8.1 Introduction

In this chapter, we discuss the complete solution of the linear nonhomogeneous matrix differential equations (LNHMDEs). What is meant by the complete solution is the full solution taking into account the initial conditions of the state as well as the initial conditions of the input. The essence of the new results presented here with respect to the existing results [13, 17, 102] is that the obtained solution displays the full solution components including the impulse response and the slow response created, not only by the initial conditions of the state, but also by the initial condition of the input. Special attention is drawn to the initial conditions of the system input. Compared to the known results [13, 17, 102], in which the initial conditions of the system input are usually assumed to be zero, our approach firstly considers this issue explicitly.

The LNHMDEs or linear nonhomogeneous regular polynomial matrix descriptions (PMDs) are described by

$\begin{array}{l}\hfill A(\rho ){\beta }_{NH}(t)=B(\rho )\mu (t),t\ge 0,\end{array}$

where ρ := d/dt is the differential operator, $A(\rho )=A_{q_1}{\rho }^{q_1}+A_{q_1-1}{\rho }^{q_1-1}+\cdots +A_1\rho +A_0\in R{[\rho ]}^{r\times r}$, rank_R(ρ)A(ρ) = r, A_i ∈ R^r×r, i = 0, 1, 2, …, q₁, q₁ ≥ 1, B(ρ) = B_lρ^l + B_l−1ρ^l−1 + ⋯ + B₁ρ + B₀ ∈ R[ρ]^r×m, B_j ∈ R^r×m, j = 0, 1, 2, …, l, l ≥ 0, ${\beta }_{NH}(t):[0,+\infty )\to R^r$ is the pseudo-state of the LNHMDE, $u(t):[0,+\infty )\to R^m$ is an i times piecewise continuously differentiable function called the input of the LNHMDE. Their homogeneous cases

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

are called the linear homogeneous matrix differential equations (LHMDEs) or homogeneous PMDs. Both regular generalized state space systems (GSSSs), which are described by

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

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

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

are special cases of the above LNHMDEs (4.1). There have been many discussions about GSSSs (see, e.g., [7–11, 103, 104]). In [12], the impulsive solution to the LHMDEs was presented in a closed form. For both the LNHMDEs and LHMDEs, Vardulakis [13] developed their solutions under the assumption that both the initial conditions of the state and the input are zero. However, as we will see later, in some real cases, the initial conditions of the state might result from a random disturbance entering the system, and a feedback controller is called for. Since the precise value of the initial conditions of the state is unpredictable and the control is likely to depend on those initial conditions of the state, so this assumption is somewhat stronger than necessary.

In this chapter, we will present a solution to Eq. (8.1) that displays the impulse response and the slow response created not only by the initial conditions of β_NH(t), but also by the initial condition of u(t). Also reformulations to the solution of the LNHMDEs in terms of the regular derivatives of u(t) are given. By defining the slow state (smooth state) and the fast state (impulsive state) of them, it is shown that the system behaviors of the LNHMDEs can be decomposed into the slow response (smooth response) and the fast response (impulsive response) completely. This approach is applied conveniently to discuss the impulse free initial conditions of Eq. (8.1).

So far there have been many discussions about the impulse free condition either of the GSSSs (see, e.g., [14, 15]) or of the LNHMDEs and the LNHMDEs (see, e.g., [13, 16]). However, the common concern in the known results is the impulse created by the appropriate initial conditions of the state alone. For LNHMDEs, a different analysis concerning this issue is carried out that considers both the impulse created by the initial conditions of u(t) and that created by the initial conditions of β_NH(t).

