12.1 Introduction

In classical network theory, a circuit representation called the chain matrix [21] was widely used to deal with the cascade connection of circuits arising in analysis and synthesis problems. Based on this, Kimura [22] developed the chain-scattering representation (CSR), which was subsequently used to provide a unified framework for $H^{\infty }$ control theory. Kimura’s approach is, however, only available to the special cases where the matrices P₂₁ and P₁₂ (refer to Eq. 11.1) satisfy some assumptions of full rank. Recently, in [31] this approach was extended to the general case in which such conditions are essentially relaxed. From an input-output consistency point of view, the generalized chain-scattering representation (GCSR) and the dual generalized chain-scattering representation (DGCSR) emerge and are there successfully used to characterize the cascade structure property and the symmetry of general plants in a general setting.

Latterly behavioral theory (see, e.g., [26, 27]) has received broad acceptance as an approach for modeling dynamical systems. One of the main features of the behavioral approach is that it does not use the conventional input-output structure in describing systems. Instead, a mathematical model is used to represent the systems in which the collection of time trajectories of the relevant variables are viewed as the behavior of the dynamical systems. This approach has been shown [26, 28] to be powerful in system modeling and analysis. In contrast to this, the classical theories such as Kalman’s state-space description and Rosenbrock’s polynomial matrix description (PMD) take the input-output representation as their starting point. In many control contexts it has proven to be very convenient to adopt the classical input/state/output framework. It is often found that the system models, in many situations, can easily be formulated into the input/state/output models such as the state space descriptions and the PMDs. Based on such input/state/output representations, the action of the controller can usually be explained in a very natural manner and the control aims can usually be attained very effectively.

As far as the issue of system modeling is concerned the computer-aided procedure, i.e., the automated modeling technology, has been well developed as a practical approach over recent years. If the physical description of a system is known, the automated modeling approach can be applied to find a set of equation to describe the dynamical behavior of the given system. It is seen that, in many cases, such as in an electrical circuit or, more generally, in an interconnection of blocks, such a physical description is more conveniently specified through the frequency behavior of the system. Consequently in these cases, a more general theoretic problem arises, i.e., if the frequency behavior description of a system is known, what is the corresponding dynamical behavior? In other words, what is the input/output or input/state/output structure in the time domain that generates the given frequency domain description. It turns out that this question can be interpreted through the notion of realization of behavior, which we shall introduce in this chapter.

In fact, as we shall see later, realization of behavior, in many cases, amounts to introduction of latent variables in the time domain. From this point of view, realization of behavior can be understood as a converse procedure of the latent variable elimination theorem [27] in one way or another. It should be emphasized that realization of behavior also generalizes the notion of transfer function matrix realization in the classical control theory framework, as the behavior equation is a more general description than the transfer function matrix. As a special case, the behavior equations determine a transfer matrix of the system if they represent an input-output system [27, 29], i.e., when the matrices describing the system satisfy a full rank condition.

Recently in [30], a realization approach was suggested that reduces high-order linear differential equations to the first-order system representations by using the method of “linearization.” From the point of view of realization in a physical sense one is, however, forced to start from the system frequency behavior description into which system behavior is generally described rather than from the high-order linear differential equations in time domain. Also a constructive procedure for autoregressive (AR) equation realization has been introduced in [114].

One of the main aims of this chapter is to present a new notion of realization of behavior. Further to the results of [31], the input-output structure of the GCSRs and the DGCSRs are thus clarified by using this approach. Subsequently the corresponding autoregressive-moving-average (ARMA) representations are proposed and are proved to be realizations of behavior for any GCSR and for any DGCSR. Once these ARMA representations are proposed, one can further find the corresponding first-order system representations by using the method of [30] or other well-developed realization approaches such as Kailath [32].

