Matrix Fraction Description in Large Scale MIMO Descriptor Systems: Matrix Polynomials Approaches

Abstract

The matrix transfer function (MTF) is fundamental to the analysis and control of multivariable descriptor systems, especially under zero initial conditions. Its importance lies in its direct relation to input–output behavior and its natural use in frequency-domain methods. Unlike classical approaches that obtain MTF through companion linearizations or indirect Weierstrass–Kronecker reductions, our method derives a closed-form MFD directly from the descriptor pencil $\left(\lambda \mathit{E}-\mathit{A}\right)$, avoiding linearizations and preserving descriptor structure. This yields (i) an explicit parameterization of state feedback gains via finite/infinite Jordan pairs, (ii) a normalization law that removes impulsive modes by design, and (iii) improved reproducibility through block-polynomial operations suited to large-scale MIMO systems. The framework further extends eigenstructure assignment to descriptor models, combining clarity of analysis with practical control design. These results establish a systematic basis for scalable methods in MIMO descriptor systems.

1. Introduction

Matrix fraction descriptions (MFD) are fundamental to the analysis and control of multivariable descriptor systems $\left(\lambda \mathit{E}-\mathit{A}\right)$, especially under zero initial conditions [1]. These systems, also referred to as singular systems, differential-algebraic systems, or generalized state-space models, are characterized by algebraic-differential equations involving a possibly singular matrix $\mathit{E}$ [2]. Unlike standard state-space models, descriptor systems exhibit both differential and algebraic dynamics, leading to finite, infinite dynamic, and infinite non-dynamic modes [3]. The presence of infinite dynamic modes is often linked with undesired impulsive responses, and thus ensuring regularity and causality becomes essential [4]. Historically, descriptor systems emerged in the early 1970s, with foundational work by Singh and Liu (1973) and continued developments in modeling constrained physical systems, such as mechanical linkages, electrical networks, and power systems [5]. Comprehensive treatments can be found in works of [6,7,8], where practical case studies and mathematical properties of descriptor systems are examined. Early studies noted the non-causal nature of descriptor systems when transfer functions are improper, making them useful for modeling spatially varying phenomena such as gravitational or electromagnetic fields. The matrix transfer function captures input–output behavior and fits naturally with frequency-domain methods [9,10]. Its form depends strongly on the structure of the system pencil $\mathit{L}\left(\lambda \right)=\lambda \mathit{E}-\mathit{A}$, whose spectral properties are explored using canonical forms like those of Weierstrass and Kronecker [11,12]. These algebraic tools, deeply rooted in matrix polynomial theory [13], provide a basis for both analysis and controller design in singular models [14].

Despite their utility, descriptor models introduce significant control-theoretic challenges. Feedback stabilization requires careful handling of the singularity in $\mathit{E}$ and the elimination of impulsive components [15]. To this end, a range of control strategies have been explored and/or need to be emerged in singular systems, such as eigenstructure assignment, normalization methods, and polynomial interpolation for exact pole placement [16,17,18]. Algebraic approaches leveraging λ-matrices, Jordan pairs, and matrix pencil factorization [19,20,21] have proven especially effective. Moreover, proportional-derivative feedback methods can be adapted to descriptor dynamics [22], superstabilization using Drazin inverse techniques [23], and robust solution methods [24] have broadened the spectrum of feasible feedback laws. Model reduction via cross-Gramian formulations [25] and Hamiltonian representations has further contributed to structure-preserving control [26]. In large-scale systems, reachability and observability properties are essential for ensuring controllability under algebraic constraints [27]. These have been studied both in continuous and discrete-time frameworks, with recent extensions to fractional-order systems and actuator failures [28,29,30]. Decentralized approaches for interconnected descriptor systems also enable scalable implementations [9]. Descriptor systems admit multiple representations depending on state variable choice. This flexibility, rooted in algebraic equivalence, means different state-space forms can describe the same physical behavior [31,32,33,34]. These results enable a systematic parameterization of the static state feedback gain matrix, extending classical finite eigenstructure assignment to include both finite and infinite Jordan pair control [35]. A robust design framework must therefore account for this variability and preserve internal structure.

Descriptor systems pose fundamental challenges due to singular pencils, impulsive modes, and the difficulty of deriving tractable transfer function representations. Existing methods often rely on indirect linearizations or partial formulations, which obscure structure and become computationally expensive in high dimensions. Recent advances in matrix polynomial theory—such as the theory of linearization [36], interpolation methods [37], recursive inversion techniques [38], λ-matrix factorization [39], and robust control [40,41]—have improved the analysis but remain fragmented. Applications in electro-mechanical systems further highlight the need for a unified approach [42,43,44]. In this work, we address these gaps by constructing the MFD directly from descriptor equations, ensuring both structural transparency and analytical tractability.

The main contribution of this paper lies in bridging the classical MFD formalism with descriptor systems using matrix polynomial theory. Specifically, we propose:

• A novel closed-form MFD derived directly from the pencil $\lambda \mathit{E}-\mathit{A}$ under state feedback, overcoming limitations in existing indirect or partial formulations.
• A novel state feedback design that will assign a desired finite Jordan pair.
• A new normalization law that transforms descriptor systems into a structured form suitable for analysis and control via polynomial methods.

This paper is organized as follows: Section 2 presents essential mathematical preliminaries on matrix polynomials. Section 3 reviews the descriptor system framework and its structural properties. In this part, we analyze the feedback control via some modern control theory. Section 4 introduces the proposed closed-form feedback control and normalization method, along with numerical examples. Finally, Section 5 concludes the paper with a summary and discussion of future extensions.

2. Matrix Polynomials Theory

The matrix polynomial problem can be cast into a block eigenvalue formulation as follows. Given a matrix $\mathit{A}$ of order $n=m\mathcal{l}$, find a matrix $\mathit{R}$ of order $m$, such that $\mathit{A}\mathit{V}=\mathit{V}\mathit{R}$, where $\mathit{V}$ is a matrix of full rank. Some of the implications of this new block eigenvalue formulation are considered.

Definition 1. 

([13,17]) Let $\mathit{A}\left(\lambda \right)={⅀}_{i=0}^{\mathcal{l}}{\mathit{A}}_i{\lambda }^{\mathcal{l}-i}\in \mathbb{C}{\left[\lambda \right]}^{m\times m}$ be a matrix polynomial of degree $\mathcal{l}$.

• The matrix polynomial $\mathit{A}\left(\lambda \right)$ is called unimodular if  $∆\left(\lambda \right)=\det \mathit{A}\left(\lambda \right)=c\in \mathbb{C}$ ;  $c\ne 0$,
• It is called regular (or nonsingular) if $rank\left(\mathit{A}\left(\lambda \right)\right)=m$, for all $\lambda \in \mathbb{C}$ or equivalently, if $\det \mathit{A}\left(\lambda \right)\not\equiv 0$. Otherwise, it is referred to as singular.
• The roots of the characteristic polynomial $∆\left(\lambda \right)$ are termed the (finite) eigenvalues of $\mathit{A}\left(\lambda \right)$.
• A rational matrix $\mathit{H}\left(\lambda \right)\in \mathbb{C}{\left(\lambda \right)}^{m\times m}$ is called biproper if

$$\underset{\lambda \to \infty }{\lim } \mathit{H}\left(\lambda \right)=\mathit{F}\in {\mathbb{C}}^{m\times m} with \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\mathit{F}\right)=m$$

where the variable  $\mathbb{C}\left[\lambda \right]$ denotes the ring of polynomials, while  $\mathbb{C}\left(\lambda \right)$ denotes the field of rational functions in  $\lambda$.

Definition 2. 

([8]) The reversal of the matrix polynomial $\mathrm{r}\mathrm{e}\mathrm{v}\left(\mathit{A}\left(\lambda \right)\right)\in \mathbb{C}{\left[\lambda \right]}^{m\times m}$ is also called the dual of $\mathit{A}\left(\lambda \right)$ and defined by

$$\mathrm{r}\mathrm{e}\mathrm{v}\left(\mathit{A}\left(\lambda \right)\right)={\lambda }^{\mathcal{l}}\mathit{A}\left(1/\lambda \right)={\mathit{A}}_{\mathcal{l}}{\lambda }^{\mathcal{l}}+\dots +{\mathit{A}}_1\lambda +{\mathit{A}}_0$$

(1)

where ${\mathit{A}}_0$ is referred to as the leading coefficient matrix, and ${\mathit{A}}_{\mathcal{l}}$ as the trailing coefficient matrix of $\mathit{A}\left(\lambda \right)$. The matrix polynomial $\mathit{A}\left(\lambda \right)$ is said to be monic if ${\mathit{A}}_0={\mathit{I}}_m$. The rank $\mathit{A}\left(\lambda \right)$ is defined as: $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\mathit{A}\left(\lambda \right)\right)=\mathrm{m}\mathrm{a}\mathrm{x}\left\{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\mathit{A}\left({\lambda }_0\right)\right) :{\lambda }_0\in \mathbb{C}\right\}$.

Infinite eigenvalues of a matrix polynomial $\mathit{A}\left(\lambda \right)$ can be analyzed via its reversal, $\mathrm{r}\mathrm{e}\mathrm{v}\left(\mathit{A}\left(\lambda \right)\right)$, since the infinite eigenvalues of $\mathit{A}\left(\lambda \right)$ correspond to the zero eigenvalues of $\mathrm{r}\mathrm{e}\mathrm{v}\left(\mathit{A}\left(\lambda \right)\right)$. When the degree $\mathcal{l}=1$, the polynomial eigenvalue problem (PEP) reduces to the classical generalized eigenvalue problem (GEP) of the form $\mathit{A}\mathbf{x}=\lambda \mathit{B}\mathbf{x}$. Additionally, a general matrix polynomial $\mathit{A}\left(\lambda \right)$ is said to be non-monic when its leading coefficient matrix ${\mathit{A}}_0$ is singular (i.e., $\mathit{A}\left(\lambda \right)$ is called non-monic if $\mathrm{d}\mathrm{e}\mathrm{t}\left({\mathit{A}}_0\right)=0$).

2.1. Invariant Pairs

Invariant pairs $\left(\mathit{X},\mathit{J}\right)\in {\mathbb{C}}^{m\times k}\times {\mathbb{C}}^{k\times k}$ with $\mathit{X}\ne 0$ are a generalization of eigenpairs for matrix polynomials (as introduced by C. Effenberger) [28]. They provide a unified theoretical framework for computing multiple eigenvalue-eigenvector pairs of a matrix polynomial. From a numerical standpoint, they offer enhanced stability compared to individual eigenpair computations, especially in the presence of multiple or closely spaced eigenvalues. (see the work of D. Kressner) [19]. The notion of invariant pairs can also be applied to more general nonlinear problems, although here we will limit our presentation to matrix polynomials.

Definition 3. 

([1,31,32]) Let $\mathit{A}\left(\lambda \right)\in \mathbb{C}{\left[\lambda \right]}^{m\times m}$ be an  $m\times m$ matrix polynomial. A pair  $\left(\mathit{X},\mathit{J}\right)\in {\mathbb{C}}^{m\times k}\times {\mathbb{C}}^{k\times k}$,  $\mathit{X}\ne 0$, is called an invariant pair if it satisfies the relation:

$$\mathit{A}\left(\mathit{X},\mathit{J}\right) :={\mathit{A}}_0\mathit{X}{\mathit{J}}^{\mathcal{l}}+\dots +{\mathit{A}}_{\mathcal{l}-1}\mathit{X}\mathit{J}+{\mathit{A}}_{\mathcal{l}}\mathit{X}=0$$

(2)

where ${\mathit{A}}_i\in {\mathbb{C}}^{m\times m}$, $i=0\dots \mathcal{l}$ and $k$ is an integer such that: $1\le k\le n=m\mathcal{l}$.

Infinite eigenvalues can be addressed by formulating invariant pairs for the reversal polynomial (1). When a matrix polynomial possesses both zero and infinite eigenvalues, separate invariant pairs must be constructed—one associated with the original matrix polynomial and another with its reversal.

Definition 4. 

([31,45]) A pair $\left(\mathit{X},\mathit{J}\right)\in {\mathbb{C}}^{m\times k}\times {\mathbb{C}}^{k\times k}$ is called minimal if there is $\sigma \in {\mathbb{N}}^{\star }$ such that:

$${\mathit{V}}_{\sigma }\left(\mathit{X},\mathit{J}\right)=\mathrm{c}\mathrm{o}\mathrm{l}{\left(\mathit{X}{\mathit{J}}^{i-1}\right)}_{i=1}^{\sigma }={\left[{\mathit{X}}^{⟙} {\left(\mathit{X}\mathit{J}\right)}^{⟙}\dots {\left(\mathit{X}{\mathit{J}}^{\sigma -1}\right)}^{⟙}\right]}^{⟙}$$

(3)

has full rank. The smallest such $\sigma$ is called minimality index of $\left(\mathit{X},\mathit{J}\right)$. The invariant pairs $\left(\mathit{X},\mathit{J}\right)$ is said to be minimal if

$$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left[\begin{array}{c}\lambda {\mathit{I}}_k-\mathit{J}\\ \mathit{X}\end{array}\right]=k \forall \lambda \in \mathbb{C}$$

(4)

Definition 5. 

([19]) An invariant pair $\left(\mathit{X},\mathit{J}\right) for a regular matrix polynomial \mathit{A}\left(\lambda \right)$ of degree $\mathcal{l}$ is said to be simple if it is minimal, and the algebraic multiplicities of the eigenvalues of $\mathit{J} exactly match those of the corresponding eigenvalues of \mathit{A}\left(\lambda \right)$.

Eigenvectors associated with multiple eigenvalues are inherently sensitive to perturbations; even minor variations in the matrix can result in instability or the disappearance of certain eigenvectors. In contrast, invariant pairs provide a more robust and reliable theoretical and numerical framework for accurately computing multiple eigenpairs of a matrix polynomial [24].

Remark 1. 

There is a strong connection between invariant pairs and the concept of standard pairs introduced in [31], particularly in relation to Jordan pairs. When an invariant pair $\left(\mathit{X},\mathit{J}\right) is simple and \mathit{J} is expressed in Jordan form, then \left(\mathit{X},\mathit{J}\right) is called a Jordan pair. If$$a spectrum of \mathit{A}\left(\lambda \right)$ includes an infinity as an eigenvalues then the Jordan pair will be composed of two main parts named the finite and infinite Jordan pairs, and denoted by $\left({\mathit{X}}_F, {\mathit{J}}_F\right)$ and $\left({\mathit{X}}_{\infty }, {\mathit{J}}_{\infty }\right)$ such that

$${\mathit{A}}_0{\mathit{X}}_F{\mathit{J}}_F^{\mathcal{l}}+\dots +{\mathit{A}}_{\mathcal{l}-1}{\mathit{X}}_F{\mathit{J}}_F+{\mathit{A}}_{\mathcal{l}}{\mathit{X}}_F=\mathbf{0} \mathrm{a}\mathrm{n}\mathrm{d} {\mathit{A}}_0{\mathit{X}}_{\infty }+\dots +{\mathit{A}}_{\mathcal{l}-1}{\mathit{X}}_{\infty }{\mathit{J}}_{\infty }^{\mathcal{l}-1}+{\mathit{A}}_{\mathcal{l}}{\mathit{X}}_{\infty }{\mathit{J}}_{\infty }^{\mathcal{l}}=\mathbf{0}$$

Polynomial eigenpairs and invariant pairs can also be defined in terms of a contour integral. Indeed, an equivalent representation for (2) is the following.

Proposition 1. 

([24]) A pair $\left(\mathit{X},\mathit{J}\right)\in {\mathbb{C}}^{m\times k}\times {\mathbb{C}}^{k\times k}$ is an invariant pair if and only if satisfies the relation:

$$\mathit{A}\left(\mathit{X},\mathit{J}\right) :={\displaystyle \frac{1}{2\pi i}}{\oint }_{\Gamma }\mathit{A}\left(\lambda \right)\mathit{X}{\left(\lambda \mathit{I}-\mathit{J}\right)}^{-1}d\lambda =\mathbf{0}$$

(5)

where $\mathsf{\Gamma }\in \mathbb{C}$ is a closed contour enclosing the spectrum of $\mathit{J}$. This contour-integral formulation allows computing invariant pairs for eigenvalues within a specific region of the complex plane.

2.2. Matrix Solvents

Now, we study the matrix solvent problem as a particular case of the invariant pair problem, and we apply to solvents some results we have obtained for invariant pairs.

Definition 6. 

([13,18]) Let $\mathit{A}\left(\lambda \right)$ be an $m\times m$ matrix polynomial. A matrix $\mathit{R}\in {\mathbb{C}}^{m\times m}$ is called a (right) solvent for $\mathit{A}\left(\mathit{\lambda }\right)$ if satisfies: $\left(\mathit{R}\right) :={\mathit{A}}_0{\mathit{R}}^{\mathcal{l}}+\dots +{\mathit{A}}_{\mathcal{l}-1}\mathit{R}+{\mathit{A}}_{\mathcal{l}}=\mathbf{0}$.