8.2 Preliminary results

Any rational matrix A(s) is equivalent [87] at $s=\infty$ to its Smith-McMillan form, having the form:

$\begin{array}{l}\hfill S_{A(s)}^{\infty }(s)=\text{block diag}\left[s^{q_1},s^{q_2},\dots ,s^{q_k},\frac{1}{s^{{\widehat{q}}_{k+1}}},\dots ,\frac{1}{s^{{\widehat{q}}_r}},0_{p-r,m-r}\right],\end{array}$

where 1 ≤ k ≤ r and $q_1\ge q_2\ge \cdots \ge q_k\ge 0,{\widehat{q}}_r\ge {\widehat{q}}_{r-1}\ge \cdots \ge {\widehat{q}}_{k+1}\ge 0.$ Any polynomial matrix $A(s)=A_0+A_{1}s+\cdots +A_{q_1}{s}^{q_1}\in R^{r\times r}[s]$ with ${\mathrm{rank}}_{\mathfrak{R}[s]}A(s)=r$ can be transformed by unimodular transformation to its Smith form

$\begin{array}{l}\hfill S_{A(s)}^C(s)=\text{diag}[1,1,\dots ,1,f_k(s),f_{k+1}(s),\dots ,f_r(s)],\end{array}$

where 1 ≤ k ≤ r, f_j(s)/f_j+1(s), j = k, k + 1, …, r − 1. We denote $x_{j,0}^i,x_{j,1}^i,\dots$, $x_{j,{\sigma }_{ij}-1}^i\in R^r(x_{j,0}^i\ne 0)$, i ∈ I, j = k, k + 1, …, r as the Jordan Chain of lengths σ_ij corresponding to the eigenvalue λ_i of A(s) and consider matrices

$\begin{array}{l}\hfill C_i=[x_{k,0}^i,\dots ,x_{k,{\sigma }_{i,k}-1}^i]\cdots [x_{r,0}^i,\dots ,x_{r,{\sigma }_{i,r}-1}^i]\in R^{r\times m_i},\end{array}$

where $m_i={\sum }_{j=k}^r{\sigma }_{ij},i=1,2,\dots$ and

$\begin{array}{l}\hfill J_i=\text{block diag}[J_{i,k},J_{i,k+1},\dots ,J_{i,r}]\in R^{m_i\times m_i},i=1,2,\dots ,\end{array}$

where $J_{i,j}\in R^{{\sigma }_{ij}\times {\sigma }_{ij}}$, i ∈ I, j = k, k + 1, …, r, is the Jordan block matrix.

8.3 A solution for the LNHMDEs

To the LNHMDEs

$\begin{array}{l}\hfill A(\rho ){\beta }_{NH}(t)=B(\rho )u(t),t\ge 0.\end{array}$

Assuming that the initial values of u(t) and its (l − 1)-derivatives at t = 0 are u(0), u⁽¹⁾(0)⋯u^(l−1)(0) and that the initial values of β_NH(t) and its (q₁ − 1)-derivatives at t = 0 are ${\beta }_{NH}(0),{\beta }_{NH}^{(1)}(0),\dots ,{\beta }_{NH}^{(q_1-1)}(0).$ Taking the Laplace transformation of Eq. (8.7), we obtain

$\begin{array}{l}\hfill A(s){\widehat{\beta }}_{NH}(s)-{\widehat{\alpha }}_{\beta }(s)=B(s)û(s)-{\widehat{\alpha }}_u(s),\end{array}$

where ${\widehat{\beta }}_{NH}(s):={\int }_0^{+\infty }{\beta }_{NH}(t)e^{-st}dt,û(s):={\int }_0^{+\infty }u(t)e^{-st}dt$. Because of the Laplace-transformation rule

$\begin{array}{l}\hfill L\left\{\frac{d^i}{d{t}^i}{\beta }_{NH}(t)\right\}=s^i{\widehat{\beta }}_{NH}(s)-s^{i-1}{\beta }_{NH}(0)-\cdots -s{\beta }_{NH}^{i-2}(0)-{\beta }_{NH}^{i-1}(0);i=0,1,\dots \end{array}$

$\begin{array}{l}\hfill L\left\{\frac{d^j}{d{t}^j}u(t)\right\}=s^iû(s)-s^{j-1}u(0)-\cdots -s{u}^{j-2}(0)-u^{j-1}(0);j=0,1,\dots \end{array}$

${\widehat{\alpha }}_{\beta }(s),{\widehat{\alpha }}_u(s)$ can be written [107] as follows