These results are interesting in that they provide a good insight into the natural relationship between the (frequency) behavior of any GCSR and any DGCSR and the (dynamical) behavior of the corresponding ARMA representations. Since no numerical computation is involved in this approach, realization of behavior is particularly accessible for situations in which the coefficient are symbolic rather than numerical. Based on the realization of behavior, one can further find the corresponding first-order system representations by using the classical realization approaches [30, 32]. Also a constructive procedure for AR equation realization suggested by Vafiadis and Karcanias [114] can be used to obtain the required first-order system representations.

The following result can be easily proposed and can be easily proved by using the definition of 1-inverse of a polynomial matrix. It is presented here for later use.

12.2 Behavior realization

This section introduce the concept of realization of behavior. Recall that in the behavioral framework the (dynamical) behavioral equations of an ARMA representation [27] are

$\begin{array}{l}\hfill R_1(\rho )u_1(t)+R_2(\rho )y_1(t)=S(\rho )\xi (t),\end{array}$

where $w^{\prime }:={\left[{(y_1(t))}^T,{(u_1(t))}^T\right]}^T$ stands for the external variables representing the dynamical behavior of the underlying dynamical system, ξ(t) which are called latent variables corresponding to auxiliary variables resulting from the modeling procedure. R₁(ρ), R₂(ρ) and S(ρ) are polynomial matrices containing the differential operator ρ = d/dt. In order to distinguish them from the existing notation y, u of Eq. (11.1), the external variables are denoted by y₁ and u₁. When S(ρ) = 0, Eq. (12.1) is termed [27] an AR representation.

In the following approach we are interested in the external behavior of the system (12.1), where we choose the underlying function space to be $C^{\infty }:=C^{\infty }(\mathcal{R},\mathfrak{R})$, this function space consists of all the infinitely differentiable functions that are defined for all time $\mathcal{R}:=[0,+\infty )$ and take values in the real number field $\mathfrak{R}$. For brevity, we write $C_k^{\infty }:=C^{\infty }(\mathcal{R},{\mathfrak{R}}^k)$. Then the dynamical external behavior of Eq. (12.1) is given by

$\begin{array}{ll}\hfill {\mathfrak{B}}_d(R_1,R_2;S)& =\left\{\left[\begin{array}{l}{y}_1(t)\\ u_1(t)\end{array}\right]\in C_{m+p}^{\infty }\left|\exists \right.\xi (t)\in C_n^{\infty }\mathrm{so}\mathrm{that}(8.1)\mathrm{is}\mathrm{valid}\right\}\hfill \\ \hfill & =\left\{\left[\begin{array}{l}{y}_1(t)\\ u_1(t)\end{array}\right]\in C_{m+p}^{\infty }\left[R_2(\rho ),R_1(\rho )\right]\left[\begin{array}{l}{y}_1(t)\\ u_1(t)\end{array}\right]\in ImS(\rho )\right\}.\hfill \end{array}$

From the above, to avoid the trivial case that the external behavior is empty, i.e., to ensure that there exist latent variables, every pair (y₁(t), u₁(t)) in the external behavior must be consistent, that is

$\begin{array}{l}\hfill (I-S(\rho )S^{-}(\rho ))(R_1(\rho )u_1(t)+R_2(\rho )y_1(t))=0,\forall \left[\begin{array}{l}{y}_1(t)\\ u_1(t)\end{array}\right]\in {\mathfrak{B}}_d(R_1,R_2;S),\end{array}$

where {1}-inverse is arbitrary. It is immediately noted that, when S(ρ) is invertible or more specially when S(ρ) = I, the above condition is automatically satisfied.

In many real cases (for example, electrical circuits), however, system behavior is usually described (see Example 12.2.1) in the frequency domain as

$\begin{array}{l}\hfill A(s)u^{*}(s)+B(s)y^{*}(s)=C(s)\eta (s),\end{array}$

where $A(s)\in \mathfrak{R}{[s]}^{q\times p},B(s)\in \mathfrak{R}{[s]}^{q\times m}$, and $C(s)\in \mathfrak{R}{[s]}^{q\times n},$ as the following example suggests, are not polynomial but rational matrices. The vector-valued signals u*(s), y*(s), and η(s) live in the square (Lebesgue) integrable functional spaces ${\mathbf{L}}_2^p,{\mathbf{L}}_2^m$, and ${\mathbf{L}}_2^n$ respectively.

