14.1 Introduction

Consider the plant

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

where $z\in \mathfrak{R}{(s)}^m$, $y\in \mathfrak{R}{(s)}^q$, $w\in \mathfrak{R}{(s)}^r$, and $u\in \mathfrak{R}{(s)}^p$ are the controlled error, the observation output, the exogenous input, and the control input, respectively. The $H_{\infty }$ control problem is to find a controller given by

$\begin{array}{l}\hfill u=Ky,\end{array}$

which internally stabilizes the closed-loop system and satisfies $\parallel \mathrm{\Phi }{\parallel }_{\infty }<1$, where Φ is the closed-loop transfer function from w to z, which is generally called a linear fractional transformation in the control literature and is denoted [22] by LF(P;K), i.e.,

$\begin{array}{l}\hfill \mathrm{\Phi }:=LF(P;K)=P_{11}+P_{12}K{(I-P_{22}K)}^{-1}{P}_{21}.\end{array}$

Concerning the above problem, one of the assumptions is the following

$\begin{array}{l}\hfill (A)\mathrm{rank}{P}_{21}=q,\mathrm{rank}{P}_{12}=p.\end{array}$

When the plant satisfies the above assumption, the $H_{\infty }$ control problem is called [37] the regular or standard case, while the case where the above assumption does not hold is referred [120] to a singular $H_{\infty }$ control problem. We do not make the above assumption, the considered problem is thus nonstandard, which includes both the regular case and the singular case. From a system point of view, since no measurement is error free and no control action is costless [117] it is physically reasonable and general (see, e.g., [90]) to assume that the dimension of z is at least that of u, while the dimension of w must be at least that of y, without loss of generality only the right noninvertibility of P₂₁ is thus considered in the following development. It should also be noted that, in Assumption (A) rank P₂₁ = q means that, rank P₂₁(jω) = q, ∀ω ∈R and rank $P_{21}(\infty )=q$.

Several approaches (e.g., [25, 36, 37, 121], and references therein) have already been proposed to solve the regular $H_{\infty }$ control problem both in the state-space framework and in the input-output operator-theoretic techniques. The singular $H_{\infty }$ control problem has attracted considerable research interests in the last few years [120, 122]. For a monograph on this subject, one can refer to [122–124] considered controller design for the singular $H_{\infty }$ control problem based on the linear matrix inequality (LMI) approach. One basic observation to the aforementioned contributions on this theme is, however, that the problem is generally attacked in the state-space framework. Although state-space techniques are powerful for computing solutions, and are growing even more so with the advent of efficient numerical techniques, they do not always give physical insight into the nature of problems and bear fundamental limitations. Such insight can be obtained much more effectively using input-output techniques that allow solutions to be revealed and understood in their most transparent forms, and without the often obscuring one details that occur in specific problems.

Recently Kimura [22, 90] has developed the CSR, and it was successfully used there to provide a unified framework of $H_{\infty }$ control theory. In this new framework, the $H_{\infty }$ control problem is essentially reduced to a J-lossless factorization of the CSR of the plant. The J-lossless conjugation then provides a powerful tool for computing the required factorization. Kimura’s CSR approach seems to be the most compact theory and the simplest method to the regular $H_{\infty }$ control problem. The known CSR approach is, however, not applicable to the singular $H_{\infty }$ control problem. This situation may improve with the advent of [125]. In [125] Kimura’s results on CSR has been extended to the general case in which the condition (A) is essentially relaxed. From an input-output consistency point of view, the GCSR emerges and is successfully used there to characterize the cascade structure property and the symmetry of general plants in a general setting.

The main motivation of our current work presented herein is to extend Kimura’s results [22, 90] on CSR approach to the $H_{\infty }$ control from the standard case to the nonstandard case by using the GCSR approach [125]. To this end, the nonstandard $H_{\infty }$ control problem is first reformulated based on the GCSR. The GCSR approach then leads naturally to a generalization of the homographic transformation. The state-space realization for this generalized homographic transformation and a number of fundamental cascade structures of the $H_{\infty }$ control systems are further studied in a unified framework of GCSR. Certain sufficient conditions for the solvability of the nonstandard $H_{\infty }$ control problem are thus established via a $(J;J^{\prime })$-lossless factorization of GCSR.

14.2 Reformulation of the nonstandard $H_{\infty }$ control problem via generalized chain-scattering representation

The main reason for using the CSR lies in its ability of representing the feedback connection as a cascade one [22, 90], in this context any feedback of an input-output system is subsequently equivalent to a termination of the corresponding CSR. The regular $H_{\infty }$ control problem, when described [22, 90] through CSR, is thus greatly simplified for the commonly used linear fraction transformation has been replaced by a much simpler form of homographic transformation. Motivated by a similar reason, the main aim of this section is to give a reformulation of the nonstandard $H_{\infty }$ control problem via GCSR on the basis of a generalization to homographic transformation.