$\begin{array}{l}\hfill {\widehat{\alpha }}_{\beta }(s)=\left[\begin{array}{l}{s}^{q_1-1}{I}_r,s^{q_1-2}{I}_r,\dots ,s{I}_r,I_r\end{array}\right]\left[\begin{array}{llll}{A}_{q_1}& 0& \cdots & 0\\ A_{q_1-1}& A_{q_1}& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ A_1& A_2& \cdots & A_{q_1}\end{array}\right]\left[\begin{array}{l}{\beta }_{NH}(0)\\ {\beta }_{NH}^{(1)}(0)\\ ⋮\\ {\beta }_{NH}^{(q_1-1)}(0)\end{array}\right]\end{array}$

$\begin{array}{l}\hfill {\widehat{\alpha }}_u(s)=[s^{l-1}{I}_r,s^{l-2}{I}_r,\dots ,s{I}_r,I_r]\left[\begin{array}{llll}{B}_l& 0& \cdots & 0\\ B_{l-1}& B_l& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ B_1& B_2& \cdots & B_l\end{array}\right]\left[\begin{array}{l}u(0)\\ u^{(1)}(0)\\ ⋮\\ u^{(l-1)}(0)\end{array}\right].\end{array}$

We denote A⁻¹(s) = H_pol(s) + H_sp(s), where H_pol(s) is the polynomial part of A⁻¹(s) and H_sp(s) is the strictly proper part of A⁻¹(s). To find a minimal realization of H_sp(s)(C, J, B) and a minimal realization of $H_{pol}(s)(C_{\infty },J_{\infty },B_{\infty })$ such that (C, J) and $(C_{\infty },J_{\infty })$ are a finite Jordan pair and an infinite Jordan pair of A(s), respectively, and

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

We obtain from [13]

where

Similarly,

where

From [13] we also obtain

$\begin{array}{l}\hfill A^{-1}(s)B(s)û(s)=[C,C_{\infty }]\mathrm{\Psi }\mathrm{\Phi }\left[\begin{array}{l}{I}_m\\ s{I}_m\\ ⋮\\ s^{{\widehat{q}}_r+1}{I}_m\end{array}\right]û(s)+C{[s{I}_n-J]}^{-1}\mathrm{\Omega }û(s),\end{array}$

where

$\begin{array}{l}\hfill \mathrm{\Omega }:=J^{l}B{B}_l+J^{l-1}B{B}_{l-1}+\cdots +JB{B}_l+B{B}_0\in R^{n\times m}.\end{array}$

Now Eq. (8.8) can be written as

$\begin{array}{l}\hfill {\widehat{\beta }}_{NH}(s)=A^{-1}(s){\widehat{\alpha }}_{\beta }(s)+A^{-1}(s)B(s)u(s)-A^{-1}(s){\widehat{\alpha }}_u(s).\end{array}$

By taking the inverse Laplace transformation of the above, we finally obtain the following theorem that represents the required complete solution of the LNHMDE (8.7) created by the nonzero initial conditions on both the control u(t) and β_NH(t).

8.4 The smooth and impulsive solution components and impulsive free initial conditions: $C_{\infty }$ is of full row rank

The main aims of this section are to analyze the fast and slow components in the solution of the LNHMDEs and then to characterize the impulsive behavior of the system. To this end, one is suggested to reformulate the solution that is given by Theorem 8.3.1 in terms of the regular derivative of u(t). This is the subject of the following result.

Consider the following notations:

$\begin{array}{l}\hfill x_s(0):=x_{s\beta }(0)-x_{su}(0),J_{\infty }{x}_f(0):=J_{\infty }{x}_{f\beta }(0)-J_{\beta }{x}_{fu}(0),\end{array}$

$\begin{array}{l}\hfill \widehat{U(t)}:=\left[\begin{array}{l}u(t)\\ u^{[1]}(t)\\ u^{[2]}(t)\\ ⋮\\ u^{[{\widehat{q}}_r+l]}(t)\end{array}\right]\in R^{({\widehat{q}}_r+l+1)m},\widehat{\delta (t)}:=\left[\begin{array}{l}\delta (t)\\ {\delta }^{(1)}(t)\\ ⋮\\ {\delta }^{{\widehat{q}}_r+l+1}(t)\end{array}\right]\in R^{({\widehat{q}}_r+l)},\end{array}$