The connection between the eigenvalues of the matrix polynomial $\mathit{A}\left(\lambda \right)$ and its solvents is established in [32]. A corollary of the generalized Bézout theorem states that if $\mathit{R}$ is a solvent of $\mathit{A}\left(\lambda \right)$, then:

$$\mathit{A}\left(\lambda \right)=\mathit{B}\left(\lambda \right)\left(\lambda \mathit{I}- \mathit{R}\right)$$

(6)

where $\mathit{B}\left(\lambda \right)$ is a matrix polynomial of degree $\mathcal{l}-1$. Consequently, any finite eigenpair of the matrix $\mathit{R}$ corresponds to a finite eigenpair of the original matrix polynomial $\mathit{A}\left(\lambda \right)$.

Proposition 2. 

([13,17]) A matrix $\mathit{R}\in {\mathbb{C}}^{m\times m}$ is a right solvent of $\mathit{A}\left(\lambda \right)$ if and only if

$$\mathit{A}\left(\mathit{R}\right) :={\displaystyle \frac{1}{2\pi i}}{\oint }_{\Gamma }\mathit{A}\left(\lambda \right){\left(\lambda \mathit{I}-\mathit{R}\right)}^{-1}d\lambda =\mathbf{0}$$

(7)

for any closed contour $\mathsf{\Gamma }\in \mathbb{C}$ that encloses the spectrum of $\mathit{R}$.

Theorem 1. 

([18,20]) Suppose $\mathit{A}\left(\lambda \right)$ has $p$ distinct eigenvalues ${\left\{{\lambda }_i\right\}}_{i=1}^p$ with $m\le p \le m\mathcal{l}$, and that the corresponding set of p eigenvectors ${\left\{{\mathbf{v}}_{\mathrm{i}}\right\}}_{i=1}^p$ satisfies the Haar condition (every subset of m of them is linearly independent). Then there are at least $p!/\left[m!\left(p-m\right)!\right]$ different solvents of $\mathit{A}\left(\lambda \right)$, and exactly this many if $p=m\mathcal{l}$, which are given by

$$\mathit{R}=\mathit{W}\mathrm{d}\mathrm{i}\mathrm{a}\left({\mu }_1\dots {\mu }_m\right){\mathit{W}}^{-1}; where \mathit{W}=\left[{\mathbf{w}}_1\dots {\mathbf{w}}_{\mathrm{m}}\right]\in {\mathbb{C}}^{m\times m} is invertible$$

(8)

where the eigenpairs ${\left\{{\mu }_i,{\mathbf{w}}_{\mathrm{i}}\right\}}_{i=1}^m$ are chosen among the eigenpairs ${\left\{{\lambda }_i,{\mathbf{v}}_{\mathrm{i}}\right\}}_{i=1}^p of \mathit{A}\left(\lambda \right)$.

Theorem 2. 

([38,39]) Let $\mathit{A}\left(\lambda \right)\in {\mathbb{C}}^{m\times m}\left[\lambda \right]$ be a $m\times m$ matrix polynomial and consider an invariant pair $\left(\mathit{X},\mathit{Y}\right)\in {\mathbb{C}}^{m\times k}\times {\mathbb{C}}^{k\times k}$ of $\mathit{A}\left(\lambda \right)$ (sometimes called admissible pairs). If the matrix $\mathit{X}$ has size $m\times m$ (i.e., $k=m$ and is invertible, then $\mathit{R}=\mathit{X}\mathit{Y}{\mathit{X}}^{-1}$ satisfies $\mathit{A}\left(\mathit{R}\right)=\mathbf{0}$) is a matrix solvent of $\mathit{A}\left(\lambda \right)$.

Proof. 

As $\left(\mathit{X},\mathit{Y}\right)\in {\mathbb{C}}^{m\times k}\times {\mathbb{C}}^{k\times k}$ is an invariant pair of $\mathit{A}\left(\lambda \right)$, we have:

$$\mathit{A}\left(\mathit{X},\mathit{Y}\right) :={\mathit{A}}_0\mathit{X}{\mathit{Y}}^{\mathcal{l}}+\dots +{\mathit{A}}_{\mathcal{l}-1}\mathit{X}\mathit{Y}+{\mathit{A}}_{\mathcal{l}}\mathit{X}=\mathbf{0}$$

Since $\mathit{X}$ is invertible, we can post-multiply by ${\mathit{X}}^{-1}$. Then we get:

$${\mathit{A}}_0\mathit{X}{\mathit{Y}}^{\mathcal{l}}{\mathit{X}}^{-1}+\dots + {\mathit{A}}_{\mathcal{l}-1}\mathit{X}\mathit{Y}{\mathit{X}}^{-1} +{\mathit{A}}_{\mathcal{l}}\mathit{X}{\mathit{X}}^{-1} = \mathbf{0} ⟺ \mathit{A}\left(\mathit{R}\right)=\mathbf{0}$$

Therefore, $\mathit{R}=\mathit{X}\mathit{Y}{\mathit{X}}^{-1}$ is a matrix solvent of $\mathit{A}\left(\lambda \right)$. □

Remark 2. 

([21]) Since $\lambda$ is a root of $\mathit{A}\left(\lambda \right)$ which entails that $1/\lambda$ is a root of $\mathrm{r}\mathrm{e}\mathrm{v}\left(\mathit{A}\left(\lambda \right)\right)$, the matrix $\mathit{S}={\mathit{R}}^{-1}=\mathit{X}{\mathit{Y}}^{-1}{\mathit{X}}^{-1}\in {\mathbb{C}}^{m\times m}$ is a solvent to the reversal matrix polynomial $\mathrm{r}\mathrm{e}\mathrm{v}\left(\mathit{A}\left(\lambda \right)\right)$.

Example 1. 

Given non-monic matrix polynomial $\mathit{A}\left(\lambda \right)={⅀}_{i=0}^3{\mathit{A}}_i{\lambda }^{3-i}$ such that:

$$\mathit{A}\left(\lambda \right)=\left[\begin{array}{cc}2{\lambda }^3+{\lambda }^2+3\lambda +1& {\lambda }^3+2{\lambda }^2+\lambda \\ 6{\lambda }^3+3{\lambda }^2-2\lambda +4& 3{\lambda }^3+4{\lambda }^2+\lambda +1\end{array}\right]=\left[\begin{array}{cc}2& 1\\ 6& 3\end{array}\right]{\lambda }^3+\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]{\lambda }^2+\left[\begin{array}{cc} 3& 1\\ -2& 1\end{array}\right]\lambda +\left[\begin{array}{cc}1& 0\\ 4& 1\end{array}\right]$$

Since ${\mathit{A}}_3=\left[1 0;4 1\right]$ is nonsingular then the latent structure of the regular non-monic matrix polynomial $\mathit{A}\left(\lambda \right)$ can be obtained using the following MATLAB command $\left[\mathit{X}\mathit{J}\right]=\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{e}\mathrm{i}\mathrm{g}\left({\mathit{A}}_3,{\mathit{A}}_2,{\mathit{A}}_1,{\mathit{A}}_0\right)$ and those of the reversal matrix polynomial $\mathrm{r}\mathrm{e}\mathrm{v}\left(\mathit{A}\left(\lambda \right)\right)$ can be obtained using $[{\mathit{X}}_r {\mathit{J}}_r] = \mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{e}\mathrm{i}\mathrm{g}({\mathit{A}}_0,{\mathit{A}}_1,{\mathit{A}}_2,{\mathit{A}}_3)$. There are six finite eigenpairs $\left({\lambda }_k, {\mathbf{x}}_{\mathrm{k}}\right)$ for $\mathrm{r}\mathrm{e}\mathrm{v}\left(\mathit{A}\left(\lambda \right)\right)$ with ${\mathit{X}}_r=\left[{\mathbf{x}}_1{\mathbf{x}}_2{\mathbf{x}}_3{\mathbf{x}}_4{\mathbf{x}}_5{\mathbf{x}}_6\right]$ and:

$${\mathbf{x}}_1,{\mathbf{x}}_2=\left[\begin{array}{c} 0.1839\mp 0.2467i\\ -0.9506\mp 0.0404i\end{array}\right]; {\mathbf{x}}_3=\left[\begin{array}{c}-0.2209\\ 0.9753\end{array}\right]; {\mathbf{x}}_4=\left[\begin{array}{c}0.0187\\ -0.9998\end{array}\right]; {\mathbf{x}}_5=\left[\begin{array}{c}0.5525\\ -0.8335\end{array}\right]; {\mathbf{x}}_6=\left[\begin{array}{c}-0.4472\\ 0.8944\end{array}\right]$$

and ${\mathit{J}}_r=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left({\lambda }_1\dots {\lambda }_6\right)$. The solvents of the reversal matrix polynomial $\mathrm{r}\mathrm{e}\mathrm{v}\left(\mathit{A}\left(\lambda \right)\right)$ are

$$\begin{array}{c}{\mathit{S}}_1=\left[{\mathbf{x}}_1 {\mathbf{x}}_2\right]\left[\begin{array}{cc}{\lambda }_1& 0\\ 0& {\lambda }_2\end{array}\right]{\left[{\mathbf{x}}_1 {\mathbf{x}}_2\right]}^{-1}=\left[\begin{array}{cc}-0.4411& -0.9333\\ 8.9238& 2.8089\end{array}\right]; {\mathit{S}}_2=\left[{\mathbf{x}}_3 {\mathbf{x}}_4\right]\left[\begin{array}{cc}{\lambda }_3& 0\\ 0& {\lambda }_4\end{array}\right]{\left[{\mathbf{x}}_3 {\mathbf{x}}_4\right]}^{-1}=\left[\begin{array}{cc}-2.0618& -0.0239\\ 5.6564& -0.6754\end{array}\right]\\ {\mathit{S}}_3=\left[{\mathbf{x}}_5 {\mathbf{x}}_6\right]\left[\begin{array}{cc}{\lambda }_5& 0\\ 0& {\lambda }_6\end{array}\right]{\left[{\mathbf{x}}_5 {\mathbf{x}}_6\right]}^{-1}=\left[\begin{array}{cc}1.5034& 0.7517\\ -2.2682& -1.1341\end{array}\right]\end{array}$$

Notice that, those obtained three solvents satisfy the matrix equation: ${⅀}_{i=0}^3{\mathit{A}}_i{\mathit{S}}_j^i=0$ for $j=\mathrm{1,2},3$. The solvents of $\mathit{A}\left(\lambda \right)$ are the inverse of the ${\mathit{S}}_j$ solvents (i.e., ${\mathit{R}}_j={\mathit{S}}_j^{-1}$) therefor: ${⅀}_{i=0}^3{\mathit{A}}_i{\mathit{R}}_j^{3-i}=\mathbf{0}$ but now only for $j=\mathrm{1,2}$. Furthermore, if $\rho \ne 0$ is a finite latent root of $\mathrm{r}\mathrm{e}\mathrm{v}\left(\mathit{A}\left(\lambda \right)\right)$, then $1/\rho$ is a finite latent root of $\mathit{A}\left(\lambda \right)$. If $\mathrm{r}\mathrm{e}\mathrm{v}\left(\mathit{A}\left(\lambda \right)\right)$ has a zero latent root (${\mathit{A}}_0$ is singular), then $\mathit{A}\left(\lambda \right)$ is said to have an unbounded latent root. A lambda-matrix $\mathit{A}\left(\lambda \right)$ is said to be degenerate (i.e., singular) if $\det \mathit{A}\left(\lambda \right)=0$ for all $\lambda$. This can only occur if ${\mathit{A}}_0$ and ${\mathit{A}}_{\mathcal{l}}$ are singular [13,17,21].

2.3. Linearization of Matrix Polynomials

For a matrix polynomial eigenvalue problem of degree $\mathcal{l}\ge 2$, the linearization method involves constructing an $\mathcal{l}m\times \mathcal{l}m$ matrix pencil $\mathit{L}\left(\lambda \right)={\mathit{A}}_c+\lambda {\mathit{E}}_c$, that shares the same eigenvalues as $\mathit{A}\left(\lambda \right)$. This will reduce the PEP to a standard linear eigenvalue problem, which can be solved using classical techniques such as the QZ algorithm.

Definition 7. 

([31,32]) Let $\mathit{A}\left(\lambda \right)$ be an $m\times m$ matrix polynomial of degree $\mathcal{l}\ge 1$. A pencil $\mathit{L}\left(\lambda \right)={\mathit{A}}_c+\lambda {\mathit{E}}_c$ with ${\mathit{A}}_c$, ${\mathit{E}}_c$$\in {\mathbb{C}}^{n\times n}$  $and$  $n=\mathcal{l}m$ is called linearization of $\mathit{A}\left(\lambda \right)$ if there exist unimodular matrix polynomials $\mathit{E}\left(\lambda \right), \mathit{F}\left(\lambda \right)\in {\mathbb{C}}^{n\times n}$, such that:

$$\mathit{E}\left(\lambda \right)\mathit{L}\left(\lambda \right)\mathit{F}\left(\lambda \right)=\mathrm{b}\mathrm{l}\mathrm{k}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\mathit{A}\left(\lambda \right),{\mathit{I}}_{\left(\mathcal{l}-1\right)m}\right)=\mathit{A}\left(\lambda \right)\oplus {\mathit{I}}_{\left(\mathcal{l}-1\right)m}$$

(9)

The linearization is not unique (see B. Bekhiti [13], K. Hariche [17]). Most of the linearizations used in practice are of the companion forms

$${\mathit{C}}_1\left(\lambda \right)=\lambda {\mathit{E}}_c-{\mathit{A}}_c$$

(10)

with

$${\mathit{A}}_c=\left[\begin{array}{cc} \begin{array}{c}⃝\\ \begin{array}{c}⋮\\ ⃝\end{array}\end{array}& \begin{array}{c}{\mathit{I}}_m\\ \\ ⃝\end{array} \ddots \begin{array}{c}⃝\\ \\ {\mathit{I}}_m\end{array}\\ -{\mathit{A}}_{\mathcal{l}}& \dots \dots -{\mathit{A}}_1\end{array}\right]; {\mathit{E}}_c=\left[\begin{array}{cc}\begin{array}{c}{\mathit{I}}_m\\ \\ ⃝\end{array}\ddots \begin{array}{c}⃝\\ \\ {\mathit{I}}_m\end{array}& \begin{array}{c}⃝\\ ⋮\\ ⃝\end{array}\\ ⃝ \dots ⃝& {\mathit{A}}_0\end{array}\right];\begin{array}{c}\begin{array}{c}{\mathit{I}}_m:m⎼\mathrm{b}\mathrm{y}⎼m \mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{y} \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{x}\\ \end{array}\\ ⃝:m\text{-}\mathrm{b}\mathrm{y}\text{-}m \mathrm{z}\mathrm{e}\mathrm{r}\mathrm{o} \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{x}\end{array}$$

The eigenvalues of $\mathit{L}\left(\lambda \right)$ coincide with those of $\mathit{A}\left(\lambda \right)$ (including multiplicities). There is a one-to-one correspondence between the eigenvectors of $\mathit{A}\left(\lambda \right)$ and $\mathit{L}\left(\lambda \right)$ under appropriate transformation. The companion form is the most classical linearization. Linearizations of matrix polynomials transform them into an equivalent matrix pencil, preserving all finite eigenvalues and (possibly) the infinite ones. Some structured linearizations (e.g., Fiedler) maintain additional properties like symmetry or sparsity. Particularly useful are those linearizations that preserve both the finite and infinite eigenstructure, ensuring an accurate spectral equivalence with the original polynomial [13,18,24].

2.4. The Spectral Data (Finite and Infinite)

When the leading coefficient matrix ${\mathit{A}}_0$ is singular then the degree of $\det \left(\mathit{A}\left(\lambda \right)\right)=\nu$ with $\nu <\mathcal{l}m$ and $\mathit{A}\left(\lambda \right)$ has $\nu$ finite eigenvalues, to which we add $\mu =\mathcal{l}m-\nu$ infinite eigenvalues. Infinite eigenvalues correspond to the zero eigenvalues of the reverse polynomial ${\lambda }^{\mathcal{l}}\mathit{A}\left({\lambda }^{-1}\right)$.

Lemma 1. 

([1,14]) If $\nu$, $\mu$ are the sum of degrees of the finite and infinite elementary divisors of a general matrix polynomial $\mathit{A}\left(\lambda \right)$ respectively, then $n =\mu +\nu =\mathcal{l}m$.