It is seen that, concerning the existence of CSR, if the input-output pair (u, y) satisfies the condition of consistency given in [125], i.e.,

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

then the CSRs are still available for the nonstandard case. It should be noted that such a condition of consistency essentially relaxes [125] the condition (A). These representations, termed as the GCSRs therein, are not unique. The set of them has, however, been parameterized in [125]. To facilitate exposition, only the special GCSR form, which is formed in terms of the Moore-Penrose inverse of P₂₁, has been chosen for use in the following development. Let us recall this result from [125].

If P₂₁ is invertible, then the CSR [22, 90] exists for the plant P, Eq. (14.4) becomes

$\begin{array}{l}\hfill \left[\begin{array}{l}z\\ w\\ \end{array}\right]=CHAIN(P)\left[\begin{array}{l}u\\ y\\ \end{array}\right],\end{array}$

where the CSR matrix is

$\begin{array}{l}\hfill CHAIN(P)=\left[\begin{array}{ll}{P}_{12}-P_{11}{P}_{21}^{-1}{P}_{22}& P_{11}{P}_{21}^{-1}\\ -P_{21}^{-1}{P}_{22}& P_{21}^{-1}\\ \end{array}\right].\end{array}$

Further denote in Eq. (14.5)

$\begin{array}{l}\hfill G^{*}:=GCHAIN(P):=\left[\begin{array}{lll}{G}_{11}& G_{12}& G_{13}\\ G_{21}& G_{22}& G_{23}\\ \end{array}\right],\end{array}$

where the sizes of the block matrices G₁₁, G₁₂, G₁₃, G₂₁, G₂₂, G₂₃ are m × p, m × q, m × r, r × p, r × q, r × r, respectively.

Now consider the terminated cascade connection (14.2) and (14.4) (refer to Fig. 14.1), by denoting Σ = [G₂₁K + G₂₂G₂₃], one can easily obtain a {1}-inverse of Σ, which is important in the sequel.

When considering the terminated cascade connection (14.2) and (14.4) (refer to Fig. 14.1), two primary issues that play a key role in our approach need to be considered. One is that the representation (14.4) together with the relation (14.2) can determine [y^T, h^T]^T. It will be seen clearly that this is always the case provided that the closed-loop system is well-posed.

The other issue is termed as output uniqueness, i.e., the condition under which the closed-loop system is able to give rise to a unique output z. These issues are central to the existence of the relevant generalized transfer function [116]. Interestingly, the determined generalized transfer function not only produces a unique output z, being a functional operator, but it is also exactly equal to the linear fractional transformation, once the condition of output uniqueness is satisfied. The following theorems establish the above interesting observations.

Unlike Kimura’s approach [22, 90] of augmentation in the regular case, in the GCSR matrix the block matrix G₂₃ is of r × r, the matrix [G₂₁K + G₂₂G₂₃] is subsequently of r × (q + r), which is nonsquare and is not invertible. This is one of the reasons that we have to utilize the matrix generalized inverses and where the difficulties arise from the present approach. It is noted that, if the condition of output uniqueness is satisfied, Eq. (14.12) determines a generalized transfer function from w to z, which is given by

$\begin{array}{l}\hfill GHM(G^{*};[K0])=[G_{11}K+G_{12}{G}_{13}]{\mathrm{\Sigma }}^{-},\end{array}$

where Σ⁻ is given by Proposition 14.2.1. As a matter of fact this generalized transfer function is an extension to the homographic transformation that is used [22, 90] in dealing with the regular $H_{\infty }$ control problem. We therefore term it as the generalized homographic transformation and denote it by GHM(G*;[K 0]) following the notation adopted in [22, 90].

It is of further interest to look at the relationship between the above proposed generalized homographic transformation and the linear fractional transformation. It is interesting that, being a functional operator, the generalized homographic transformation is equal to the linear fractional transformation. This observation is stated as the following theorem.

The above theorems are interesting, not least for the way in which they reduce the original closed-loop system (14.1) and (14.2) into a wave scatter GCHAIN(P) terminated by a load K. This situation is illustrated in Fig. 14.1. More important than this, however, is that the implied mechanism enables us to pose the nonstandard $H_{\infty }$ control problem in the framework of GCSRs in the following manner.

Nonstandard $H_{\infty }$ problem reformulation: Find a controller (load) K such that the terminated cascade connection (14.2) and (14.4) (refer to Fig. 14.1) is well-posed, internally stable, the output uniqueness condition is satisfied and

$\begin{array}{l}\hfill \parallel GHM(G^{*};[K0]){\parallel }_{\infty }<1.\end{array}$

One further observation arising from Theorem 14.2.3 is that the $H_{\infty }$ norm of the closed-loop system can be given equivalently in terms of the generalized homographic transformation. It will also be seen clearly in the next section that the internal stability of the terminated cascade connection (14.2) and (14.4) can be characterized in terms of the A-matrix in the corresponding state-space realization to the generalized homographic transformation. These two observations are exactly the rationale behind the above reformulation of the nonstandard $H_{\infty }$ control problem.