Example 12.2.1

Consider the system. The loop equations of Fig. 12.1 are written as

$\begin{array}{ll}\hfill \left(R+\frac{2}{sC}\right)I_1(s)-\frac{1}{sC}{I}_2(s)& =E_{in}(s)\hfill \end{array}$

  (12.4)

$\begin{array}{ll}\hfill -\frac{1}{sC}{I}_1(s)+\left(R+\frac{1}{sC}\right)I_2(s)& =0.\hfill \end{array}$

  (12.5)

In order to obtain the transfer function $Z_t(s)=\frac{E_0(s)}{E_{in}(s)}$, one needs to solve for I₂(s) to obtain

$\begin{array}{l}\hfill I_2(s)=\frac{E_{in}(s)Cs}{s^2{R}^2{C}^2+3sRC+1}\end{array}$

and then note that the output voltage is

$\begin{array}{l}\hfill E_0(s)=R{I}_2(s).\end{array}$

This suggests that I₁(s) can be regarded as latent variable, the original loop equations thus can be written into

$\begin{array}{l}\hfill \left[\begin{array}{l}1\\ 0\end{array}\right]E_{in}(s)+\left[\begin{array}{l}\frac{1}{sC}\\ R+\frac{1}{sC}\end{array}\right]I_2(s)=\left[\begin{array}{l}R+\frac{2}{sC}\\ \frac{1}{sC}\end{array}\right]I_1(s),\end{array}$

  (12.6)

which is just in the form of Eq. (12.3). The coefficient matrices of the external variables and the latent variables are seen to be rational but not necessarily polynomial.

As a special case when C(s) = 0 and B(s) is invertible, Eq. (12.3) determines a transfer function G(s) = −B⁻¹(s)A(s).

The frequency behavior of Eq. (12.3) is given by

$\begin{array}{ll}\hfill {\mathfrak{B}}_f(A,B;C)& =\left\{\left[\begin{array}{l}{y}^{*}(s)\\ u^{*}(s)\end{array}\right]\in {\mathbf{L}}_2^{m+p}\left|\exists \right.\eta (s)\in {\mathbf{L}}_2^n\mathrm{so}\mathrm{that}(8.3)\mathrm{is}\mathrm{valid}\right\}\hfill \\ \hfill & =\left\{\left[\begin{array}{l}{y}^{*}(s)\\ u^{*}(s)\end{array}\right]\in {\mathbf{L}}_2^{m+p}\left|\left[B(s),A(s)\right]\left[\begin{array}{l}{y}^{*}(s)\\ u^{*}(s)\end{array}\right]\in ImC(s)\right.\right\}.\hfill \end{array}$

  (12.7)

In here the matrix B(s) is not necessarily invertible. It will be seen later that this is the reason why the realization of behavior comes out to be a generalization of realization of transfer function. Instead of this condition, we only make a relaxed assumption that every u* in the behavior is consistent, i.e.,

$\begin{array}{l}\hfill (I-B(s)B^{-}(s))A(s)u^{*}(s)=0,\forall u^{*}(s)\in {\mathfrak{B}}_f(A,B),\end{array}$

where B⁻(s) is any {1}-inverse of B(s). Again it should be noted that, when B(s) is invertible or more specially when B(s) = I, the above condition is automatically satisfied. By denoting

$\begin{array}{l}\hfill \mathcal{L}({\mathcal{B}}_d)(R_1,R_2;S)=\left\{\left[\begin{array}{l}{ŷ}_1(s)\\ {û}_1(s)\end{array}\right]\left|\left[\begin{array}{l}{ŷ}_1(s)\\ {û}_1(s)\end{array}\right]\right.=\mathcal{L}\left(\left[\begin{array}{l}{y}_1(t)\\ u_1(t)\end{array}\right]\right),\right.\end{array}$

$\begin{array}{l}\hfill \left.\left[\begin{array}{l}{y}_1(t)\\ u_1(t)\end{array}\right]\in {\mathcal{B}}_d(R_1,R_2;S),{\rho }^i{y}_1(0)=0,{\rho }^i{u}_1(0)=0,{\rho }^i\xi (0)=0,i=0,1,\dots \right\},\end{array}$

where $\mathcal{L}\left(f(t)\right)$ denotes the Laplace transformation of f(t), the definition of realization of behavior follows.