The finite Jordan pair (see Gohberg I, Chapter 7 [31]) of $\mathrm{r}\mathrm{e}\mathrm{v}\left(\mathit{A}\left(\lambda \right)\right)$ corresponding to the zero structure at $\lambda =0$, is defined as the infinity Jordan pair $\left({\mathit{X}}_{\infty }, {\mathit{J}}_{\infty }\right)$, of $\mathit{A}\left(\lambda \right)$. As a result, the finite and the infinity Jordan pairs of $\mathit{A}\left(\lambda \right)$ satisfy the following properties:

$$\left\{\begin{array}{c}{\mathsf{\Omega }}_F=\mathrm{c}\mathrm{o}\mathrm{l}{\left({\mathit{X}}_F{\mathit{J}}_F^{i-1}\right)}_{i=1}^{\mathcal{l}}=\left[\begin{array}{c}{\mathit{X}}_F\\ \begin{array}{c}{\mathit{X}}_F{\mathit{J}}_F\\ ⋮\end{array}\\ {\mathit{X}}_F{\mathit{J}}_F^{\mathcal{l}-1}\end{array}\right]; \\ {⅀}_{i=0}^{\mathcal{l}}{\mathit{A}}_i{\mathit{X}}_F{\mathit{J}}_F^{\mathcal{l}-i}=\mathbf{0} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k} {\mathsf{\Omega }}_F=\nu \end{array}\right\} \mathrm{a}\mathrm{n}\mathrm{d}\left\{\begin{array}{c}{\mathsf{\Omega }}_{\infty }=\mathrm{c}\mathrm{o}\mathrm{l}{\left({\mathit{X}}_{\infty }{\mathit{J}}_{\infty }^{\mathcal{l}-i}\right)}_{i=1}^{\mathcal{l}}=\left[\begin{array}{c}{\mathit{X}}_{\infty }{\mathit{J}}_{\infty }^{\mathcal{l}-1}\\ \begin{array}{c}⋮\\ {\mathit{X}}_{\infty }{\mathit{J}}_{\infty }\end{array}\\ {\mathit{X}}_{\infty }\end{array}\right] \\ {⅀}_{i=0}^{\mathcal{l}}{\mathit{A}}_i{\mathit{X}}_{\infty }{\mathit{J}}_{\infty }^i=\mathbf{0} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k} {\mathsf{\Omega }}_{\infty }=\mu \end{array}\right\}$$

(11)

Furthermore, the structure of the infinity Jordan pair of $\mathit{A}\left(\lambda \right)$ is closely related (see Vardulakis in [8]) to its Smith-McMillan form at $\lambda =\infty$. Particularly,

$${\mathit{J}}_{\infty }=\mathrm{b}\mathrm{l}\mathrm{k}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left\{{\mathit{J}}_i^{\infty }\right\}\in {\mathbb{R}}^{\mu \times \mu }; \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} {\mathit{J}}_i^{\infty }=\left[\begin{array}{c}0\\ \begin{array}{c}0\\ ⋮\end{array}\\ 0\end{array} \begin{array}{c}1\\ \begin{array}{c}0\\ \ddots \end{array}\\ \dots \end{array} \begin{array}{c}0\\ \begin{array}{c}\ddots \\ \ddots \end{array}\\ 0\end{array} \begin{array}{c}0\\ \begin{array}{c}⋮\\ 1\end{array}\\ 0\end{array}\right]$$

(12)

Theorem 3. 

([1,14,32]) Let $\left({\mathit{X}}_F,{\mathit{J}}_F\right)$ and $\left({\mathit{X}}_{\infty },{\mathit{J}}_{\infty }\right)$ be the finite and infinite Jordan pairs of $\mathit{A}\left(\lambda \right)$, with ${\mathit{X}}_F\in {\mathbb{R}}^{m\times \nu }$, ${\mathit{J}}_F\in {\mathbb{R}}^{\nu \times \nu }$, ${\mathit{X}}_{\infty }\in {\mathbb{R}}^{m\times \mu }$, ${\mathit{J}}_{\infty }\in {\mathbb{R}}^{\mu \times \mu }$ and $\mu =\mathcal{l}m-\nu$. These pairs satisfy the following properties:

• $\mathrm{deg}\left(\det \left(\mathit{A}\left(\lambda \right)\right)\right)=\nu$
• $\det \left({\lambda }^{\mathcal{l}}\mathit{A}\left({\lambda }^{-1}\right)\right)$ has a zero at $\lambda =0$ with multiplicity $\mu$.
• ${⅀}_{i=0}^{\mathcal{l}}{\mathit{A}}_i{\mathit{X}}_F{\mathit{J}}_F^{\mathcal{l}-i}=\mathbf{0};{⅀}_{i=0}^{\mathcal{l}}{\mathit{A}}_i{\mathit{X}}_{\mathrm{\infty }}{\mathit{J}}_{\mathrm{\infty }}^i=\mathbf{0}$
• $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k} {\mathsf{\Omega }}_F=\nu ; and \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k} {\mathsf{\Omega }}_{\mathrm{\infty }}=\mu$
• ${\left[{\mathit{A}}_{\mathcal{l}}^{⟙}\dots {\mathit{A}}_1^{⟙}{\mathit{A}}_0^{⟙}\right]}^{⟙}\in \mathrm{n}\mathrm{u}\mathrm{l}\mathrm{l}\left({\mathit{Q}}^{⟙}\right); with \mathit{Q}=\left[{\mathit{Q}}_F{\mathit{Q}}_{\mathrm{\infty }}\right]=\left[\begin{array}{c}{\mathit{X}}_F\\ \begin{array}{c}{\mathit{X}}_F{\mathit{J}}_F\\ ⋮\end{array}\\ {\mathit{X}}_F{\mathit{J}}_F^{\mathcal{l}}\end{array}\begin{array}{c}{\mathit{X}}_{\mathrm{\infty }}{\mathit{J}}_{\mathrm{\infty }}^{\mathcal{l}}\\ \begin{array}{c}⋮\\ {\mathit{X}}_{\mathrm{\infty }}{\mathit{J}}_{\mathrm{\infty }}\end{array}\\ {\mathit{X}}_{\mathrm{\infty }}\end{array}\right]$

In addition, a realization of ${\mathit{A}}^{-1}\left(\lambda \right)$ is given by

$$\begin{array}{c}{\mathit{A}}^{-1}\left(\lambda \right)=\left[{\mathit{X}}_F {\mathit{X}}_{\infty }\right]{\left[\begin{array}{cc}\lambda {\mathit{I}}_{\nu }-{\mathit{J}}_F& 0\\ 0& \lambda {\mathit{J}}_{\infty }-{\mathit{I}}_{\mu }\end{array}\right]}^{-1}\left[\begin{array}{c}{\mathit{Y}}_F\\ {\mathit{Y}}_{\infty }\end{array}\right]\mathit{with} {\mathit{Y}}_F\in {\mathbb{R}}^{\nu \times m}, {\mathit{Y}}_{\infty }\in {\mathbb{R}}^{\mu \times m}\\ \left[\begin{array}{c}{\mathit{Y}}_F\\ {\mathit{Y}}_{\infty }\end{array}\right]=\left[\begin{array}{c}{\mathit{I}}_{\nu }\\ 0 \end{array} \begin{array}{c}0\\ {\mathit{J}}_{\infty }^{\mathcal{l}-1}\end{array}\right]{\left[{\mathsf{\Gamma }}_1 {\mathsf{\Gamma }}_2\right]}^{-1}\left[\begin{array}{c}{0}_m\\ ⋮\\ {\mathit{I}}_m\end{array}\right] and \mathsf{\Gamma }=\left[{\mathsf{\Gamma }}_1 {\mathsf{\Gamma }}_2\right]=\left[\left.\begin{array}{c}{\mathit{X}}_F\\ \begin{array}{c}⋮\\ {\mathit{X}}_F{\mathit{J}}_F^{\mathcal{l}-2}\end{array}\\ {\mathit{A}}_0{\mathit{X}}_F{\mathit{J}}_F^{\mathcal{l}-1}\end{array}\right|\begin{array}{c}{\mathit{X}}_{\infty }{\mathit{J}}_{\infty }^{\mathcal{l}-2}\\ \begin{array}{c}⋮\\ {\mathit{X}}_{\infty }\end{array}\\ -{⅀}_{i=0}^{\mathcal{l}}{\mathit{A}}_{\mathcal{l}-i}{\mathit{X}}_{\infty }{\mathit{J}}_{\infty }^{\mathcal{l}-1-i}\end{array}\right]\end{array}$$

A matrix polynomial $\mathit{A}\left(\lambda \right)={⅀}_{i=0}^{\mathcal{l}}{\mathit{A}}_i{\lambda }^{\mathcal{l}-i}$ has no eigenvalue at infinity if and only if the leading coefficient ${\mathit{A}}_0$ is nonsingular—this is always the case for monic polynomials where ${\mathit{A}}_0=\mathit{I}$. Conversely, if the trailing coefficient ${\mathit{A}}_{\mathcal{l}}$ is singular, then $\mathit{A}\left(\lambda \right)$ has a zero eigenvalue, and this translates to an eigenvalue at infinity for $\mathrm{r}\mathrm{e}\mathrm{v}\left(\mathit{A}\left(\lambda \right)\right)$. The multiplicity of the eigenvalue at infinity is then defined via the zero eigenvalue of $\mathit{A}\left(\lambda \right)$. For a finite eigenvalue $\beta$, $\mathit{A}\left(\beta \right)$ becomes rank-deficient, and the associated nonzero vectors ${\mathbf{u}}_{\mathit{i}}$ satisfying $\mathit{A}\left(\beta \right){\mathbf{u}}_{\mathit{i}} = 0$ form the (right) eigenvectors. The geometric multiplicity is $m_{\mathrm{g}}=m-\mathrm{rank}\left(\mathit{A}\left(\beta \right)\right)$, and the algebraic multiplicity is the multiplicity of $\beta$ as a root of $\mathit{det}\mathit{A}\left(\lambda \right)$. In the case of multiple eigenvalues, generalized eigenvectors arise via Jordan chains. When ${\mathit{A}}_0$ is singular, the Jordan structure decomposes into a finite part $\left({\mathit{X}}_F,{\mathit{J}}_F\right)$ for finite eigenvalues and an infinite part $\left({\mathit{X}}_{\infty },{\mathit{J}}_{\infty }\right)$, where ${\mathit{J}}_{\infty }$ contains Jordan blocks associated with the eigenvalue at infinity [1,8,14].

Theorem 4. 

([1,31]) Let $\mathit{A}\left(\lambda \right)$ be a regular matrix polynomial and let $\left({\mathit{X}}_F,{\mathit{J}}_F\right)$ and $\left({\mathit{X}}_{\infty },{\mathit{J}}_{\infty }\right)$ be its finite and infinite Jordan pairs, respectively. Then the pair $\left(\left[{\mathit{X}}_F {\mathit{X}}_{\infty }\right], {\mathit{J}}_F\oplus {\mathit{J}}_{\infty }\right)$ is a decomposable pair for $\mathit{A}\left(\lambda \right)$.

Example 2. 

Given the following finite and infinite Jordan pairs of a regular matrix polynomial $\mathit{A}\left(\lambda \right)$, construct its coefficients ${\mathit{A}}_i$.

$${\mathit{X}}_F=\left[\begin{array}{c}1 0 0 1\\ 0 0 0 8\end{array}\right]; {\mathit{J}}_F=\mathrm{b}\mathrm{l}\mathrm{k}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\left[\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 0& 0& 1\end{array}\right],-1\right); {\mathit{X}}_{\infty }=\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right]; {\mathit{J}}_{\infty }=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$$

The pair $\left(\left[{\mathit{X}}_F {\mathit{X}}_{\infty }\right], {\mathit{J}}_F\oplus {\mathit{J}}_{\infty }\right)$ is a decomposable pair means that:

$${\mathit{A}}_0{\mathit{X}}_F{\mathit{J}}_F^3+{\mathit{A}}_1{\mathit{X}}_F{\mathit{J}}_F^2+{\mathit{A}}_2{\mathit{X}}_F{\mathit{J}}_F^1+{\mathit{A}}_3{\mathit{X}}_F=\mathbf{0}; {\mathit{A}}_0{\mathit{X}}_{\infty }+{\mathit{A}}_1{\mathit{X}}_{\infty }{\mathit{J}}_{\infty }^1+{\mathit{A}}_2{\mathit{X}}_{\infty }{\mathit{J}}_{\infty }^2+{\mathit{A}}_3{\mathit{X}}_{\infty }{\mathit{J}}_{\infty }^3=\mathbf{0}$$

In more compact form we write $\left[{\mathit{A}}_3 {\mathit{A}}_2 {\mathit{A}}_1 {\mathit{A}}_0\right]\left[\mathrm{c}\mathrm{o}\mathrm{l}{\left({\mathit{X}}_F{\mathit{J}}_F^{i-1}\right)}_{i=1}^{\mathcal{l}+1} ⋮ \mathrm{c}\mathrm{o}\mathrm{l}{\left({\mathit{X}}_{\infty }{\mathit{J}}_{\infty }^{\mathcal{l}-i}\right)}_{i=0}^{\mathcal{l}}\right]$. The coefficients ${\mathit{A}}_i$ of the regular non-monic matrix polynomial are constructed from the null space of ${\mathit{Q}}^{⟙}$: ${\left[{\mathit{A}}_3^{⟙} {\mathit{A}}_2^{⟙} {\mathit{A}}_1^{⟙} {\mathit{A}}_0^{⟙}\right]}^{⟙}\in \mathrm{n}\mathrm{u}\mathrm{l}\mathrm{l}\left({\mathit{Q}}^{⟙}\right)$. The coefficients ${{\mathit{A}}_i}^{\prime }s$ are:

$${\mathit{A}}_0=\left[\begin{array}{cc} 0.0966& 0\\ -0.1986& 0\end{array}\right]; {\mathit{A}}_1=\left[\begin{array}{cc}-0.2898& 0\\ 0.5959& 0\end{array}\right]; {\mathit{A}}_2=\left[\begin{array}{cc} 0.2898& 0.5876\\ -0.5959& 0.4086\end{array}\right]; {\mathit{A}}_3=\left[\begin{array}{cc}-0.0966& 0.6842\\ 0.1986& 0.2100\end{array}\right]$$

2.5. Generalized Eigenvalue Problem and Degenerate Matrix Polynomials

A vector $\mathbf{x} \in {\mathbb{C}}^n$, with $\mathbf{x} \ne \mathbf{0}$, is called an eigenvector of the ordered pair of matrices $\left(\mathit{A},\mathit{E}\right)\in {\mathbb{C}}^{n\times n}$ if there exist a nonzero complex scalars $\mu$ and $\nu$ such that

$$\mu \mathit{A}\mathbf{x}=\nu \mathit{E}\mathbf{x}$$

(13)

The scalars $\mu$ and $\nu$ are generally not unique, but their ratio $\lambda =\nu / \mu$ is well-defined (except in the singular case $\mathit{A}\mathbf{x}=\mathit{E}\mathbf{x}=\mathbf{0}$, which leads to a singular situation). If $\mu \ne 0$, Equation (13) simplifies to the generalized eigenvalue problem $\mathit{A}\mathbf{x} =\lambda \mathit{E}\mathbf{x}$ where $\lambda \in \mathbb{C}$ is the eigenvalue associated with the eigenvector $\mathbf{x}$. If $\mu =0$ and $\nu \ne 0$, then the eigenvalue is considered infinite, and the pair $\left(\mathit{A},\mathit{E}\right)$ is said to have an infinite eigenvalue (the eigenvalue of the ordered pair $\left(\mathit{A},\mathit{E}\right)$ associated with $\mathbf{x}$ is $\infty$). When $\mathit{E}=\mathit{I}$, this framework reduces to the classical eigenvalue problem.

Proposition 3. 

([4,5]) Let $\left(\mathit{A},\mathit{E}\right)\in {\mathbb{C}}^{n\times n}$ be an ordered pair of matrices, and let $\lambda \in \mathbb{C}\setminus \left\{0\right\}$. Then the following equivalences hold:

1. 

The scalar $\lambda$ is an eigenvalue of $\left(\mathit{A},\mathit{E}\right) if and only if 1/\lambda is an eigenvalue of \left(\mathit{E},\mathit{A}\right)$.

2. 

Infinity $\lambda =\mathrm{\infty }$ is an eigenvalue of $\left(\mathit{A},\mathit{E}\right) if and only if 0 is an eigenvalue of \left(\mathit{E},\mathit{A}\right)$.

3. 

The pair $\left(\mathit{A},\mathit{E}\right) has an infinite eigenvalue if and only if the matrix \mathit{E}$ is singular.

When the matrix $\mathit{E}\in {\mathbb{C}}^{n\times n}$ is nonsingular, the eigenvalues of the matrix pair $\left(\mathit{A},\mathit{E}\right)$ are identical to those of $\mathit{A}{\mathit{E}}^{-1}$ and ${\mathit{E}}^{-1}\mathit{A}$. In this context, if $\mathbf{x}$ is an eigenvector of the pair associated with eigenvalue $\lambda$, then $\mathbf{x}$ is also an eigenvector of ${\mathit{E}}^{-1}\mathit{A}$, while $\mathit{E}\mathbf{x}$ becomes an eigenvector of $\mathit{A}{\mathit{E}}^{-1}$, both corresponding to the same eigenvalue $\lambda$. The expression $\mathit{A}-\lambda \mathit{E}$ is commonly referred to as a matrix pencil, and the terms matrix pencil and matrix pair are often used interchangeably. A scalar $\lambda \in \mathbb{C}$ is said to be an eigenvalue of the pair $\left(\mathit{A},\mathit{E}\right)$ if the matrix $\mathit{L}\left(\mathit{\lambda }\right)=\left(\mathit{A}-\mathit{\lambda }\mathit{E}\right)$ is singular, i.e., $\det \left(\lambda \mathit{E}-\mathit{A}\right)=0$ which defines the characteristic equation of the pair. The function $∆\left(\lambda \right)=\det \left(\lambda \mathit{E}-\mathit{A}\right)$ is a polynomial in $\lambda$ of degree less than or equal to $n$, known as the characteristic polynomial. However, it may happen that $∆\left(\lambda \right)\equiv 0$ for all $\lambda$ (i.e., $∆\left(\lambda \right)$ is identically zero), such as when a nonzero vector $\mathbf{x}$ satisfies $\mathit{A}\mathbf{x}=\mathit{E}\mathbf{x}=\mathbf{0}$. In such a case, the pencil $\mathit{L}\left(\mathit{\lambda }\right)$ is singular for all $\lambda$, and every complex number is an eigenvalue. The pair is then called singular. Otherwise, if $∆\left(\lambda \right)\not\equiv 0$ (i.e., is not identically zeros), the pair is classified as regular pair [3,25].