14.3 Solvability of nonstandard $H_{\infty }$ control problem

This section mainly studies the state-space realization for the generalized homographic transformation and some fundamental cascade structures of the $H_{\infty }$ control systems in the framework of GCSR. Certain sufficient conditions for the solvability of the nonstandard $H_{\infty }$ control problem are then established via $(J,J^{\prime })$-lossless factorization of GCSR.

If the state-space realizations for the GCSR and the controller K are given by

$\begin{array}{l}\hfill G^{*}=\left[\begin{array}{llll}A& B_1& B_2& B_3\\ C_1& D_{11}& D_{12}& D_{13}\\ C_2& D_{21}& D_{22}& D_{23}\\ \end{array}\right]\end{array}$

and

$\begin{array}{l}\hfill K=\left[\begin{array}{ll}{A}_K& B_K\\ C_K& D_K\\ \end{array}\right]\end{array}$

respectively. The above relations (14.16) and (14.17) are then written as

$\begin{array}{l}\hfill \left.\begin{array}{cc}& \dot{x}=Ax+B_{1}u+B_{2}y+B_{3}h\\ & z=C_{1}x+D_{11}u+D_{12}y+D_{13}h\\ & z=C_{2}x+D_{21}u+D_{22}y+D_{23}h\\ \end{array}\right\}\end{array}$

and

$\begin{array}{l}\hfill \left.\begin{array}{cc}& \dot{\xi }=A_K\xi +B_{K}y\\ & u=C_K\xi +D_{K}y\\ \end{array}\right\}.\end{array}$

Elimination of u from the above relations yields

$\begin{array}{l}\hfill \left.\begin{array}{cc}\left[\begin{array}{l}\dot{x}\\ \dot{\xi }\\ \end{array}\right]& =\left[\begin{array}{ll}A& B_1{C}_K\\ 0& A_K\\ \end{array}\right]\left[\begin{array}{l}x\\ \xi \\ \end{array}\right]+\left[\begin{array}{l}\widehat{B}\\ \widehat{B_K}\\ \end{array}\right]\left[\begin{array}{l}y\\ h\\ \end{array}\right]\\ \left[\begin{array}{l}z\\ w\\ \end{array}\right]& =\left[\begin{array}{ll}{C}_1& D_{11}{C}_K\\ C_2& D_{21}{C}_K\\ \end{array}\right]\left[\begin{array}{l}x\\ \xi \\ \end{array}\right]+\left[\begin{array}{l}{D}_1\\ D_2\\ \end{array}\right]\left[\begin{array}{l}y\\ h\\ \end{array}\right]\\ \end{array}\right\}\end{array}$

where we denote

$\begin{array}{l}\hfill \begin{array}{c}\left[\begin{array}{l}\widehat{B}\\ D_1\\ D_2\\ \end{array}\right]=\left[\left[\begin{array}{ll}{B}_1& B_2\\ D_{11}& D_{12}\\ D_{21}& D_{22}\\ \end{array}\right]\left[\begin{array}{l}{D}_K\\ I\\ \end{array}\right]\begin{array}{l}{B}_3\\ D_{13}\\ D_{23}\end{array}\right]\\ \widehat{B_K}=[B_{k}0].\end{array}\end{array}$

In a similar manner described in [90] and considering the output uniqueness condition, from the above relations one obtains the state-space realization of the generalized homographic transformation GHM(G*;[K 0]) given by

$\begin{array}{l}\hfill GHM(G^{*};[K0])=\left[\begin{array}{ll}A(G^{*};[K0])& B(G^{*};[K0])\\ C(G^{*};[K0])& D(G^{*};[K0])\\ \end{array}\right],\end{array}$

where

$\begin{array}{l}\hfill \left.\begin{array}{cc}A(G^{*};[K0])& =\left[\begin{array}{ll}A& B_1{C}_K\\ 0& A_K\\ \end{array}\right]-\left[\begin{array}{l}\widehat{B}\\ \widehat{B_K}\\ \end{array}\right]\times D_2[C_2{D}_{21}{C}_K]\\ B(G^{*};[K0])& =\left[\begin{array}{l}\widehat{B}\\ \widehat{B_K}\\ \end{array}\right]D_2^{-}\\ C(G^{*};[K0])& =[C_1-D(G^{*};[K0])C_2,(D_{11}-D(G^{*};[K0])C_k]\\ D(G^{*};[K0])& =D_1{D}_2^{-}\\ \end{array}\right\}\end{array}$

where $D_2^{-}$ is a {1}-inverse of D₂ given by

$\begin{array}{l}\hfill D_2^{-}=\left[\begin{array}{l}{(D_{21}{D}_K+D_{22})}^{-1}(I-D_{23}{D}_{23}^{†})\\ D_{23}^{†}\\ \end{array}\right].\end{array}$

Now we are ready to introduce the following definition.