When T(ρ) = ρE − A with E singular, the above PMD is termed a singular system, while when T(ρ) = ρI − A, the above description is known as the conventional state space system. It is clearly seen that in the above special cases, realization of behavior is equivalent to realization of transfer function in the classical sense.

12.3 Realization of behavior for GCSRs and DGCSRs

Before developing the realization of behavior for GCSRs, we will establish a realization of behavior for the general plant. Given the general plant described by

$\begin{array}{l}\hfill \left[\begin{array}{l}z(s)\\ y(s)\end{array}\right]=P(s)\left[\begin{array}{l}w(s)\\ u(s)\end{array}\right]=\left[\begin{array}{ll}{P}_{11}(s)& P_{12}(s)\\ P_{21}(s)& P_{22}(s)\end{array}\right]\left[\begin{array}{l}w(s)\\ u(s)\end{array}\right],\end{array}$

where P_ij(s) (i = 1, 2; j = 1, 2) are all rational matrices with dimensions m_i × k_j (i = 1, 2; j = 1, 2), consider the rational matrix $P(s)\in \mathfrak{R}{[s]}^{(m_1+m_2)\times (k_1+k_2)}$. It is well-known [4] that there always exist nonunique polynomial pairs $P_1(s)\in \mathfrak{R}{[s]}^{(m_1+m_2)\times (m_1+m_2)}$ and $P_2(s)\in \mathfrak{R}{[s]}^{(m_1+m_2)\times (k_1+k_2)}$ such that

$\begin{array}{l}\hfill P(s)=P_1^{-1}(s)P_2(s).\end{array}$

It should be noted in here P₁(s) and P₂(s) do not need to be coprime. In this way, the following result is obtained.

Recall now Theorem 11.2.2. If the input-output pair (u(s), y(s)) is consistent about w to the plant P, then the GCSR is represented by

$\begin{array}{ll}\hfill \left[\begin{array}{l}z(s)\\ w(s)\end{array}\right]& =GCHAIN(P;P_{21}^{-})\left[\begin{array}{l}u(s)\\ y(s)\\ h(s)\end{array}\right]\hfill \\ \hfill & =GCHAI{N}^{*}(P;P_{21}^{-})\left[\begin{array}{l}u(s)\\ y(s)\end{array}\right]+\mathrm{\Delta }GCHAIN(P;P_{21}^{-})h(s),\hfill \end{array}$

where we denote the GCSR matrix

The above GCSR gives rise to the frequency behavior

$\begin{array}{ll}\hfill & {\mathcal{B}}_f(I,-GCHAIN(P;P_{21}^{-}))\hfill \\ \hfill & :=\left\{\left[\begin{array}{l}{y}^{*}(s)\\ u^{*}(s)\end{array}\right]:=\left[\frac{\begin{array}{l}z(s)\\ w(s)\end{array}}{\begin{array}{l}u(s)\\ y(s)\end{array}}\right]\left|\begin{array}{l}\\ y^{*}(s)=GCHAI{N}^{*}(P;P_{21}^{-})u^{*}(s)\\ \\ +\mathrm{\Delta }GCHAIN(P;P_{21}^{-})h(s),\\ h(s)\mathrm{is}\mathrm{arbitrary},\\ (u(s),y(s))\mathrm{is}\mathrm{consistent}\mathrm{to}P\end{array}\right.\right\}.\hfill \end{array}$

It can be proved that every GCSR gives rise to the same frequency behavior, in other words, the frequency behavior of GCSRs is independent of the particular {1}-inverse. The following theorem establishes this observation.