Theorem 5. 

([8,19]) If the pair $\left(\mathit{A},\mathit{E}\right)$ is singular (i.e., $\det \left(\lambda \mathit{E}-\mathit{A}\right)=0 \forall \lambda \in \mathbb{C}$) then this pair of matrices is devisable on the left or the right by singular square matrix ${\mathit{F}}_L or {\mathit{F}}_R$ respectively.

$$\lambda \mathit{E}-\mathit{A}={\mathit{F}}_L\left(\lambda {\mathit{E}}_L-{\mathit{A}}_L\right)=\left(\lambda {\mathit{E}}_R-{\mathit{A}}_R\right) {\mathit{F}}_R$$

(14)

Lemma 2. 

([13,21]) Two matrix polynomials $\mathit{A}\left(\lambda \right)$ and $\mathit{B}\left(\lambda \right)\in {\mathbb{C}\left[\lambda \right]}^{m\times m}$ are related by the left transformation $\mathit{A}\left(\lambda \right)={\mathit{T}}_L\mathit{B}\left(\lambda \right)$ or by the right transformation $\mathit{A}\left(\lambda \right)=\mathit{B}\left(\lambda \right){\mathit{T}}_R$ with ${\mathit{T}}_L$ and ${\mathit{T}}_R\in {\mathbb{C}}^{m\times m}$, then they satisfy the following statements:

• If ${\mathit{T}}_L$ or ${\mathit{T}}_R is nonsingular then both \mathit{A}\left(\lambda \right)$ and $\mathit{B}\left(\lambda \right)$ have the same set of solvents.
• If ${\mathit{T}}_L$ or ${\mathit{T}}_R is singular then \mathit{A}\left(\lambda \right)$ is a degenerate $\lambda$-matrix with all singular ${{\mathit{A}}_i}^{\prime }s$.
• If ${\mathit{T}}_L$ or ${\mathit{T}}_R is singular then solvents of \mathit{B}\left(\lambda \right)$ are solvents of $\mathit{A}\left(\lambda \right)$ but not vice versa.
• Degenerate non-monic matrix polynomial may have complete set of finite solvents.

Proof. 

Let us prove the last statement in the lemma. Assume that the ${\mathcal{l}}^{th}$ degree non-monic matrix polynomial $\mathit{A}\left(\lambda \right)\in {\mathbb{C}\left[\lambda \right]}^{m\times m}$ has a complete set of solvents ${\mathit{R}}_i$ with $i=1,\dots ,\mathcal{l}$ then:

$$\left\{\begin{array}{c}\begin{array}{c}{\mathit{A}}_1{\mathit{R}}_1^{\mathcal{l}-1}+\dots +{\mathit{A}}_{\mathcal{l}-1}{\mathit{R}}_1+{\mathit{A}}_{\mathcal{l}}=-{\mathit{A}}_0{\mathit{R}}_1^{\mathcal{l}}\\ {\mathit{A}}_1{\mathit{R}}_2^{\mathcal{l}-1}+\dots +{\mathit{A}}_{\mathcal{l}-1}{\mathit{R}}_2+{\mathit{A}}_{\mathcal{l}}=-{\mathit{A}}_0{\mathit{R}}_2^{\mathcal{l}}\end{array}\\ \begin{array}{c}\begin{array}{c} \\ ⋮\end{array}\\ {\mathit{A}}_1{\mathit{R}}_{\mathcal{l}}^{\mathcal{l}-1}+\dots +{\mathit{A}}_{\mathcal{l}-1}{\mathit{R}}_{\mathcal{l}}+{\mathit{A}}_{\mathcal{l}}=-{\mathit{A}}_0{\mathit{R}}_{\mathcal{l}}^{\mathcal{l}}\end{array}\end{array}\right\}⟺\left[\begin{array}{cc} \begin{array}{c}⃝\\ \begin{array}{c}⋮\\ ⃝\end{array}\end{array}& \begin{array}{c}{\mathit{I}}_m\\ \\ ⃝\end{array} \ddots \begin{array}{c}⃝\\ \\ {\mathit{I}}_m\end{array}\\ -{\mathit{A}}_{\mathcal{l}}& \dots \dots -{\mathit{A}}_1\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{\mathit{I}}_m\\ \begin{array}{c}{\mathit{R}}_1\\ ⋮ \end{array}\end{array}\\ {\mathit{R}}_1^{\mathcal{l}-1}\end{array}\ddots \begin{array}{c}\begin{array}{c}{\mathit{I}}_m\\ \begin{array}{c}{\mathit{R}}_{\mathcal{l}}\\ ⋮ \end{array}\end{array}\\ {\mathit{R}}_{\mathcal{l}}^{\mathcal{l}-1}\end{array}\right]=\left[\begin{array}{cc}\begin{array}{c}{\mathit{I}}_m\\ \\ ⃝\end{array}\ddots \begin{array}{c}⃝\\ \\ {\mathit{I}}_m\end{array}& \begin{array}{c}⃝\\ ⋮\\ ⃝\end{array}\\ ⃝ \dots ⃝& {\mathit{A}}_0\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{\mathit{I}}_m\\ \begin{array}{c}{\mathit{R}}_1\\ ⋮ \end{array}\end{array} \\ {\mathit{R}}_1^{\mathcal{l}-1}\end{array}\ddots \begin{array}{c}\begin{array}{c}{\mathit{I}}_m\\ \begin{array}{c}{\mathit{R}}_{\mathcal{l}}\\ ⋮ \end{array}\end{array}\\ {\mathit{R}}_{\mathcal{l}}^{\mathcal{l}-1}\end{array}\right]\left[\begin{array}{c}{\mathit{R}}_1\\ \\ ⃝\end{array}\ddots \begin{array}{c}⃝\\ \\ {\mathit{R}}_{\mathcal{l}}\end{array}\right]$$

Collecting these set of equations in a compact matrix form we get ${\mathit{A}}_c{\mathit{V}}_R={\mathit{E}}_c{\mathit{V}}_R{\mathit{\Lambda }}_R$ with ${\mathit{A}}_c$, ${\mathit{E}}_c$ are given in Equation (10) and ${\mathit{V}}_R$ is the block Vandermond matrix, ${\mathit{\Lambda }}_R$ is the block diagonal matrix

$${\mathit{V}}_R=\left[{\mathit{X}}_1{\mathit{X}}_2\dots {\mathit{X}}_{\mathcal{l}}\right]; {\mathit{X}}_i=\mathrm{c}\mathrm{o}\mathrm{l}{\left({\mathit{R}}_i^k\right)}_{k=0}^{\mathcal{l}-1} \mathrm{a}\mathrm{n}\mathrm{d} {\mathit{\Lambda }}_R={\mathit{R}}_1\oplus {\mathit{R}}_2\oplus \dots \oplus {\mathit{R}}_{\mathcal{l}}$$

(15)

Finally we notice that the pair matrix $\left({\mathit{A}}_{\mathit{c}},{\mathit{E}}_{\mathit{c}}\right)$ are related by the following transformation ${\mathit{A}}_c={\mathit{E}}_c\left({\mathit{V}}_R{\mathit{\Lambda }}_R{\mathit{V}}_R^{-1}\right)$ or equivalently

$$\lambda {\mathit{E}}_c-{\mathit{A}}_c={\mathit{E}}_c\left[\lambda {\mathit{I}}_n-{\mathit{V}}_R{\mathsf{\Lambda }}_R{\mathit{V}}_R^{-1}\right] ⟺ \det \left(\lambda {\mathit{E}}_c-{\mathit{A}}_c\right)=\mathrm{d}\mathrm{et}{\mathit{E}}_c\det \left(\lambda {\mathit{I}}_n-{\mathit{V}}_R{\mathsf{\Lambda }}_R{\mathit{V}}_R^{-1}\right)$$

(16)

Since the matrix ${\mathit{E}}_c$ is singular than $\det \left(\lambda {\mathit{E}}_c-{\mathit{A}}_c\right)=0$ for all $\lambda \in \mathbb{C}$, therefore the $\lambda$-matrix $\mathit{A}\left(\lambda \right)$ is degenerate and the fourth statement is proved. □

Example 3. 

To verify the fourth statement, let us consider the following 2nd order non-monic matrix polynomial $\mathit{A}\left(\lambda \right)$ with the next coefficients.

$$\begin{array}{c}{\mathit{A}}_0=\left[\begin{array}{cc}3& 6\\ 2& 4\end{array}\right]; {\mathit{A}}_1=\left[\begin{array}{cc}23.7402& 26.4331\\ 15.8268& 17.6220\end{array}\right]; {\mathit{A}}_2=\left[\begin{array}{cc}35.0551& 27.5906\\ 23.3701& 18.3937\end{array}\right]\end{array}$$

One can check that ${\mathit{R}}_1$ and ${\mathit{R}}_2$ are complete set of finite solvents of $\mathit{A}\left(\lambda \right)$ even if $\det \left(\mathit{A}\left(\lambda \right)\right)=0; \forall \lambda$

$${\mathit{R}}_1=\left[\begin{array}{cc}-2.20& -0.60\\ 0.40& -0.80\end{array}\right]; {\mathit{R}}_2=\left[\begin{array}{cc}-4.20& 0.30\\ -0.80& -2.80\end{array}\right]$$

2.6. Higher Order System Described by Matrix Differential Equations

We consider the linear time-invariant dynamical system of the form:

$${\mathit{A}}_0{\mathbf{x}}^{\left(\mathcal{l}\right)}\left(t\right)+{\mathit{A}}_1{\mathbf{x}}^{\left(\mathcal{l}-1\right)}\left(t\right)+\dots + {\mathit{A}}_{\mathcal{l}}\mathbf{x}\left(t\right)=\mathit{B}\mathbf{u}\left(t\right) \mathrm{and} \mathbf{y}\left(t\right)=\mathit{C}\mathbf{x}\left(t\right)+\mathit{D}\mathbf{u}\left(t\right)$$

(17)

where $\mathbf{u}\left(t\right):\mathbb{R}\to {\mathbb{C}}^m$ is the input control, $\mathbf{y}\left(t\right):\mathbb{R}\to {\mathbb{C}}^p$ is the output measurement, $\mathbf{x}\left(t\right):\mathbb{R}\to {\mathbb{C}}^r$ is the state function and ${\mathit{A}}_i\in {\mathbb{C}}^{r\times r}$, $\mathit{B}\in {\mathbb{C}}^{r\times m}$, $\mathit{C}\in {\mathbb{C}}^{p\times r}$, $\mathit{D}\in {\mathbb{C}}^{p\times m}$ are the coefficient matrices.

Applying the Laplace transformation to (17) yields the $\mathbf{y}\left(\lambda \right)=\left[\mathit{C}{\mathit{A}}^{-1}\left(\lambda \right)\mathit{B}+\mathit{D}\right]\mathbf{u}\left(\lambda \right)$. If the leading coefficient matrix ${\mathit{A}}_0$ is nonsingular ($\det \left({\mathit{A}}_0\right)\ne 0$) then the associated matrix polynomial $\mathit{A}\left(\lambda \right)={⅀}_{k=0}^{\mathcal{l}}{\mathit{A}}_{\mathcal{l}-k}{\lambda }^k$ is said to be monic else we say that $\mathit{A}\left(\lambda \right)$ is non-monic matrix polynomial. The eigenvalue of $\mathit{A}\left(\lambda \right)$ is complex number ${\lambda }_i\in \mathbb{C}$ such that $\det \left(\mathit{A}\left({\lambda }_i\right)\right)=0$ in which case the left null space of $\mathit{A}\left({\lambda }_i\right)$ is called the left eigenspace, the right null space of $\mathit{A}\left({\lambda }_i\right)$ is called the right eigenspace, each vector in the left eigenspace is called left eigenvector and each vector in the right eigenspace is called right eigenvector corresponding to ${\lambda }_i$ [13,18].

• Controllability of Higher Order Systems: The state-space system in (17) or the tuple $\left({\mathit{A}}_0,{\mathit{A}}_1,\dots ,{\mathit{A}}_{\mathcal{l}},\mathit{B}\right), \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} {\mathit{A}}_i\in {\mathbb{C}}^{r\times r} \mathrm{a}\mathrm{n}\mathrm{d} \mathit{B}\in {\mathbb{C}}^{r\times m}$ is called controllable if it is possible to derive the system into any desired state at any given time by the proper selection of the input. This is fundamentally characterized by a rank condition [27].

  $$\mathbf{y}\left(\lambda \right)=\left[\mathit{C}{\mathit{A}}^{-1}\left(\lambda \right)\mathit{B}+\mathit{D}\right]\mathbf{u}\left(\lambda \right) \mathrm{i}\mathrm{s} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e} \mathrm{i}\mathrm{f}\mathrm{f}: \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left[\mathit{A}\left(\lambda \right) \mathit{B}\right]=r \forall \lambda \in \mathbb{C}$$

  (18)
• Observability of Higher Order Systems: A linear autonomous system is considered observable if each distinct initial condition leads to a unique output trajectory, allowing the reconstruction of the initial state from output measurements. In contrast to controllability (which is concerned with the set of reachable states from a given input) observability addresses the ability to infer the system’s internal state based solely on external outputs. Similarly, this is also characterized by a rank condition

  $$\mathbf{y}\left(\lambda \right)=\left[\mathit{C}{\mathit{A}}^{-1}\left(\lambda \right)\mathit{B}+\mathit{D}\right]\mathbf{u}\left(\lambda \right) \mathrm{i}\mathrm{s} \mathrm{o}\mathrm{b}\mathrm{s}\mathrm{e}\mathrm{r}\mathrm{v}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e} \mathrm{i}\mathrm{f}\mathrm{f}: \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}{\left[{\mathit{A}}^{⟙}\left(\lambda \right) {\mathit{C}}^{⟙}\right]}^{⟙}=r \forall \lambda \in \mathbb{C}$$

  (19)

  for observability is analogous to (18) [19,29].
• Stabilizability of Higher Order Systems: A system described by a general state-space realization (17) is stabilizable if it is possible to design an input that ensures the state converges to zero asymptotically, regardless of the initial condition. Although controllability guarantees stabilizability, the reverse is not necessarily true [13,27]. Verifying stabilizability typically reduces to checking a certain conditions.

  $$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left[\mathit{A}\left(\lambda \right) \mathit{B}\right]=r \forall \lambda \in {\mathbb{C}}_{+}$$

  (20)

This condition mirrors the controllability criterion, with the distinction that the rank test is required only over the closed right half of the complex plane.

• In discrete-time systems, the stability depends on the moduli of the eigenvalues of the associated $\mathit{A}\left(\lambda \right)$, not their real parts. The system is stable if all eigenvalues lie strictly inside the unit circle, i.e., the spectral radius of $\mathit{A}\left(\lambda \right)$ is less than one.

  $$\rho \left(\mathit{A}\left(\lambda \right)\right)=\max \left\{\left|\lambda \right|: \lambda \in \mathbb{C} \mathrm{s}.\mathrm{t}. \det \left(\mathit{A}\left(\lambda \right)\right)=0\right\}$$

  (21)

  This spectral radius $\rho$ determines the asymptotic decay rate of the system [8,24].
• The equivalent characterizations of controllability and observability in discrete-time systems mirror those of the continuous case. However, for stabilizability, the rank conditions must be verified on or outside the unit circle.

  $$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left[\mathit{A}\left(\lambda \right) \mathit{B}\right]=r \forall \lambda \in \mathbb{C} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \left|\lambda \right|\ge 1$$

  (22)
• Without loss of generality, the rank tests can be equivalently formulated by replacing the matrix polynomial $\mathit{A}\left(\lambda \right)$ with the linear pencil ${\mathit{C}}_1\left(\lambda \right)=\lambda {\mathit{E}}_c-{\mathit{A}}_c$, and the input matrix $\mathit{B}$ with the structured matrix ${\mathit{B}}_c={\left[\mathbf{0},\dots ,\mathbf{0},\mathit{I}\right]}^{⟙}$, see [27].

• Robustness of Higher Order Systems: Since state-space models often approximate complex systems affected by uncertainties, it is essential that fundamental properties remain robust under small perturbations [5,24]. One way to quantify this robustness is through the distance to stability, defined as follows

  $$\beta \left(\mathit{A}\left(\lambda \right),\mathit{\gamma }\right)=\inf \left\{{‖∆\mathit{A}‖}_2: \mathrm{s}.\mathrm{t}. \mathit{A}\left(\mathit{\gamma },∆\mathit{A},\lambda \right) \mathrm{i}\mathrm{s} \mathrm{u}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\right\}$$

  (23)

  where ${‖∆\mathit{A}‖}_2={‖∆{\mathit{A}}_{\mathcal{l}},∆{\mathit{A}}_{\mathcal{l}-1},\dots ,∆{\mathit{A}}_0‖}_2$, $\mathit{A}\left(\mathit{\gamma },∆\mathit{A},\lambda \right)=\left({\mathit{A}}_{\mathcal{l}}+{\gamma }_{\mathcal{l}}∆{\mathit{A}}_{\mathcal{l}},\dots ,{\mathit{A}}_0+{\gamma }_0∆{\mathit{A}}_0\right)$. The vector $\mathit{\gamma }=\left[{\gamma }_{\mathcal{l}},\dots ,{\gamma }_1,{\gamma }_0\right]$ consists of nonnegative scalers not all zero.