Theorem 14.3.1

For the terminated cascade connection (Fig. 14.2) of a GCSR G* = GCHAIN(P₁) and a CSRU = CHAIN(P₂), if the condition of output uniqueness is satisfied, then we have

$\begin{array}{l}\hfill GHM(G^{*};[HM(U;K)0])=GHM\left(G^{*}\left[\begin{array}{ll}U& 0\\ 0& I\\ \end{array}\right];[K0]\right).\end{array}$

  (14.24)

(J, J′)-lossless factorization [22, 90] is a general notion that includes the well-known inner-outer factorization and the spectral factorization. If a matrix $G(s)\in R{L}_{(m+r)\times (p+q)}^{\infty }$ is represented as a product G(s) = Θ(s)Π(s), where $\mathrm{\Theta }(s)\in R{L}_{(m+r)\times (p+q)}^{\infty }$ is (J_mr, J_pq)-lossless and Π(s) is unimodular in $R{H}_{(m+q)\times (p+q)}^{\infty }$, then G(s) is said to have a (J_mr, J_pq)-lossless factorization. It is established [22, 90] that the problem of regular $H_{\infty }$ control can be reduced to finding a special class of (J, J′)-lossless factorization of the CSR. On the bases of Definition 14.3.1 and Theorems 14.3.1 and 14.3.3, concerning the solvability of the nonstandard $H_{\infty }$ control problem, we are now ready to propose the following important result.

As far as the output uniqueness condition (14.34) is concerned, Theorem 14.2.2 tells us that it is a must for the generalized homographic transformations to exist.

From Lemma 4.4 in [90] if the matrix Θ(s) is (J, J′)-lossless, then there exists a lossless matrix P₁ such that Θ(s) is a CSR of P₁, i.e., Θ(s) = CHAIN(P₁). Subsequently, if $K\in B{H}^{\infty }$, taking into account Theorem 4.15 in [90], then we find that

$\begin{array}{ll}\hfill GHM(\mathrm{\Theta };[K0])& =GHM(CHAIN(P_1);[K0])\hfill \\ \hfill & =HM(CHAIN(P_1);K)\in B{H}^{\infty }.\hfill \end{array}$

Based on the above observations, by using Theorems 14.3.1 and 14.3.3, it follows that

$\begin{array}{ll}\hfill GHM(G^{*},[K0])& =GHM\left(\mathrm{\Theta }\left[\begin{array}{ll}{\mathrm{\Pi }}_{11}(s)& 0\\ {\mathrm{\Pi }}_{21}(s)& {\mathrm{\Pi }}_{22}(s)\\ \end{array}\right];[HM({\mathrm{\Pi }}_{11}^{-1};S)0]\right)\hfill \\ \hfill & =GHM\left(\mathrm{\Theta }\left[\begin{array}{ll}{\mathrm{\Pi }}_{11}(s)& 0\\ {\mathrm{\Pi }}_{21}(s)& {\mathrm{\Pi }}_{22}(s)\\ \end{array}\right]\left[\begin{array}{ll}{\mathrm{\Pi }}_{11}^{-1}& 0\\ 0& I\\ \end{array}\right];[S0]\right)\hfill \\ \hfill & =GHM\left(\mathrm{\Theta }\left[\begin{array}{ll}I& 0\\ {\mathrm{\Pi }}_{21}(s){\mathrm{\Pi }}_{11}^{-1}& {\mathrm{\Pi }}_{22}(s)\\ \end{array}\right];[S0]\right)\hfill \\ \hfill & =GHM(\mathrm{\Theta };[S0])\hfill \\ \hfill & =GHM(CHAIN(P_1);[S0])\hfill \\ \hfill & =HM(CHAIN(P_1);S)\in B{H}^{\infty }\forall S\in B{H}_{(p+q)\times (p+q)}^{\infty }.\hfill \end{array}$

As far as the issue of internal stability is concerned in the above equaling process and considering Definition 14.3.1, one can see that the $H_{\infty }$ controller only cancels out all the stable poles and zeros from the (J, J′)-lossless factorization. The internal stability is therefore left invariant.

Due to the fact that one needs to use Theorem 4.15 of [90] in proposing the above theorem, we have assumed that GCSR G* = GCHAIN(P) has no zeros or poles on the jω-axis. However, this condition is not as restrictive as the standard assumption (A) that we are trying to relax in this approach. To see this, take the following simple example.