begin{array}{l}\hfill A(s)A^{-1}(s)=I,\end{array}$

it follows that

$\begin{array}{l}\hfill \sum _{j=k+1}^{q_1}{A}_{j}C{J}^{j-1-k}{Z}_1-\sum _{j=0}^k{A}_j{C}_{\infty }{J}_{\infty }^{k-j}{Z}_2=\left\{\begin{array}{ll}I,& k=0\\ 0,& k>0.\end{array}\right.\end{array}$

Hence

$\begin{array}{l}\hfill A(\rho )\xi (t)=Bu(t),\end{array}$

which finishes the proof.

Remark 9.3.4

According to Gohberg et al. [17], the solution of PMD (Eq. 9.12) is following (without the impulse solution part):

$\begin{array}{l}\hfill \xi (t)=C{e}^{Jt}{x}_s(0-)+C{\int }_{0-}^t{e}^{J(t-\tau )}{Z}_{1}Bu(\tau )d\tau -C_{\infty }^{\prime }\sum _{i=0}^{\widehat{q_r}+q_1-1}{(J_{\infty }^{\prime })}^i{Z}_2^{\prime }B{u}^{(i)}(t),\end{array}$

where ($C_{\infty }^{\prime },J_{\infty }^{\prime }$) is the infinite Jordan pair in the sense of Gohberg et al. [17]. $Z_1,Z_2^{\prime }$ are given by Eq. (9.9). By using Theorem 9.2.1, Proposition 9.3.1 and noticing that $J_{\infty }$ is nilpotent with an index of nil-potency $\widehat{q_r}+1$, one finds that

It is thus clearly seen that the redundant information included in the solution of Gohberg et al. [17] due to using the generalized infinite Jordan pairs does not appear in our solution any more. Also this helps to clarify the mechanism of decoupling in the solution of Gohberg et al. [17].

By comparing our solution with the solution of Gohberg et al. [17] and the solution of Vardulakis [13], it can be seen that, besides the fact that it displays the impulsive solution part, our solution can be carried out more efficiently in actual computation. The main reason for this is that the refined resolvent decomposition does not contain any redundant information, since the infinite Jordan pairs used in our solution are of minimal dimensions, and the matrices Z₁ and Z₂ are formulated explicitly. The refined resolvent decomposition facilitates computation of the inverse matrix of A(s) because the dimension of the matrices used are minimal. Rather than calculating the inverse matrix A⁻¹(s) with an overly large dimension, it is suggested first to calculate the easily available elementary matrix P to delete all redundant information and then to obtain the inverse matrix with the minimal dimension. Once the refined resolvent decomposition is obtained, the generalized infinite Jordan pair and the elementary matrix P are not necessary for the calculation of the solution of the regular PMD. This thus presents another advantage of this method, which is algorithmically attractive when applied in actual computation.

9.4 Algorithm and examples

As mentioned before, the refined resolvent decomposition plays a key role in our approach, the difficulty in obtaining the solution of the regular PMDs depends on the calculation of this refined resolvent decomposition. To help clarify the construction of this refined resolvent decomposition, for the implementation of Theorem 9.3.1, the following algorithm is now given, which is useful for symbolic computational packages Maple.