The vector $\mathit{\gamma }$ is introduced to restrict and scale perturbations across selected coefficient matrices. Setting certain ${\gamma }_i=0$ excludes those terms from perturbation, while choosing $\mathit{\gamma }=\left[{‖{\mathit{A}}_{\mathcal{l}}‖}_2 ,\dots ,{‖{\mathit{A}}_1‖}_2 ,{‖{\mathit{A}}_0‖}_2\right]$ allows for relative perturbations. Let ${\mathbb{C}}_g$ and ${\mathbb{C}}_b$ denote the stable and unstable regions, respectively, with boundary $\partial {\mathbb{C}}_b$. Then, the distance to instability is

$$\beta \left(\mathit{A}\left(\lambda \right),\mathit{\gamma }\right)=\underset{\lambda \in \partial {\mathbb{C}}_{\mathrm{b}}}{\inf }\mathcal{V}\left(\lambda ,\mathit{A},\mathit{\gamma }\right)$$

(24)

with $\mathcal{V}=\inf \left\{{‖∆\mathit{A}‖}_2: \mathrm{s}.\mathrm{t}. \det \left(\left(\mathit{A}+∆\mathit{A}\right)\left(\lambda \right)\right)=0\right\}$ and $∆\mathit{A}=\mathit{A}\left(\lambda \right)={⅀}_{k=0}^{\mathcal{l}}\left({\gamma }_k∆{\mathit{A}}_k\right){\lambda }^{\mathcal{l}-k}$. The resulting optimization task corresponds to a structured singular value problem, allowing the final expression to be simplified as follows.

$$\beta \left(\mathit{A}\left(\lambda \right),\mathit{\gamma }\right)=\underset{\lambda \in \partial {\mathbb{C}}_{\mathrm{b}}}{\inf }{\sigma }_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(\mathit{A}\left(\lambda \right)/p_{\mathit{\gamma }}\left(\left|\lambda \right|\right)\right)$$

(25)

where $p_{\mathit{\gamma }}\left(x\right)=\sqrt{{⅀}_{k=0}^{\mathcal{l}}{\gamma }_k^2{x}^{2k}}$.

3. Descriptor Systems in Modern Control Theory

Descriptor linear systems form a fundamental branch within modern control theory and have gained significant traction over the past few decades. Despite the theoretical depth and wide applicability of the subject, only a limited number of comprehensive references exist, notably the works of Campbell [2] and Liyi Dai [3]. Such systems naturally arise in diverse disciplines including power grids, electrical circuits, aerospace systems, chemical processes, socioeconomic modeling, biological networks, and time-series analysis. In particular, many electrical and dynamic network models are best represented by descriptor systems. In this work, we focus on linear differential-algebraic equations with constant coefficients, expressed in the form:

$$\mathit{E}\dot{\mathbf{x}}\left(t\right)=\mathit{A}\mathbf{x}\left(t\right)+\mathit{B}\mathbf{u}\left(t\right) \mathrm{a}\mathrm{n}\mathrm{d} \mathbf{y}\left(t\right)=\mathit{C}\mathbf{x}\left(t\right)+\mathit{D}\mathbf{u}\left(t\right)$$

(26)

with $\mathit{E},\mathit{A}\in {\mathbb{C}}^{n\times n}$ and $\mathit{B}\in {\mathbb{C}}^{n\times m}$, and for linear discrete time descriptor system we have

$$\mathit{E}\mathbf{x}\left(k+1\right)=\mathit{A}\mathbf{x}\left(k\right)+\mathit{B}\mathbf{u}\left(k\right) \mathrm{a}\mathrm{n}\mathrm{d} \mathbf{y}\left(k\right)=\mathit{C}\mathbf{x}\left(k\right)+\mathit{D}\mathbf{u}\left(k\right)$$

(27)

where $\mathbf{x}\left(k\right)\in {\mathbb{R}}^n$ is the state; $\mathbf{u}\left(k\right)\in {\mathbb{R}}^m$ is the control input, $\mathbf{y}\left(k\right)\in {\mathbb{R}}^p$ is the output. The matrix $\mathit{E}\in {\mathbb{R}}^{n\times n}$ may be singular, we shall assume that $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\mathit{E}\right)= r\le n$. $\mathit{A}$, $\mathit{B}$ and $\mathit{C}$ are known real constant matrices with appropriate dimensions.

3.1. Structural Properties of Descriptor Systems

The properties of systems (26) and (27) are well understood for more than one century, in particular by the work of Weierstra β 1868 and Kronecker (1890) [11,12]. The reason is that Equations (26) and (27) can be treated by purely algebraic techniques. In the following, we describe the main aspects of this approach.

Lemma 3. 

([3,25]) A descriptor system is said to be completely controllable if and only if:

■

$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\left[\lambda \mathit{E}-\mathit{A},\mathit{B}\right]\right)=n$ for all finit $\lambda \in \mathbb{C}$ and $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\left[\mathit{E} \mathit{B}\right]\right)=n$

A descriptor system is said to be completely observable if and only if:  

■

$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left({\left[\lambda {\mathit{E}}^{⟙}-{\mathit{A}}^{⟙},{\mathit{C}}^{⟙}\right]}^{⟙}\right)=n$ for all finit $\lambda \in \mathbb{C}$ and $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left({\left[{\mathit{E}}^{⟙}{\mathit{C}}^{⟙}\right]}^{⟙}\right)=n$

We shall assume that $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\mathit{E}\right)= r\le n$. Also we assume that $\mathit{A}$,$\mathit{B}$ and $\mathit{C}$ are known real constant matrices with appropriate dimensions. For the sake of simplicity, we use $\left(\mathit{E},\mathit{A},\mathit{B},\mathit{C}\right)$ to denote the descriptor system in (26) and (27). We define the generalized spectral abscissa of the pair $\left(\mathit{E},\mathit{A}\right)$ as $\rho \left(\mathit{E},\mathit{A}\right)=\underset{\lambda \in s|\det \left(s\mathit{EA}\right)=0}{\max }\mathrm{R}\mathrm{e}\left(\lambda \right)$. For discrete time case we have $\rho \left(\mathit{E},\mathit{A}\right)=\underset{\lambda \in z|\det \left(z\mathit{EA}\right)=0}{\max }\left|\lambda \right|$ and the pair $\left(\mathit{E},\mathit{A}\right)$ is said to be stable if $\rho \left(\mathit{E},\mathit{A}\right)<1$.

Definition 8. 

([4,7,16]) The pair $\left(\mathit{E},\mathit{A}\right)$ is said to be regular if $\det \left(\lambda \mathit{E}-\mathit{A}\right)$ is not identically zero. The pair $\left(\mathit{E},\mathit{A}\right)$ is said to be impulse-free if $\mathrm{d}\mathrm{e}\mathrm{g}\left(\det \left(\lambda \mathit{E}-\mathit{A}\right)\right) = \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\mathit{E}\right)$. The pair $\left(\mathit{E},\mathit{A}\right)$ is said to be stable if all the roots of $\det \left(\lambda \mathit{E}-\mathit{A}\right)=0$ have negative real parts. The pair $\left(\mathit{E},\mathit{A}\right)$ is said to be admissible if it is regular, impulse-free and stable.

Lemma 4. 

([1,2,3,4,8]) The pair $\left(\mathit{E},\mathit{A}\right)$ is regular if and only if there exist two nonsingular matrices $\mathit{M}$ and $\mathit{N}$ such that: $\mathit{M}\mathit{E}\mathit{N}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\mathit{I},\mathcal{J}\right) and \mathit{M}\mathit{A}\mathit{N}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}({\mathit{A}}_1,\mathit{I})$ where $\mathcal{J}\in {\mathbb{R}}^{k\times k}$ is a nilpotent matrix which satisfy the following properties $\det \left(\mathcal{J}\right)=\mathrm{t}\mathrm{r}\left(\mathcal{J}\right)=0$, $\det \left(\lambda \mathit{I}-\mathcal{J}\right)={\lambda }^k$ and $\det \left(\lambda \mathcal{J}-\mathit{I}\right)={\left(-1\right)}^k \forall \lambda \in \mathbb{C}$.

This is called: the Slow-Fast or the Weierstrass-Kronecker canonical decompositions.

Lemma 5. 

([3,5]) Suppose that the pair $\left(\mathit{E},\mathit{A}\right)$ is regular, and two nonsingular matrices $\mathit{M} and \mathit{N}$ are found such that slow-fast decomposition holds, then we have:

The pair $\left(\mathit{E},\mathit{A}\right)$ is impulse-free if and only if $\mathcal{J}=\mathbf{0}$. And is stable if and only if $\rho \left(\mathit{A}\right)<0$.

The pair $\left(\mathit{E},\mathit{A}\right)$ is admissible if and only if $\mathcal{J}=\mathbf{0}$ and $\rho \left(\mathit{A}\right)<0$.

Theorem 6. 

([25]) Given the linear descriptor system $\left(\mathit{E},\mathit{A},\mathit{B},\mathit{C}\right) with \mathit{E},\mathit{A}\in {\mathbb{C}}^{n\times n}$, $\mathit{B}\in {\mathbb{C}}^{n\times m}$, $\mathit{C}\in {\mathbb{C}}^{p\times n}$ and $\left(\mathit{E},\mathit{A}\right)$ is regular, then there exist two nonsingular matrices $\mathit{M}$ and $\mathit{N}$ such that

$${\mathit{E}}_r=\mathit{M}\mathit{E}\mathit{N}={\mathit{I}}_{n_1}⨁\mathcal{J}; {\mathit{A}}_r=\mathit{M}\mathit{A}\mathit{N}={\mathit{A}}_1⨁{\mathit{I}}_{n_2}; {\mathit{B}}_r=\mathit{M}\mathit{B}={\left[{\mathit{B}}_1^{⟙}{\mathit{B}}_2^{⟙}\right]}^{⟙}; {\mathit{C}}_r=\mathit{C}\mathit{N}=\left[{\mathit{C}}_1{\mathit{C}}_2\right]$$

where $n_1+n_2=n$, and the matrix $\mathcal{J}\in {\mathbb{C}}^{n_2\times n_2}$ is nilpotent.

According to Theorem 5, the linear descriptor system (26) can be decomposed into the following systems:

$$\begin{array}{c}\begin{array}{l}\begin{array}{l}\begin{array}{l}\mathrm{T}\mathrm{h}\mathrm{e} \mathrm{s}\mathrm{l}\mathrm{o}\mathrm{w} \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}:{\dot{\mathbf{x}}}_1\left(t\right)={\mathit{A}}_1{\mathbf{x}}_1\left(t\right)+{\mathit{B}}_1\mathbf{u}\left(t\right) \mathrm{a}\mathrm{n}\mathrm{d} {\mathbf{y}}_1\left(t\right)={\mathit{C}}_1{\mathbf{x}}_1\left(t\right)+{\mathit{D}}_1\mathbf{u}\left(t\right)\hfill \\ \mathrm{T}\mathrm{h}\mathrm{e} \mathrm{f}\mathrm{a}\mathrm{s}\mathrm{t} \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}:\mathcal{J}{\dot{\mathbf{x}}}_2\left(t\right)={\mathbf{x}}_2\left(t\right)+{\mathit{B}}_2\mathbf{u}\left(t\right) \mathrm{a}\mathrm{n}\mathrm{d} {\mathbf{y}}_2\left(t\right)={\mathit{C}}_2{\mathbf{x}}_2\left(t\right)+{\mathit{D}}_2\mathbf{u}\left(t\right)\end{array}\\ \end{array}\hfill \end{array}\end{array}$$

(28)

And the output systems is given by $\mathbf{y}\left(t\right)={\mathbf{y}}_1\left(t\right)+{\mathbf{y}}_2\left(t\right)$. This system is called the standard decomposition form. The controllability matrix of linear descriptor system related to the slow and fast subsystems are given by ${\mathsf{\Omega }}_{c1}=\mathcal{C}\left({\mathit{A}}_1,{\mathit{B}}_1\right)=\mathrm{r}\mathrm{o}\mathrm{w}{\left({\mathit{A}}_1^{i-1}{\mathit{B}}_1\right)}_{i=1}^{n_1}$ and ${\mathsf{\Omega }}_{c2}=\mathcal{C}\left(\mathcal{J},{\mathit{B}}_2\right)=\mathrm{r}\mathrm{o}\mathrm{w}{\left({\mathcal{J}}^{i-1}{\mathit{B}}_2\right)}_{i=1}^{n_2}$, respectively.

Theorem 7. 

([2,4]) The regular linear descriptor system is completely controllable if and only if $\mathcal{R}={\mathcal{R}}_1⨁{\mathcal{R}}_2={\mathbb{R}}^n$ where $⨁$ is the external direct sum of two linear spaces of vectors and ${\mathcal{R}}_1=\mathrm{I}\mathrm{m}\left(\mathrm{r}\mathrm{o}\mathrm{w}{\left({\mathit{A}}_1^{i-1}{\mathit{B}}_1\right)}_{i=1}^{n_1}\right)={\mathbb{R}}^{n_1}; and {\mathcal{R}}_2=\mathrm{I}\mathrm{m}\left(\mathrm{r}\mathrm{o}\mathrm{w}{\left({\mathcal{J}}^{i-1}{\mathit{B}}_2\right)}_{i=1}^{n_2}\right)={\mathbb{R}}^{n_2}$.

The slow dynamics subsystem is controllable if and only if $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left[\mathrm{r}\mathrm{o}\mathrm{w}{\left({\mathit{A}}_1^{i-1}{\mathit{B}}_1\right)}_{i=1}^{n_1}\right]=n_1$ or equivalently $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\left[\lambda \mathit{I}-{\mathit{A}}_1,{\mathit{B}}_1\right]\right)=n_1$ for all finit $\lambda \in \mathbb{C}$. The fast dynamics subsystem is controllable if and only if $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left[\mathrm{r}\mathrm{o}\mathrm{w}{\left({\mathcal{J}}^{i-1}{\mathit{B}}_2\right)}_{i=1}^{n_2}\right]=n_2$ or equivalently $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\left[\mathcal{J},{\mathit{B}}_2\right]\right)=n_2$.

The linear descriptor system is completely controllable if and only if both its slow and fast sub systems are both controllable. The observability matrices of slow and fast subsystems of linear descriptor system are given by ${\mathsf{\Omega }}_{o1}=\mathcal{Q}\left({\mathit{A}}_1,{\mathit{C}}_1\right)=\mathrm{c}\mathrm{o}\mathrm{l}{\left({\mathit{C}}_1{\mathit{A}}_1^{i-1}\right)}_{i=1}^{n_1}$ and ${\mathsf{\Omega }}_{o2}=\mathcal{Q}\left(\mathcal{J},{\mathit{C}}_2\right)=\mathrm{c}\mathrm{o}\mathrm{l}{\left({\mathit{C}}_2{\mathcal{J}}^{i-1}\right)}_{i=1}^{n_2}$, respectively.

Theorem 8. 

([3]) The regular linear descriptor system is completely observable if and only if ${\mathcal{O}}_1⨁{\mathcal{O}}_2=\left\{\mathbf{0}\right\}$ with ${\mathcal{O}}_1=\mathrm{k}\mathrm{e}\mathrm{r}\left[\mathrm{c}\mathrm{o}\mathrm{l}{\left({\mathit{C}}_1{\mathit{A}}_1^{i-1}\right)}_{i=1}^{n_1}\right]=\left\{\mathbf{0}\right\} and {\mathcal{O}}_2=\mathrm{k}\mathrm{e}\mathrm{r}\left[\mathrm{c}\mathrm{o}\mathrm{l}{\left({\mathit{C}}_2{\mathcal{J}}^{i-1}\right)}_{i=1}^{n_2}\right]=\left\{\mathbf{0}\right\}$. The slow subsystem is observable if and only if $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left[\mathrm{c}\mathrm{o}\mathrm{l}{\left({\mathit{C}}_1{\mathit{A}}_1^{i-1}\right)}_{i=1}^{n_1}\right]=n_1$ or equivalently $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left({\left[\lambda \mathit{I}-{\mathit{A}}_1^{⟙},{\mathit{C}}_1^{⟙}\right]}^{⟙}\right)=n_1$ for all finit $\lambda \in \mathbb{C}$. The fast dynamics subsystem is controllable if and only if $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left[\mathrm{c}\mathrm{o}\mathrm{l}{\left({\mathit{C}}_2{\mathcal{J}}^{i-1}\right)}_{i=1}^{n_2}\right]=n_2$ or equivalently $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left({\left[{\mathcal{J}}^{⟙}{\mathit{C}}_2^{⟙}\right]}^{⟙}\right)=n_2$.