From Lemma 11.1.2, there exists a matrix K(s) such that

$\begin{array}{l}\hfill P_{21}^{-}(s)=P_{21}^g(s)+K(s)-P_{21}^g(s)P_{21}(s)K(s)P_{21}(s)P_{21}^g(s).\end{array}$

Also the input-output pair (u(s), y(s)), if being consistent to the plant P, must satisfy (Theorem 11.2.1)

$\begin{array}{l}\hfill (I-P_{21}(s)P_{21}^g(s))[-P_{22}(s),I]\left[\begin{array}{l}u(s)\\ y(s)\end{array}\right]=0.\end{array}$

This follows that

$\begin{array}{l}\hfill y(s)-P_{22}(s)u(s)=P_{21}(s)P_{21}^g(s)(y(s)-P_{22}(s)u(s)).\end{array}$

Given any

$\begin{array}{l}\hfill \left[\begin{array}{l}{y}^{*}(s)\\ u^{*}(s)\end{array}\right]\in {\mathcal{B}}_f(I,-GCHAIN(P;P_{21}^{-})),\end{array}$

there should be a rational vector h₁(s) such that

$\begin{array}{ll}\hfill \left[\begin{array}{l}z(s)\\ w(s)\end{array}\right]& =y^{*}(s)=GCHAI{N}^{*}(P;P_{21}^{-})u^{*}(s)+\mathrm{\Delta }GCHAIN(P;P_{21}^{-})h_1(s)\hfill \\ \hfill & =\left[\begin{array}{ll}{P}_{12}(s)-P_{11}(s)P_{21}^{-}(s)P_{22}(s)& P_{11}(s)P_{21}^{-}(s)\\ -P_{21}^{-}(s)P_{22}(s)& P_{21}^{-}(s)\end{array}\right]\left[\begin{array}{l}u(s)\\ y(s)\end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{l}{P}_{11}(s)(I-P_{21}^{-}(s)P_{21}(s))\\ I-P_{21}^{-}(s)P_{21}(s)\end{array}\right]h_1(s).\hfill \end{array}$

By substituting Eqs. (12.16), (12.17) into Eq. (12.18), one yields

$\begin{array}{ll}\hfill z(s)& =(P_{12}(s)-P_{11}(s)P_{21}^{-}(s)P_{22}(s))u(s)+P_{11}(s)P_{21}^{-}(s)y(s)\hfill \\ \hfill & +P_{11}(s)(I-P_{21}^{-}(s)P_{21}(s))h_1(s)\hfill \\ \hfill & =[P_{12}(s)-P_{11}(s)(P_{21}^g(s)+K(s)-P_{21}^g(s)P_{21}(s)K(s)P_{21}(s)P_{21}^g(s))P_{22}(s)]u(s)\hfill \\ \hfill & +P_{11}(s)(P_{21}^g(s)+K(s)-P_{21}^g(s)P_{21}(s)K(s)P_{21}(s)P_{21}^g(s))y(s)\hfill \\ \hfill & +P_{11}(s)[I-(P_{21}^g(s)+K(s)-P_{21}^g(s)P_{21}(s)K(s)P_{21}(s)P_{21}^g(s))P_{21}(s)]h_1(s)\hfill \\ \hfill & =(P_{12}(s)-P_{11}(s)P_{21}^g(s)P_{22}(s))u(s)+P_{11}(s)P_{21}^g(s)y(s)\hfill \\ \hfill & +P_{11}(s)(I-P_{21}^g(s)P_{21}(s))[K(s)(y(s)-P_{22}(s)u(s))\hfill \\ \hfill & +(I-K(s)P_{21}(s))h_1(s)];\hfill \end{array}$