$\begin{array}{l}\hfill \mathcal{U}(0):=\left[\begin{array}{lllll}0& 0& \cdots & 0& 0\\ u(0)& 0& \cdots & 0& 0\\ u^{[1]}(0)& u(0)& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ u^{[{\widehat{q}}_r+l-1]}(0)& u^{[{\widehat{q}}_r+l-2]}(0)& \cdots & u^{[1]}(0)& u(0)\end{array}\right]\in R^{({\widehat{q}}_r+l+1)m\times ({\widehat{q}}_r+l)},\end{array}$

$\begin{array}{l}\hfill \widehat{J_{\infty }}{x}_f(0):=[J_{\infty }{x}_f(0),J_{\infty }^2{x}_f(0),\dots ,J_{\infty }^{{\widehat{q}}_r}{x}_f(0)]\in R^{\mu \times {\widehat{q}}_r},\end{array}$

$\begin{array}{l}\hfill \widehat{{Ĵ}_{\infty }{x}_f(0)}:=[\widehat{J_{\infty }}{x}_f(0),0_{\mu \times l}]\in R^{\mu \times ({\widehat{q}}_r+l)},\end{array}$

$\begin{array}{l}\hfill {\mathrm{\Psi }}_1:=[J^{l-1}B,J^{l-2}B,\dots ,B]\in R^{n\times Ir},{\mathrm{\Psi }}_2:=[B_{\infty },J_{\infty }{B}_{\infty },\dots ,J_{\infty }^{{\widehat{q}}_r}{B}_{\infty }]\in R^{\mu \times ({\widehat{q}}_r+1)r},\end{array}$

$\begin{array}{l}\hfill {\mathrm{\Phi }}_1(B):=\left[\begin{array}{llllllll}{B}_l& 0& \cdots & 0& 0& 0& \cdots & 0\\ B_{l-1}& B_l& \cdots & 0& 0& 0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ B_1& B_2& \cdots & B_l& 0& 0& \cdots & 0\end{array}\right]\in R^{Ir\times ({\widehat{q}}_r+l+1)m},\end{array}$

$\begin{array}{l}\hfill {\mathrm{\Phi }}_2(B):=\left[\begin{array}{llllllll}{B}_0& B_1& \cdots & B_{l-1}& B_l& 0& \cdots & 0\\ 0& B_0& \cdots & B_{l-2}& B_{l-1}& B_l& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& \ddots & \ddots & ⋮\\ 0& 0& \cdots & B_0& B_1& \cdots & B_{l-1}& B_l\end{array}\right]\in R^{({\widehat{q}}_r+1)r\times ({\widehat{q}}_r+l+1)m}.\end{array}$

Proof

From Eqs. (10.9), (8.12), we know

from Eq. (8.6), we have

$\begin{array}{l}\hfill \left[\begin{array}{l}u(t)\\ u^{(1)}(t)\\ ⋮\\ u^{({\widehat{q}}_r+l)}(t)\end{array}\right]=\widehat{U(t)}+\mathcal{U}(0)\widehat{\delta (t)},\end{array}$

  (8.15)

if $C_{\infty }$ has full row rank, we have $C_{\infty }{C}_{\infty }^{(1)}=I_{r\times r}$. Substituting the above into Eq. (10.11), we obtain