We now present the admissibility condition using Linear Matrix Inequalities (LMIs), without requiring matrix decomposition. To proceed, we first introduce key preliminary results needed for the analysis.

Lemma 6. 

([10]) Given any real square matrix $\mathcal{X}$ with appropriate dimensions. The matrix measure $\mu \left(\mathcal{X}\right)$ defined as $\mu \left(\mathcal{X}\right)=\underset{\theta \to 0}{\lim }\left[\left(\theta \mathcal{X}+\mathit{I}\right)-1\right]/\theta$ has the properties.

$$\left(i\right)-‖\mathcal{X}‖\le {\lambda }_{max}\left(\mathcal{X}\right)\le \mu \left(\mathcal{X}\right)\le ‖\mathcal{X}‖; \mathrm{a}\mathrm{n}\mathrm{d} \left(ii\right) \mu \left(\mathcal{X}\right)={\displaystyle \frac{1}{2}}{\lambda }_{max}\left({\mathcal{X}}^{⟙}+\mathcal{X}\right)$$

(29)

We are now ready to present a necessary and sufficient condition for the admissibility of the descriptor system given in (26) and (27), formulated in terms of LMIs. While a similar condition was reported in [46], we provide an alternative derivation here.

Theorem 9. 

([4,5]) The pair $\left(\mathit{E},\mathit{A}\right)$ is admissible if and only if there exists a matrix $\mathit{P}$ such that

$${\mathit{E}}^{⟙}\mathit{P}={\mathit{P}}^{⟙}\mathit{E}\ge 0; {\mathit{P}}^{⟙}\mathit{A}+{\mathit{A}}^{⟙}\mathit{P}<0$$

(30)

the pair $\left({\mathit{E}}^{⟙},{\mathit{A}}^{⟙}\right)$ is admissible if and only if there exists a matrix $\mathit{P}$ such that

$$\mathit{E}\mathit{P}={\mathit{P}}^{⟙}{\mathit{E}}^{⟙}\ge 0; \mathit{P}\mathit{A}+{\mathit{A}}^{⟙}{\mathit{P}}^{⟙}<0$$

(31)

for discrete time case the pair $\left(\mathit{E},\mathit{A}\right)$ is admissible if and only if there exists a matrix $\mathit{P}={\mathit{P}}^{⟙}$ such that ${\mathit{E}}^{⟙}\mathit{P}\mathit{E}\ge 0$ and ${\mathit{A}}^{⟙}\mathit{P}\mathit{A}- {\mathit{E}}^{⟙}\mathit{P}\mathit{E}<0$. In this case, the matrix $\mathit{P}$ is nonsingular.

Theorem 10. 

([5]) The pair $\left(\mathit{E},\mathit{A}\right)$ is admissible if and only if there exist matrices $\mathit{P}>0$ and $\mathit{Q}$ such that

$${\left(\mathit{P}\mathit{E}+\mathit{S}\mathit{Q}\right)}^{⟙}\mathit{A}+{\mathit{A}}^{⟙}\left(\mathit{P}\mathit{E}+\mathit{S}\mathit{Q}\right)<\mathbf{0}$$

(32)

where $\mathit{S}\in {\mathbb{R}}^{n\times \left(n-r\right)}$ is any matrix with full column rank and satisfies ${\mathit{E}}^{⟙}\mathit{S}=\mathbf{0}$. For discrete time case the pair $\left(\mathit{E},\mathit{A}\right)$ is admissible if and only if there exist matrices $\mathit{P}>0$ and $\mathit{Q}$ such that

$${\mathit{A}}^{⟙}\mathit{P}\mathit{A}-{\mathit{E}}^{⟙}\mathit{P}\mathit{E}+\mathit{Q}{\mathit{S}}^{⟙}\mathit{A}+{\mathit{A}}^{⟙}\mathit{S}{\mathit{Q}}^{⟙}<\mathbf{0}$$

(33)

where $\mathit{S}\in {\mathbb{R}}^{n\times \left(n-r\right)}$ is any matrix with full column rank and satisfies ${\mathit{E}}^{⟙}\mathit{S}=\mathbf{0}$.

Given the fundamental role of input-output transmission zeros in system stability and their intrinsic connection to the system’s characteristic roots, it is both relevant and necessary to examine them thoroughly.

Definition 9. 

Consider the square descriptor linear system. Any finite $\alpha \in \mathbb{C}$ satisfying $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left[\alpha \mathit{E}-\mathit{A}, \mathit{B}\right]<n$ is called an input transmission zero of the system, and any finite $\beta \in \mathbb{C}$ satisfying $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}{\left[\beta {\mathit{E}}^{⟙}-{\mathit{A}}^{⟙}, {\mathit{C}}^{⟙}\right]}^{⟙}<n$ is called an output transmission zero of the system.

Proposition 4. 

([4,5]) The finite poles of a square descriptor linear system coincide with its input (output) transmission zeros. Furthermore, the admissibility of a continuous-time descriptor system is closely linked to a class of generalized Lyapunov matrix equations, typically expressed in the following form:

$${\mathit{E}}^{⟙}\mathit{X}\mathit{A}+{\mathit{A}}^{⟙}\mathit{X}\mathit{E}=-{\mathit{E}}^{⟙}\mathit{Y}\mathit{E}$$

(34)

where $\mathit{Y}= {\mathit{Y}}^{⟙}>0$.

Theorem 11. 

([5,16]) Let $\left(\mathit{E},\mathit{A}\right)$ be regular. Then the following hold:

1. 

If there exist  $\mathit{X}={\mathit{X}}^{⟙}\ge 0$ and  $\mathit{Y}={\mathit{Y}}^{⟙}>0$ satisfying (34), then$\left(\mathit{E},\mathit{A}\right)$ is admissible.

2. 

If  $\left(\mathit{E},\mathit{A}\right)$ is admissible, then for each  $\mathit{Y}>0$ there exists$\mathit{X}>0$ satisfying (34).

A stability condition for continuous-time linear descriptor systems (26) was established by Lewis F.L. [7,16], using a generalized Lyapunov matrix equation of the form ${\mathit{A}}^{⟙}\mathit{X}\mathit{E}+{\mathit{E}}^{⟙}\mathit{X}\mathit{A}+{\mathit{E}}^{⟙}{\mathit{C}}^{⟙}\mathit{C}\mathit{E}=\mathbf{0}$, where $\mathit{C}$ represents the output measurement matrix.

3.2. Feedback Design in Descriptor Linear Systems

Controlling a system involves modifying its dynamics via a controller to ensure it behaves according to specified objectives. As such, the design of an appropriate controller is fundamental. To enable real-time adaptation, most controllers are implemented using feedback configurations.

➢

Static Feedback: The simplest form of static state feedback of the system (26) takes the form $\mathbf{u}\left(t\right)=\mathit{K}\mathbf{x}\left(t\right)+\mathbf{r}\left(t\right)$, where $\mathit{K}\in {\mathbb{R}}^{m\times n}$ is the feedback gain matrix to be designed, and, $\mathbf{r}$ is an external input [38,39]. When this controller is applied, the system dynamics are modified accordingly, leading to a closed-loop system

$$\dot{\mathbf{x}}\left(t\right)={\mathit{A}}_{new}\mathbf{x}\left(t\right)+\mathit{B}\mathbf{r}\left(t\right) \mathrm{a}\mathrm{n}\mathrm{d} \mathbf{y}\left(t\right)={\mathit{C}}_{new}\mathbf{x}\left(t\right)+\mathit{D}\mathbf{r}\left(t\right)$$

(35)

with ${\mathit{A}}_{new}=\mathit{A}+\mathit{B}\mathit{K}$, ${\mathit{C}}_{new}=\mathit{C}+\mathit{D}\mathit{K}$. This resulting system (35) is referred to as the closed-loop system under state feedback. The feedback matrix $\mathit{K}$ must be designed to ensure the closed-loop system satisfies specific performance and stability requirements.

➢

Derivative Feedback: A pure generalized derivative feedback law can be formulated as $\mathbf{u}\left(t\right)=-\mathit{K}\dot{\mathbf{x}}\left(t\right)+\mathbf{r}\left(t\right)$ where $\mathit{K}\in {\mathbb{R}}^{m\times n}$ denotes the derivative feedback gain matrix [15,22]. Applying this control law yields the closed-loop dynamics:

$${\mathit{E}}_{new}\dot{\mathbf{x}}\left(t\right)=\mathit{A}\mathbf{x}\left(t\right)+\mathit{B}\mathbf{r}\left(t\right) \mathrm{a}\mathrm{n}\mathrm{d} \mathbf{y}\left(t\right)=\mathit{C}\mathbf{x}\left(t\right)+{\mathit{C}}_{new}\dot{\mathit{x}}\left(t\right)+\mathit{D}\mathbf{r}\left(t\right)$$

(36)

with ${\mathit{E}}_{new}=\mathit{E}+\mathit{B}\mathit{K}$, and ${\mathit{C}}_{new}=-\mathit{D}\mathit{K}$.

➢

P-D Feedback: A general feedback strategy for descriptor systems involves both state and derivative components, and is given by $\mathbf{u}\left(t\right)={\mathit{K}}_p\mathbf{x}\left(t\right)+{\mathit{K}}_d\dot{\mathbf{x}}\left(t\right)+\mathbf{r}\left(t\right)$. ${\mathit{K}}_p\in {\mathbb{R}}^{m\times n}$ and ${\mathit{K}}_d\in {\mathbb{R}}^{m\times n}$ are two parameter matrices to be designed, and are often called the proportional and derivative feedback gain matrices [13].

➢

Dynamic Feedback: A generic dynamical compensator for linear descriptor system is represented in MFD as $\mathit{C}\left(\lambda \right)={\mathit{D}}_c^{-1}\left(\lambda \right){\mathit{N}}_c\left(\lambda \right)$, or in state equation

$${\mathit{E}}_{cp}\dot{\mathit{\vartheta }}\left(t\right)={\mathit{A}}_{cp}\mathit{\vartheta }\left(t\right)+{\mathit{B}}_{cp}\mathsf{\epsilon }\left(t\right) \mathrm{a}\mathrm{n}\mathrm{d} \mathbf{y}\left(t\right)={\mathit{C}}_{cp}\mathit{\vartheta }\left(t\right)+{\mathit{D}}_{cp}\mathsf{\epsilon }\left(t\right)$$

(37)

where $\mathit{\vartheta }\left(t\right)$ is the controller state. If the matrix ${\mathit{E}}_{cp}$ is square and nonsingular (implying ${\mathit{D}}_c\left(\lambda \right)$ is a monic matrix polynomial) the compensator is termed a normal dynamical compensator. Otherwise, it is referred to as a descriptor dynamical compensator [18].

➢

Feedback via Lyapunov Method: The stabilizing controller design can be formulated as a convex optimization problem characterized by linear matrix inequalities (LMI) [46].

Now, we first consider the design of a state feedback controller for continuous descriptor systems such that the closed-loop system is admissible. Then the design of dynamic output feedback controllers is investigated.

Theorem 12. 

([5]) Consider the continuous descriptor system. There exists a state feedback controller such that the closed-loop system is admissible if and only if there exist matrices $\mathit{P}>0$, $\mathit{Q}$ and $\mathit{Y}$ such that

$$\left(\mathit{E}{\mathit{P}}^{⟙}+{\mathit{Q}}^{⟙}{\mathit{S}}^{⟙}\right){\mathit{A}}^{⟙}+\mathit{A}\left(\mathit{P}{\mathit{E}}^{⟙}+\mathit{S}\mathit{Q}\right)+\mathit{B}\mathit{Y}+{\mathit{Y}}^{⟙}{\mathit{B}}^{⟙}<0$$

(38)

where $\mathit{S}\in {\mathbb{R}}^{n\times \left(n-r\right)}$ is any matrix with full column rank and satisfies ${\mathit{E}}^{⟙}\mathit{S}=\mathbf{0}$. In this case, we can assume that the matrix $\mathsf{\Omega }=\left(\mathit{P}{\mathit{E}}^{⟙}+\mathit{S}\mathit{Q}\right)$ is nonsingular, then a stabilizing state feedback controller can be chosen as $\mathbf{u}\left(t\right)=\mathit{Y} {\mathsf{\Omega }}^{-1}\mathbf{x}\left(t\right)$.

Now, we present the results of the stabilizing discrete descriptor systems. The class of linear discrete descriptor systems to be considered is $\mathit{E}\mathbf{x}\left(k+1\right)=\mathit{A}\mathbf{x}\left(k\right)+\mathit{B}\mathbf{u}\left(k\right)$. For the discrete-time descriptor system, we assume that all the state variables are available for state feedback, and consider the following state feedback controller: $\mathbf{u}\left(k\right)=\mathit{K}\mathbf{x}\left(k\right)$, $\mathit{K}\in {\mathbb{R}}^{m\times n}$. The purpose of the stabilization problem is the design of a state feedback controller such that the closed loop system is admissible [8,9,20].

Theorem 13. 

([4,5]) Consider the discrete-time descriptor system. There exists a state feedback controller such that the closed-loop system is admissible if and only if there exist a scalar $\delta >0$, matrices $\mathit{P}>0$, $\mathit{Q}$ such that

$$\mathit{Q}{\mathit{S}}^{⟙}\mathit{A}+{\mathit{A}}^{⟙}\mathit{S}{\mathit{Q}}^{⟙}{\mathit{E}}^{⟙}\mathit{P}\mathit{E}+{\mathit{A}}^{⟙}\mathit{P}\mathit{A}-\left(\mathit{Q}{\mathit{S}}^{⟙}+{\mathit{A}}^{⟙}\mathit{P}\right)\mathit{B}\left({\mathit{B}}^{⟙}\mathit{P}\mathit{B}+\delta \mathit{I}\right){\mathit{B}}^{⟙}{\left(\mathit{Q}{\mathit{S}}^{⟙}+{\mathit{A}}^{⟙}\mathit{P}\right)}^{⟙}<0$$

(39)

where $\mathit{S}\in {\mathbb{R}}^{n\times \left(n-r\right)}$ is any matrix with full column rank and satisfies ${\mathit{E}}^{⟙}\mathit{S}=\mathbf{0}$. In this case the stabilizing state feedback controller is $\mathbf{u}\left(t\right)=-\left({\mathit{B}}^{⟙}\mathit{P}\mathit{B}+\delta \mathit{I}\right){\mathit{B}}^{⟙}{\left(\mathit{Q}{\mathit{S}}^{⟙}+{\mathit{A}}^{⟙}\mathit{P}\right)}^{⟙}\mathbf{x}\left(t\right)$.

Under the controllability/admissibility conditions of Theorems 12 and 13, Algorithm 1 constructs $\mathit{K}$ from the desired block spectral factors, so the closed loop satisfies regularity and stability, and (when combined with Section 4.3) becomes impulse-free by normalization.

4. The Proposed State Feedback Control Design Procedure

Given a large scale MIMO descriptor system of the form $\mathit{E}\mathbf{x}\left(t\right)=\mathit{A}\mathbf{x}\left(t\right)+\mathit{B}\mathbf{u}\left(t\right)$, and $\mathbf{y}\left(t\right)=\mathit{C}\mathbf{x}\left(t\right)+\mathit{D}\mathbf{u}\left(t\right)$ where $\mathbf{x}\left(t\right)\in {\mathbb{R}}^n$ is the state, $\mathbf{u}\left(t\right)\in {\mathbb{R}}^m$ is the control input, $\mathbf{y}\left(t\right)\in {\mathbb{R}}^p$ is the output. The matrix $\mathit{E}\in {\mathbb{R}}^{n\times n}$ may be singular $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\mathit{E}\right)=r$. $\mathit{A}$, $\mathit{B}$, $\mathit{C}$ and $\mathit{D}$ are known real constant matrices with an appropriate dimensions. For the sake of simplicity, sometimes we use $\left(\mathit{E},\mathit{A},\mathit{B},\mathit{C}\right)$ to denote the descriptor system.

Assumptions 1. 

Let we assume that: (i) $n/p$ and $n/m$ be an integer numbers. (ii) $n-m\le \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\mathit{E}\right)=r<n$. (iii) The triple $\left(\mathit{E},\mathit{A},\mathit{B}\right)$ is completely controllable.

Problem 1. 

Let matrices $\mathit{A},\mathit{E}\in {\mathbb{R}}^{n\times n}$ and $\mathit{B}\in {\mathbb{R}}^{n\times m}$ satisfy Assumption 1 find a parameterization for all the matrices $\mathit{K}\in {\mathbb{R}}^{m\times n}$ satisfying: $\exists {\lambda }_i\in \mathbb{C}$ and ${\mathit{v}}_i\in {\mathbb{C}}^n$ such that $\det \left({\lambda }_i\mathit{E}-{\mathit{A}}_d\right)=\det \left({\lambda }_i\mathit{E}-\mathit{A}+\mathit{B}\mathit{K}\right)$ and $\left({\lambda }_i\mathit{E}-\mathit{A} + \mathit{B}\mathit{K}\right){\mathbf{v}}_i=\mathbf{0}$. In other words, characterize the set

$$\mathcal{L}=\left\{\mathit{K}\in {\mathbb{R}}^{m\times n}| \sigma \left(\mathit{E},{\mathit{A}}_d\right)=\sigma \left(\mathit{E},\mathit{A}-\mathit{B}\mathit{K}\right)\right\}$$

where $\sigma \left(\mathit{E},\mathit{A}\right)$ denotes the spectrum of $\left(\mathit{E},\mathit{A}\right)$ and ${\mathit{A}}_d$ stands for the desired matrix.