$\begin{array}{ll}\hfill w(s)& =-P_{21}^{-}(s)P_{22}(s)u(s)+P_{21}^{-}(s)y(s)+(I-P_{21}^{-}(s)P_{21}(s))h_1(s)\hfill \\ \hfill & =-(P_{21}^g(s)+K(s)-P_{21}^g(s)P_{21}(s)K(s)P_{21}(s)P_{21}^g(s))P_{22}(s)u(s)\hfill \\ \hfill & +(P_{21}^g(s)+K(s)-P_{21}^g(s)P_{21}(s)K(s)P_{21}(s)P_{21}^g(s))y(s)\hfill \\ \hfill & +[I-(P_{21}^g(s)+K(s)-P_{21}^g(s)P_{21}(s)K(s)P_{21}(s)P_{21}^g(s))P_{21}(s)]h_1(s)\hfill \\ \hfill & =-P_{21}^g(s)P_{22}(s)u(s)+P_{21}^g(s)y(s)\hfill \\ \hfill & +(I-P_{21}^g(s)P_{21}(s))[K(s)(y(s)-P_{22}(s)u(s))+(I-K(s)P_{21}(s))h_1(s)].\hfill \end{array}$

By letting

$\begin{array}{l}\hfill h_2(s):=K(s)(y(s)-P_{22}(s)u(s))+(I-K(s)P_{21}(s))h_1(s),\end{array}$

the above formulations about z(s) and w(s) can be written into the following matrix form

$\begin{array}{l}\hfill \left[\begin{array}{l}z(s)\\ w(s)\end{array}\right]=GCHAI{N}^{*}(P;P_{21}^g)u^{*}(s)+\mathrm{\Delta }GCHAIN(P;P_{21}^g)h_2(s),\end{array}$

which displays the fact that

$\begin{array}{l}\hfill \left[\begin{array}{l}{y}^{*}(s)\\ u^{*}(s)\end{array}\right]\in {\mathcal{B}}_f(I,-GCHAIN(P;P_{21}^g)).\end{array}$

Subsequently

$\begin{array}{l}\hfill {\mathcal{B}}_f(I,-GCHAIN(P;P_{21}^{-}))\subseteq {\mathcal{B}}_f(I,-GCHAIN(P;P_{21}^g)).\end{array}$

This finishes the proof.

By the virtue of the above theorem, the frequency behavior of any GCSR thus can be simply denoted by ${\mathcal{B}}_f(I,-GCHAIN(P)).$

One of the remaining aims of this section is to show how the frequency behavior of GCSRs can be realized as the dynamical behavior of an ARMA representation through the approach of realization of behavior. To this end, the general plant (Eq. 12.11) is rewritten into

$\begin{array}{l}\hfill \left[\begin{array}{l}z(s)\\ y(s)\end{array}\right]=\left[\begin{array}{ll}\frac{P_{11}^{*}(s)}{g(s)}& \frac{P_{12}^{*}(s)}{g(s)}\\ \frac{P_{21}^{*}(s)}{g(s)}& \frac{P_{22}^{*}(s)}{g(s)}\end{array}\right]\left[\begin{array}{l}w(s)\\ u(s)\end{array}\right],\end{array}$

where g(s) is the least common (monic) multiple of the denominator polynomials of all the entries in P(s), and P_ij*(s)/g(s) = P_ij(s), i = 1, 2; j = 1, 2. It is immediately noted that the above decomposition of P is a special case of Eq. (12.12). By letting

$\begin{array}{c}\hfill \left[\begin{array}{l}{z}_c(s)\\ y_c(s)\end{array}\right]=g(s)I_{m_1+m_2}\left[\begin{array}{l}z(s)\\ y(s)\end{array}\right],\hfill \end{array}$

$\begin{array}{c}\hfill P^{*}(s)=\left[\begin{array}{ll}{P}_{11}^{*}(s)& P_{12}^{*}(s)\\ P_{21}^{*}(s)& P_{22}^{*}(s)\end{array}\right],\hfill \end{array}$

where $I_{m_1+m_2}$ is the identity matrix with dimension m₁ + m₂. Eq. (12.19) thus takes the form

$\begin{array}{l}\hfill \left[\begin{array}{l}{z}_c(s)\\ y_c(s)\end{array}\right]=P^{*}(s)\left[\begin{array}{l}w(s)\\ u(s)\end{array}\right].\end{array}$

It should be noted that in here P*(s) is a polynomial matrix and g(s) is a polynomial.