$\begin{array}{ll}\hfill {\beta }_{NH}(t)& =C{e}^{Jt}{x}_s(0)+C{\int }_0^t{e}^{J(t-\tau )}\mathrm{\Omega }u(\tau )d\tau \hfill \\ \hfill & -C_{\infty }\widehat{J_{\infty }}{x}_f(0)\widehat{\delta (t)}+(C{\mathrm{\Psi }}_1{\mathrm{\Phi }}_1(B)+C_{\infty }{\mathrm{\Psi }}_2{\mathrm{\Phi }}_2(B))(\widehat{U(t)}+\mathcal{U}(0)\widehat{\delta (t)})\hfill \\ \hfill & =C{e}^{Jt}{x}_s(0)+C{\int }_0^t{e}^{J(t-\tau )}\mathrm{\Omega }u(\tau )d\tau \hfill \\ \hfill & +C_{\infty }({\mathrm{\Psi }}_2{\mathrm{\Phi }}_2(B)+C_{\infty }^{(1)}C{\mathrm{\Psi }}_1{\mathrm{\Phi }}_1(B))\widehat{U(t)}\hfill \\ \hfill & +C_{\infty }[({\mathrm{\Psi }}_2{\mathrm{\Phi }}_2(B)+C_{\infty }^{(1)}C{\mathrm{\Psi }}_1{\mathrm{\Phi }}_1(B))\mathcal{U}(0)-\widehat{\hat{J_{\infty }}{x}_f(0)}]\widehat{\delta (t)}\hfill \\ \hfill & =[C,C_{\infty }]\left[\begin{array}{l}{\beta }_{s(NH)}(t)\\ {\beta }_{f(NH)}(t)\end{array}\right].\hfill \end{array}$

Now we shall generalize the concept of the state to the LNHMDEs as [13] has done for the LHMDEs and the GSSS.

For the LHMDEs

$\begin{array}{l}\hfill A(\rho ){\beta }_H(t)=0,\end{array}$

its solution is

$\begin{array}{l}\hfill {\beta }_H(t)=[C,C_{\infty }]\left[\begin{array}{l}{x}_s(t)\\ x_f(t)\end{array}\right],\end{array}$

where

$\begin{array}{ll}\hfill x_s(t)& =e^{Jt}{x}_s(0),\hfill \\ \hfill x_f(t)& =-\sum _{i=1}^{{\widehat{q}}_r}\delta {(t)}^{(i-1)}{J}_{\infty }^i{x}_f(0).\hfill \end{array}$

It is clear that the impulsive behavior of β_H(t) at t = 0 only depends on x_f(0), which is only related to the initial conditions of β_H(t) and its derivatives at t = 0. Thus the set of impulse free initial conditions for LHMDE A(ρ)β_H(t) = 0, t ≥ 0 is [13–16]

$\begin{array}{l}\hfill H_{I(H)}=\left\{x(0)=\left[\begin{array}{l}{x}_s(0)\\ x_f(0)\end{array}\right]:x_s(0)\in R^n,x_f(0)\in Ker{J}_{\infty }\right\}=R^n\oplus Ker{J}_{\infty }.\end{array}$

However, for the LNHMDEs, this issue becomes much more complicated, for in this case not only the initial conditions of β_NH(t), but also the initial conditions of u(t) influence the solution structure. The following result provides an answer to this difficulty.

8.5 The smooth and impulsive solution components and impulsive free initial conditions: $C_{\infty }$ is not of full row rank

This section is devoted to analyze the solution components and impulsive free initial conditions of the LNHMDEs without assuming that $C_{\infty }$ is of full row rank.

If $C_{\infty }$ is not of full row rank, say its row rank is r₁ < r, then there exists a transformation called “row compression,” which reduces $C_{\infty }$ to the form

$\begin{array}{l}\hfill P{C}_{\infty }=\left[\frac{C_{\infty }^{*}}{0}\right],\end{array}$

where $C_{\infty }^{*}$is of full row rank r₁, it thus satisfies

$\begin{array}{l}\hfill C_{\infty }^{*}{(C_{\infty }^{*})}^{(1)}=I_{r_1}.\end{array}$

If one denotes

$\begin{array}{l}\hfill P{\beta }_{NH}(t):=\left[\frac{{\beta }_{NH1}(t)}{{\beta }_{NH2}(t)}\right],PC:=\left[\frac{C_1}{C_2}\right],\end{array}$

where

$\begin{array}{l}\hfill {\beta }_{NH1}(t)\in R^{r_1\times l},{\beta }_{NH2}(t)\in R^{(r-r_1)\times 1}\end{array}$

$\begin{array}{l}\hfill C_1\in R^{r_1\times n},C_2\in R^{(r-r_1)\times n}\end{array}$

then one has the following result, which designates the fast and the slow components of the system solution structure.