4.1. Matrix Fraction Description from State Space

Consider now the system described by matrices $\left(\mathit{E},\mathit{A},\mathit{B},\mathit{C}\right)$ then we have

$$\mathit{C}{\left(\lambda \mathit{E}-\mathit{A}\right)}^{-1} \mathit{B}=\mathit{N}\left(\lambda \right){\mathit{D}}^{-1}\left(\lambda \right) ⟺ \mathit{C}{\left(\lambda \mathit{E}-\mathit{A}\right)}^{-1} \mathit{B}\mathit{D}\left(\lambda \right)=\mathit{N}\left(\lambda \right)$$

Now, since $\mathit{N}\left(\lambda \right)\in {\mathbb{C}}^{p\times m}\left[\lambda \right]$ is related to the output, and $\mathit{C}\in {\mathbb{C}}^{p\times n}$, so it is always possible to write: $\mathit{N}\left(\lambda \right)=\mathit{C}\mathcal{N}\left(\lambda \right)$ for some polynomial matrix $\mathcal{N}\left(\lambda \right)\in {\mathbb{C}}^{n\times m}\left[\lambda \right]$.

$$\begin{array}{c}\begin{array}{c}\mathit{C}{\left(\lambda \mathit{E}-\mathit{A}\right)}^{-1} \mathit{B}=\mathit{N}\left(\lambda \right){\mathit{D}}^{-1}\left(\lambda \right) ⟺ \mathit{B}\mathit{D}\left(\lambda \right)=\left(\lambda \mathit{E}-\mathit{A}\right)\mathcal{N}\left(\lambda \right)\end{array}\end{array}$$

We arrive the following final matrix equation:

$$\lambda \mathit{E}\mathcal{N}\left(\lambda \right)=\mathit{B}\mathit{D}\left(\lambda \right)+\mathit{A}\mathcal{N}\left(\lambda \right); \mathit{D}\left(\lambda \right)\in \mathbb{C}{\left[\lambda \right]}^{m\times m};\mathit{N}\left(\lambda \right)\in \mathbb{C}{\left[\lambda \right]}^{p\times m};\mathcal{N}\left(\lambda \right)\in \mathbb{C}{\left[\lambda \right]}^{n\times m}$$

(40)

where $\mathit{D}\left(\lambda \right)={⅀}_{i=0}^{\mathcal{l}}{\mathit{D}}_i{\lambda }^{\mathcal{l}-i};\mathit{N}\left(\lambda \right)={⅀}_{i=1}^{\mathcal{l}}{\mathit{N}}_i{\lambda }^{\mathcal{l}-i}$ and $\mathcal{N}\left(\lambda \right)={⅀}_{i=1}^{\mathcal{l}}{\mathcal{N}}_i{\lambda }^{\mathcal{l}-i}$

Expanding Equation (40), gives:

$$\mathit{B}{\mathit{D}}_i-\mathit{E}{\mathcal{N}}_{i+1}+\mathit{A}{\mathcal{N}}_i=\mathbf{0} i=1,\dots ,\mathcal{l}-1$$

(41)

with the boundary conditions $\mathit{B}{\mathit{D}}_0-\mathit{E}{\mathcal{N}}_1=\mathbf{0}$ and $\mathit{B}{\mathit{D}}_{\mathcal{l}}+\mathit{A}{\mathcal{N}}_{\mathcal{l}}=\mathbf{0}$. From Equation (41) solving from the above with back substitutions we obtain:

$$\left\{\begin{array}{l}{\mathbb{W}}_1\mathbb{D}=\mathbf{0} \hfill \\ {\mathbb{W}}_2\mathbb{N}=\left({\mathit{I}}_{\mathcal{l}+1}\otimes \mathit{B}\right)\mathbb{D}\end{array}\right. ⟺ \left[\begin{array}{cc} {\mathbb{W}}_1& 0\\ -\mathbb{B}& {\mathbb{W}}_2\end{array}\right]\left[\begin{array}{c}\mathbb{D}\\ \mathbb{N}\end{array}\right]=\left[\begin{array}{c}\mathbf{0}\\ \mathbf{0}\end{array}\right] \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \mathbb{B}=\left({\mathit{I}}_{\mathcal{l}+1}\otimes \mathit{B}\right)$$

(42)

where the operator $\otimes$ is the Kronecker product and

$$\begin{array}{l}{\mathbb{W}}_1=\left[\mathit{B} \left(\mathit{E}{\mathit{A}}^{-1}\right)\mathit{B} {\left(\mathit{E}{\mathit{A}}^{-1}\right)}^2\mathit{B} \dots {\left(\mathit{E}{\mathit{A}}^{-1}\right)}^{\mathcal{l}}\mathit{B}\right];\mathbb{D}={\left[{\mathit{D}}_0^{⟙} {\mathit{D}}_1^{⟙} {\mathit{D}}_2^{⟙}\dots {\mathit{D}}_{\mathcal{l}}^{⟙}\right]}^{⟙}\hfill \\ {\mathbb{W}}_2=\left[\begin{array}{c}\begin{array}{c}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}-\mathit{A}& \mathit{E}\\ 0& -\mathit{A}\end{array}& \begin{array}{c}0 \\ \mathit{E}\end{array}\end{array}& \begin{array}{c} \\ \end{array}\end{array}& \begin{array}{c}\mathit{⃝}\\ \end{array}\end{array}\\ \end{array}\\ \ddots \ddots \\ \begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc} & \\ \mathit{⃝}& \end{array}& \begin{array}{c} \\ \end{array}\end{array}& \begin{array}{c}-\mathit{A}\\ 0\end{array}\end{array}& \begin{array}{c}\mathit{E}\\ -\mathit{A}\end{array}\end{array}\end{array}\right];\mathbb{N}=\left[\begin{array}{c}\begin{array}{c}{\mathcal{N}}_1\\ {\mathcal{N}}_2\end{array}\\ ⋮\\ {\mathcal{N}}_{\mathcal{l}}\end{array}\right]; \left[\begin{array}{c}\begin{array}{c}{\mathit{N}}_1\\ {\mathit{N}}_2\end{array}\\ ⋮\\ {\mathit{N}}_{\mathcal{l}}\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}\mathit{C}{\mathcal{N}}_1\\ \mathit{C}{\mathcal{N}}_2\end{array}\\ ⋮\\ \mathit{C}{\mathcal{N}}_{\mathcal{l}}\end{array}\right] and \mathbb{B}\mathbb{D}=\left[\begin{array}{c}\begin{array}{c}\mathit{B}{\mathit{D}}_0\\ \mathit{B}{\mathit{D}}_1\end{array}\\ ⋮\\ \mathit{B}{\mathit{D}}_{\mathcal{l}}\end{array}\right]\end{array}$$

Key properties of the non-monic matrix polynomial $\mathit{D}\left(\lambda \right)\in \mathbb{C}{\left[\lambda \right]}^{m\times m}$ in the context of descriptor systems are summarized below, including its eigenstructure, factorizations, and relations to system pencils.

■

The number of finite eigenvalues $\eta$ of the non-monic matrix polynomial $\mathit{D}\left(\lambda \right)$ is given by: $\eta =n+\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left({\mathit{D}}_0\right)-m = m(\mathcal{l}-1)+ \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left({\mathit{D}}_0\right)$.

■

The numerator matrix $\mathbb{N}$ can be written as $\mathbb{N}={\left[{\mathbb{W}}_2^{⟙}{\mathbb{W}}_2\right]}^{-1}{\mathbb{W}}_2.{\mathsf{c}\mathsf{o}\mathsf{l}\left(\mathit{B}{\mathit{D}}_i\right)}_{i=\mathbf{0}}^{\mathcal{l}}$ or $\mathbb{N}={\mathbb{W}}_2^{\#}.{\mathsf{c}\mathsf{o}\mathsf{l}\left(\mathit{B}{\mathit{D}}_i\right)}_{i=\mathbf{0}}^{\mathcal{l}}$ where $\#$ stands for the pseudo inverse.

■

The finite eigenvalues of the non-monic matrix polynomial $\mathit{D}\left(\lambda \right)$ are the roots: $\det \left({⅀}_{i=0}^{\mathcal{l}}{\mathit{D}}_i{\lambda }^{\mathcal{l}-i}\right)=0$ or equivalently $\left[{\lambda }_i\right]=\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{e}\mathrm{i}\mathrm{g}\left({\mathit{D}}_{\mathcal{l}},\dots ,{\mathit{D}}_0\right)$.

■

Applying the Schur determinant formula, $\det \left(\mathit{X}\right)=\det \left({\mathit{X}}_{11}\left[{\mathit{X}}_{22}-{\mathit{X}}_{21}{\mathit{X}}_{11}^{-1}{\mathit{X}}_{12}\right]\right)$ to a block-structured pencil, one obtains the identity $\det \left(\lambda \mathit{E}-\mathit{A}\right)=0 ⟺ \det \left(\mathit{D}\left(\lambda \right)\right)=0$.

■

The non-monic $\lambda$-matrix $\mathit{D}\left(\lambda \right)$ can be factorized as a product of linear factors $\mathit{D}\left(\lambda \right)=\left[\lambda {\mathit{D}}_0-{\mathit{Q}}_{\mathcal{l}}\right]\left[\lambda \mathit{I}-{\mathit{Q}}_{\mathcal{l}-1}\right]\dots \left[\lambda \mathit{I}-{\mathit{Q}}_1\right]$ so $\det \left(\mathit{D}\left(\lambda \right)\right)=\det \left(\lambda {\mathit{D}}_0-{\mathit{Q}}_{\mathcal{l}}\right)\dots \det \left(\lambda \mathit{I}-{\mathit{Q}}_1\right)$ where all the eigenvalues of (${\mathit{Q}}_i$ for $i=\mathrm{1,2},\dots ,\mathcal{l}-1$) are an eigenvalues of $\mathit{D}\left(\lambda \right)$ and roots of $\det \left(\lambda {\mathit{D}}_0-{\mathit{Q}}_{\mathcal{l}}\right)$ are also an eigenvalues of $\mathit{D}\left(\lambda \right)$. However, it is important to note that the eigenvalues of ${\mathit{Q}}_{\mathcal{l}}$ are not necessarily eigenvalues of $\mathit{D}\left(\lambda \right)$.

■

If $\mathit{D}\left(\lambda \right)$ is a non-monic then can be factorized into $\mathit{D}\left(\lambda \right)={\mathit{D}}_0\mathit{S}\left(\lambda \right)$ with $\mathit{S}\left(\lambda \right)$ is any general matrix polynomial. All block roots of $\mathit{S}\left(\lambda \right)$ are also a block roots of $\mathit{D}\left(\lambda \right)$, but not all eigenvalues of $\mathit{D}\left(\lambda \right)$ are eigenvalues of $\mathit{S}\left(\lambda \right)$.

■

If ${\mathit{D}}_0$ is nonsingular matrix then, the eigenvalues of $\mathit{D}\left(\lambda \right)$ and $\mathit{S}\left(\lambda \right)$ will coincides.

It is well known that the state-space representation of a descriptor system is not unique, owing to the flexibility in the choice of the state vector. In conventional linear systems, this non-uniqueness is captured by the notion of algebraic equivalence, typically expressed through similarity transformations between system matrices [13,18]. For linear descriptor systems, a similar rationale applies; however, the broader structure of such systems necessitates a more general concept known as restricted system equivalence. Unlike normal systems, where equivalence is established via a single similarity matrix, descriptor system equivalence involves two nonsingular transformation matrices, ${\mathit{T}}_L$ and ${\mathit{T}}_R$, acting on both sides of the system equations.

$$\begin{array}{c}\begin{array}{c}{\mathit{A}}_c={\mathit{T}}_L^{-1}\mathit{A}{\mathit{T}}_R=\left[\begin{array}{cc} \begin{array}{c}⃝\\ \begin{array}{c}⋮\\ ⃝\end{array}\end{array}& \begin{array}{c}{\mathit{I}}_m\\ \\ ⃝\end{array} \ddots \begin{array}{c}⃝\\ \\ {\mathit{I}}_m\end{array}\\ -{\mathit{D}}_{\mathcal{l}}& \dots \dots -{\mathit{D}}_1\end{array}\right]; {\mathit{E}}_c={\mathit{T}}_L^{-1}\mathit{E}{\mathit{T}}_R=\left[\begin{array}{cc}\begin{array}{c}{\mathit{I}}_m\\ \\ ⃝\end{array}\ddots \begin{array}{c}⃝\\ \\ {\mathit{I}}_m\end{array}& \begin{array}{c}⃝\\ ⋮\\ ⃝\end{array}\\ ⃝ \dots ⃝& {\mathit{D}}_0\end{array}\right]; {\mathit{B}}_c={\mathit{T}}_L^{-1}\mathit{B}=\left[\begin{array}{c}\begin{array}{c}⃝\\ ⋮\end{array}\\ ⃝\\ {\mathit{I}}_m\end{array}\right];{\mathit{C}}_c^{⟙}={\left[\mathit{C}{\mathit{T}}_R\right]}^{⟙}=\left[\begin{array}{c}\begin{array}{c}{\mathit{N}}_{\mathcal{l}}^{⟙}\\ ⋮\end{array}\\ {\mathit{N}}_2^{⟙}\\ {\mathit{N}}_1^{⟙}\end{array}\right]\end{array}\\ \end{array}$$

with $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left({\mathit{D}}_0\right)<m$ and ${\mathit{D}}_i\in {\mathbb{R}}^{m\times m}$.

Facts on those Obtained Structured Matrix Equations: From the above equations we have ${\mathbb{W}}_1={\mathrm{r}\mathrm{o}\mathrm{w}\left({\left(\mathit{E}{\mathit{A}}^{-1}\right)}^i\mathit{B}\right)}_{i=\mathbf{0}}^{\mathcal{l}}$, $\mathbb{D}={\mathrm{c}\mathrm{o}\mathrm{l}\left({\mathit{D}}_i\right)}_{i=\mathbf{0}}^{\mathcal{l}}$, $\mathbb{N}={\mathrm{c}\mathrm{o}\mathrm{l}\left({\mathcal{N}}_i\right)}_{i=1}^{\mathcal{l}}$, $\mathbb{B}\mathbb{D}={\mathrm{c}\mathrm{o}\mathrm{l}\left(\mathit{B}{\mathit{D}}_i\right)}_{i=\mathbf{0}}^{\mathcal{l}}$ and ${\mathbb{W}}_2={\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}}_{\mathcal{l}}\left(-\mathit{A},\mathit{E}\right)$ where: the operator ${\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}}_k\left(-\mathit{A},\mathit{E}\right)$ denote the block companion-like matrix with $-\mathit{A}$ on the diagonal and $\mathit{E}$ on the super-diagonal.