As is shown before, the realization of behavior for a general plant is rather straightforward, while the realization of behavior for GCSR is more obscure, as in this case the introduction of latent variables is necessary. To propose a realization of behavior for GCSRs, consider the following ARMA representation

$\begin{array}{l}\hfill \left[\begin{array}{ll}{P}_{22}^{*}(\rho )& -g(\rho )I\\ P_{12}^{*}(\rho )& 0\\ 0& 0\end{array}\right]\left[\begin{array}{l}u(t)\\ y(t)\end{array}\right]+\left[\begin{array}{ll}0& 0\\ -g(\rho )I& 0\\ 0& -I\end{array}\right]\left[\begin{array}{l}z(t)\\ w(t)\end{array}\right]\\ \hfill =\left[\begin{array}{l}{P}_{21}^{*}(\rho )\\ P_{11}^{*}(\rho )\\ I\end{array}\right]x(t),t\ge 0.\end{array}$

The above ARMA representation is in fact

$\begin{array}{l}\hfill \left\{\begin{array}{l}{P}_{21}^{*}(\rho )x(t)=[P_{22}^{*}(\rho ),-g(\rho )I]\left[\begin{array}{l}u(t)\\ y(t)\end{array}\right]\\ \left[\begin{array}{ll}g(\rho )I& 0\\ 0& I\end{array}\right]\left[\begin{array}{l}z(t)\\ w(t)\end{array}\right]=\left[\begin{array}{l}-P_{11}^{*}(\rho )\\ -I\end{array}\right]x(t)+\left[\begin{array}{ll}{P}_{12}^{*}(\rho )& 0\\ 0& 0\end{array}\right]\left[\begin{array}{l}u(t)\\ y(t)\end{array}\right],\end{array}\right.\end{array}$

where all the identity matrices and all the zero block matrices are of appropriate dimensions. $x(t)\in C_{k_1}^{\infty }$ are the latent variables. It is noted that every condition pair

$\begin{array}{l}\hfill \left[\begin{array}{l}{y}_1(t)\\ u_1(s)\end{array}\right]:=\left[\begin{array}{l}z(t)\\ w(t)\\ u(t)\\ y(t)\end{array}\right]\in {\mathcal{B}}_d\end{array}$

is consistent about the latent variables x(t) is equivalent that every input-output pair (u(s), y(s)) is consistent about w.

The following technical result will be used in the proof of Theorem 12.3.3. It is stated as a lemma.

Now we are ready to state and prove the following result.

Due to the consistency of $\left[\begin{array}{l}û(s)\\ ŷ(s)\end{array}\right]$, Eq. (12.26) determines the latent variables $\widehat{x}(s)$. By solving the latent variables $\widehat{x}$ in Eq. (12.26) and then substituting into Eq. (12.27), one obtains

$\begin{array}{ll}\hfill & \left[\begin{array}{ll}g(s)I& 0\\ 0& I\end{array}\right]\left[\begin{array}{l}\widehat{z}(s)\\ ŵ(s)\end{array}\right]\hfill \\ \hfill & =\left[\begin{array}{ll}{P}_{12}^{*}(s)-P_{11}^{*}(s){(P_{21}^{*})}^{-}(s)P_{22}^{*}(s)& g(s)P_{11}^{*}(s){(P_{21}^{*})}^{-}(s)\\ -{(P_{21}^{*})}^{-}(s)P_{22}^{*}(s)& g(s){(P_{21}^{*})}^{-}(s)\end{array}\right]\left[\begin{array}{l}û(s)\\ ŷ(s)\end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{l}{P}_{11}^{*}(s)[I-{(P_{21}^{*})}^{-}(s)P_{21}^{*}(s)]\\ I-{(P_{21}^{*})}^{-}(s)P_{21}^{*}(s)\end{array}\right]h(s),\hfill \end{array}$