• $\mathbb{D}\in \mathrm{n}\mathrm{u}\mathrm{l}\mathrm{l}\left({\mathbb{W}}_1\right)$ and $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left({\mathbb{W}}_1\right)=n$
• If $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left({\mathbb{W}}_2\right)=n\mathcal{l}$ then $\mathbb{N}$ is a solution of the matrix equation ${\mathbb{W}}_2\mathbb{N}=\mathbb{B}\mathbb{D}$
• $\det \left(\lambda \mathit{E}-\mathit{A}\right)=\det \left(\lambda {\mathit{E}}_c-{\mathit{A}}_c\right)=0⟺\det \left(\mathit{D}\left(\lambda \right)\right)=0$

4.2. State Feedback Controller Design

In this section, we shall deal with the stabilization problem for descriptor systems. The purpose is the design of controllers such that the closed-loop system is regular, stable and impulse-free (in the continuous case) or causal (in the discrete case). The objectives in this section are to design a state feedback control $\mathbf{u}\left(t\right)=\mathbf{r}\left(t\right)-\mathit{K}\mathbf{x}\left(t\right)$, or obtaining a gain matrix $\mathit{K}$ in order to relocate the original admissible pair to a desired location. Consider the descriptor system governed in the Laplace domain by the closed-loop equation: $\left(\mathit{\lambda }\mathit{E}-\mathit{A}+\mathit{B}\mathit{K}\right)\mathbf{x}\left(\mathit{\lambda }\right)+\mathit{B}\mathbf{r}\left(\mathit{\lambda }\right)=0$ where $\mathit{K}\in {\mathbb{R}}^{m\times n}$ is the state feedback gain matrix. The closed-loop system matrix is then: ${\mathit{A}}_d=\mathit{A}-\mathit{B}\mathit{K}$. Let ${\mathit{T}}_L$ and ${\mathit{T}}_R$ be the left and right basis transformation matrices such that the system is expressed in a transformed coordinate system as: ${\mathit{A}}_c={\mathit{T}}_L^{-1}\mathit{A}{\mathit{T}}_R \mathrm{a}\mathrm{n}\mathrm{d} {\mathit{B}}_c={\mathit{T}}_L^{-1}\mathit{B}$. This framework give us:

$${\mathit{A}}_c-{\mathit{A}}_{cd}={\mathit{T}}_L^{-1}\left(\mathit{A}-{\mathit{A}}_{\mathit{d}}\right){\mathit{T}}_R={\mathit{T}}_L^{-1}\mathit{B}\mathit{K}{\mathit{T}}_R={\mathit{B}}_{\mathit{c}}\mathit{K}{\mathit{T}}_R$$

(43)

Denoting ${\mathit{K}}_c=\mathit{K}{\mathit{T}}_R$, it follows that ${\mathit{A}}_c-{\mathit{A}}_{cd}={\mathit{B}}_{\mathit{c}}{\mathit{K}}_{\mathit{c}}$, and using ${\mathit{B}}_c^{⟙}{\mathit{B}}_{\mathit{c}}=\mathit{I}$, the transformed gain matrix can be explicitly computed as ${\mathit{K}}_c={\mathit{B}}_c^{⟙}{\mathit{T}}_L^{-1}\left(\mathit{A}-{\mathit{A}}_{\mathit{d}}\right){\mathit{T}}_R$. As previously discussed, the transformed matrices ${\mathit{A}}_c$ and ${\mathit{A}}_{cd}$ admit structured decompositions inherited from the block companion form of the descriptor matrix polynomial. Consequently, the feedback gain can be explicitly recovered from the difference between their corresponding lower block components:

$${\mathit{K}}_c={\mathit{B}}_c^{⟙}{\mathit{T}}_L^{-1}\left(\mathit{A}-{\mathit{A}}_d\right){\mathit{T}}_R={\mathit{B}}_c^{⟙}\left({\mathit{A}}_{cd}-{\mathit{A}}_{cd}\right)=\left[ ⃝ \dots ⃝ ,\mathit{I}\right]\left[\begin{array}{cc} \begin{array}{c}⃝\\ \begin{array}{c}⋮\\ ⃝\end{array}\end{array}& \begin{array}{c}{\mathit{I}}_m\\ \\ ⃝\end{array} \ddots \begin{array}{c}⃝\\ \\ {\mathit{I}}_m\end{array} \\ {\mathit{D}}_{d\mathcal{l}}-{\mathit{D}}_{\mathcal{l}}& \dots \dots {\mathit{D}}_{d1}-{\mathit{D}}_1\end{array}\right]$$

(44)

Therefore, ${\mathit{K}}_c=\left[\left({\mathit{D}}_{d\mathcal{l}}-{\mathit{D}}_{\mathcal{l}}\right)\dots \left({\mathit{D}}_{d1}-{\mathit{D}}_1\right)\right]=\mathrm{r}\mathrm{o}\mathrm{w}{\left({\mathit{D}}_{di}-{\mathit{D}}_i\right)}_{i=\mathcal{l}}^{i=1}$. The above derivation is summarized in the Algorithm 1.

Algorithm 1: Feedback Gain Computation via Matrix Fraction Description (MFD)
Input: Assumption:	$\mathrm{State}-\mathrm{space} \mathrm{matrices} \left(\mathit{E},\mathit{A},\mathit{B},\mathit{C}\right)$ $\mathrm{with} \mathit{E}$ possibly singular (descriptor) Assume that all states are available and measurable.
Procedure: ➢ Obtain the MFD from the general state space representation using the equation $\left[\begin{array}{cc} {\mathbb{W}}_1& 0\\ -\mathbb{B}& {\mathbb{W}}_2\end{array}\right]\left[\begin{array}{c}\mathbb{D}\\ \mathbb{N}\end{array}\right]=\left[\begin{array}{c}\mathbf{0}\\ \mathbf{0}\end{array}\right] \mathrm{o}\mathrm{r} \mathrm{i}\mathrm{n} \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{d} \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m} \left[\begin{array}{cc} {\mathrm{r}\mathrm{o}\mathrm{w}\left({\left(\mathit{E}{\mathit{A}}^{-1}\right)}^i\mathit{B}\right)}_{i=\mathbf{0}}^{\mathcal{l}}& 0\\ -{\mathrm{b}\mathrm{l}\mathrm{k}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}_{\left(\mathcal{l}+1\right)}\left(\mathit{B}\right) & {\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}}_{\mathcal{l}}\left(-\mathit{A},\mathit{E}\right)\end{array}\right]\left[\begin{array}{c}\mathbb{D}\\ \mathbb{N}\end{array}\right]=\left[\begin{array}{c}\mathbf{0}\\ \mathbf{0}\end{array}\right]$ ➢ $\mathrm{Design} \mathrm{the} \mathrm{desired} \mathrm{matrix} \mathrm{polynomial} {\mathit{D}}_d\left(\lambda \right)={\mathit{D}}_{d0}{\lambda }^{\mathcal{l}}+{⅀}_{i=1}^{\mathcal{l}}{\mathit{D}}_{di}{\lambda }^{\mathcal{l}-i}$ using block spectral factors such that ${\mathit{D}}_d\left(\lambda \right)=\left(\lambda {\mathit{D}}_{d0}-{\mathit{Q}}_{d\mathcal{l}}\right)\left(\lambda \mathit{I}-{\mathit{Q}}_{d\left(\mathcal{l}-1\right)}\right)\dots \left(\lambda \mathit{I}-{\mathit{Q}}_{d1}\right)$ $\mathrm{where} {\mathit{Q}}_{di}$ $\mathrm{for} i=\mathrm{1,2},\dots ,\mathcal{l}-1$ are block spectral factors correspond to finite eigenvalues and ${\mathit{Q}}_{d\mathcal{l}}$ is the block spectral factor which satisfies $\det \left({\lambda }_i{\mathit{D}}_{d0}-{\mathit{Q}}_{d\mathcal{l}}\right)=0$ for infinite eigenvalues ${\lambda }_i$. ➢ Construct the feedback gain matrix ${\mathit{K}}_c=\mathrm{r}\mathrm{o}\mathrm{w}{\left({\mathit{D}}_{di}-{\mathit{D}}_i\right)}_{i=\mathcal{l}}^{i=1}\in {\mathbb{R}}^{m\times n}$ ➢ Find the left and right transformations ${\mathit{T}}_L=[{\mathit{T}}_{L1},\dots , {\mathit{T}}_{L\mathcal{l}}]$ and ${\mathit{T}}_R=[{\mathit{T}}_{R1},\dots , {\mathit{T}}_{R\mathcal{l}}]$ by solving the matrix equations: ${\mathit{A}}_c={\mathit{T}}_L^{-1}\mathit{A}{\mathit{T}}_R; {\mathit{E}}_c={\mathit{T}}_L^{-1}\mathit{E}{\mathit{T}}_R; {\mathit{B}}_c={\mathit{T}}_L^{-1}\mathit{E}; {\mathit{C}}_c=\mathit{C}{\mathit{T}}_R$ ➢ Recover the feedback gain in the original basis using $\mathit{K}={\mathit{K}}_c{\mathit{T}}_R^{-1}$. Output: State feedback matrix $\mathit{K}\in {\mathbb{R}}^{m\times n}$

Example 4. 

Consider the following linear descriptor system (chemical reactor model) parameterized by the next matrices as shown below

$$\left\{\begin{array}{c}\mathit{E}=\left[\begin{array}{c}0.9733\\ \begin{array}{c}0.9210\\ -0.5862\end{array}\\ -0.2658\end{array} \begin{array}{c}0.0030\\ \begin{array}{c}0.8964\\ 0.0660\end{array}\\ 0.0299\end{array} \begin{array}{c}-0.0355\\ \begin{array}{c}1.2245\\ 0.2205\end{array}\\ -0.3534\end{array} \begin{array}{c}-0.0091\\ \begin{array}{c}0.3127\\ -0.1990\end{array}\\ 0.9098\end{array}\right]; \mathit{B}=\left[\begin{array}{c}-0.0603\\ \begin{array}{c}2.1963\\ -2.1467\end{array}\\ -0.3789\end{array} \begin{array}{c}-0.7049\\ \begin{array}{c}-4.3673\\ 4.6119\end{array}\\ 4.2975\end{array}\right]\\ \mathit{A}=\left[\begin{array}{c}9.10190\\ \begin{array}{c}48.6555\\ -51.0707\end{array}\\ -53.3387\end{array} \begin{array}{c}11.0811\\ \begin{array}{c}53.8052\\ -57.7658\end{array}\\ -58.9560\end{array} \begin{array}{c}9.6066\\ \begin{array}{c}52.6208\\ -55.8321\end{array}\\ -55.1056\end{array} \begin{array}{c}3.67530\\ \begin{array}{c}13.2208\\ -14.3055\end{array}\\ -17.0750\end{array}\right]; {\mathit{C}}^{⟙}=\left[\begin{array}{c}4.6241\\ \begin{array}{c}5.3466\\ 4.3667\end{array}\\ 1.6594\end{array} \begin{array}{c}5.2382\\ \begin{array}{c}6.4039\\ 5.4293\end{array}\\ 1.7563\end{array}\right]\end{array}\right\}$$

Now we compute the matrix ${\mathbb{W}}_1=\left[\mathit{B} \left(\mathit{E}{\mathit{A}}^{-1}\right)\mathit{B} {\left(\mathit{E}{\mathit{A}}^{-1}\right)}^2\mathit{B} \right]$ according to (42) we get the following full rank matrix

$${\mathbb{W}}_1=\left[\begin{array}{c}-0.0603\\ \begin{array}{c} 2.1963\\ -2.1467\end{array}\\ -0.3789\end{array} \begin{array}{c}-0.7049\\ \begin{array}{c}-4.3673\\ 4.6119\end{array}\\ 4.2975\end{array} \begin{array}{c}-1.0301\\ \begin{array}{c} 0.8468\\ 0.6612\end{array}\\ 0.7243\end{array} \begin{array}{c} 0.0524\\ \begin{array}{c}-0.1377\\ -0.0645\end{array}\\ 0.0527\end{array} \begin{array}{c}-0.7924\\ \begin{array}{c} 1.6570\\ 0.3125\end{array}\\ 1.6579\end{array} \begin{array}{c} 0.2415\\ \begin{array}{c}-0.3351\\ -0.0966\end{array}\\ -0.4438\end{array}\right]$$

Let us denote $\mathbb{D}=\mathrm{n}\mathrm{u}\mathrm{l}\mathrm{l}\left({\mathbb{W}}_1\right)$, then the matrix fraction description of the corresponding descriptor system can be obtained easily using the following formula

$${\mathit{D}}_0=\mathbb{D}\left(1:2, :\right)=\left[\begin{array}{cc}0.1521& 0.0727\\ 0.0678& 0.0324\end{array}\right]; {\mathit{D}}_1=\mathbb{D}\left(3:4, :\right)=\left[\begin{array}{cc}-0.0253& 0.0942\\ -0.9758& 0.1237\end{array}\right]; {\mathit{D}}_2=\mathbb{D}\left(5:6, :\right)=\left[\begin{array}{cc}-0.0661& 0.1481\\ 0.1229& 0.9734\end{array}\right]$$

The numerator matrix coefficients are obtained from the following matrix equation: ${\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}}_{\mathcal{l}}\left(-\mathit{A},\mathit{E}\right).\mathbb{N}={\mathrm{c}\mathrm{o}\mathrm{l}\left(\mathit{B}{\mathit{D}}_i\right)}_{i=\mathbf{0}}^{\mathcal{l}}$. In MATLAB language we can write

$$\mathbb{N}=\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{e}\left(\left[\mathit{E} zeros\left(\mathrm{4,4}\right);-\mathit{A} \mathit{E}; zeros\left(\mathrm{4,4}\right)-\mathit{A}\right], {\left[{\left(\mathit{B}{\mathit{D}}_0\right)}^{⟙} {\left(\mathit{B}{\mathit{D}}_1\right)}^{⟙} {\left(\mathit{B}{\mathit{D}}_2\right)}^{⟙}\right]}^{⟙}\right)$$

Then we obtain

$${\mathcal{N}}_1=\mathbb{N}\left(1:4, :\right)=\left[\begin{array}{c}-0.0459\\ \begin{array}{c}-0.3446\\ 0.2298\end{array}\\ 0.3442\end{array} \begin{array}{c}-0.0264\\ \begin{array}{c}-0.0110\\ 0.0120\end{array}\\ 0.1202\end{array}\right]; {\mathcal{N}}_2=\mathbb{N}\left(5:8, :\right)=\left[\begin{array}{c}-0.0690\\ \begin{array}{c}-0.1180\\ 0.1731\end{array}\\ 0.0965\end{array} \begin{array}{c} 0.0967\\ \begin{array}{c} 0.1723\\ -0.1393\end{array}\\ -0.2058\end{array}\right]$$

The canonical state equation of this descriptor system is ${\mathit{E}}_c = \left[{\mathit{I}}_m ⃝ ; ⃝ {\mathit{D}}_0\right]$, ${\mathit{A}}_c= \left[ ⃝ {\mathit{I}}_m;-{\mathit{D}}_2 -{\mathit{D}}_1\right]$, ${\mathit{B}}_c =\left[ ⃝ ; {\mathit{I}}_m\right]$, ${\mathit{C}}_c=\mathit{C}\left[{\mathcal{N}}_2 {\mathcal{N}}_1\right]$, ${\mathit{D}}_c= ⃝$. The characteristic non-monic matrix polynomial corresponding to this linear descriptor system is given by $\mathit{D}\left(\lambda \right)={\mathit{D}}_0{\lambda }^2 +{\mathit{D}}_1\lambda +{\mathit{D}}_2$, our objectives here are to design stabilizing state feedback gain matrix meeting a desired performance. The desired non-monic matrix polynomial is ${\mathit{D}}_{\mathit{d}}\left(\lambda \right)={\mathit{D}}_{d0}{\lambda }^2 +{\mathit{D}}_{d1}\lambda +{\mathit{D}}_{d2}=\left({\mathit{D}}_0\lambda -{\mathit{Q}}_{d2}\right)\left(\lambda {\mathit{I}}_m-{\mathit{Q}}_{d1}\right)$ expand the last matrix equation and equalize the same power terms we get: ${\mathit{D}}_{d0}={\mathit{D}}_0$, ${\mathit{D}}_{d1}=-\left({\mathit{D}}_0{\mathit{Q}}_{d1}+{\mathit{Q}}_{d2}\right)$ and ${\mathit{D}}_{d2}={\mathit{Q}}_{d2}{\mathit{Q}}_{d1}$. The matrices $\left({\mathit{Q}}_{di} \mathrm{f}\mathrm{o}\mathrm{r} i=\mathrm{1,2}\right)$ are called spectral factors of ${\mathit{D}}_{\mathit{d}}\left(\lambda \right)$, and

$$\det \left({\mathit{D}}_d\left(\lambda \right)\right)=0 ⟺ \left\{\begin{array}{c}\det \left({\lambda }_i{\mathit{I}}_m-{\mathit{Q}}_{d1}\right)=0 \mathrm{f}\mathrm{o}\mathrm{r} i=\mathrm{1,2}\\ \det \left({\lambda }_i{\mathit{D}}_0-{\mathit{Q}}_{d2}\right)=0 \mathrm{f}\mathrm{o}\mathrm{r} i=3 \end{array}\right.$$

The desired non-monic matrix polynomial ${\mathit{D}}_{\mathbf{d}}\left(\lambda \right)$ contains $\eta =m(\mathcal{l}-1)+\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left({\mathit{D}}_0\right)=3$ finite eigenvalues, the desired eigenvalues to be assigned to this descriptor system are ${\lambda }_i=\left\{-1,-2,-3\right\}$.

$$\begin{array}{c}{\mathit{Q}}_{d1}={\mathbf{r}}_1\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\left[-1,-2\right]\right){\mathbf{r}}_1^{-1}, \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} {\mathbf{r}}_1=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\left(\mathrm{2,2}\right) \\ {\mathit{Q}}_{d2}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\left[x,y\right]\right), \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \det \left({\lambda }_i{\mathit{D}}_0-{\mathit{Q}}_{d2}\right)=0 \mathrm{a}\mathrm{n}\mathrm{d} {\lambda }_i=-3\end{array}$$

This leads to the following values $y=\left[-1+\sqrt{1-12{\mathit{D}}_0\left(\mathrm{1,1}\right){\mathit{D}}_0\left(\mathrm{2,2}\right)}\right]/2{\mathit{D}}_0\left(\mathrm{1,1}\right);$ and $x=3/y$. Numerical application gives the following result

$${\mathit{Q}}_{d1}=\left[\begin{array}{cc}-2.1351& 0.2476\\ -0.6194& -0.8649\end{array}\right]; \mathrm{a}\mathrm{n}\mathrm{d} {\mathit{Q}}_{d2}=\left[\begin{array}{cc}-30.3879& 0\\ 0& -0.0987\end{array}\right]$$

Now, apply the control input $\mathbf{u}\left(t\right)=-{\mathit{K}}_c{\mathbf{x}}_c\left(t\right)+{\mathbf{u}}_e\left(t\right)$ we obtain the closed loop non-monic matrix polynomial ${\mathit{D}}_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}}\left(\lambda \right)={\mathit{D}}_0{\lambda }^2+\left({\mathit{D}}_1-{\mathit{K}}_{c1}\right)\lambda +\left({\mathit{D}}_2-{\mathit{K}}_{c2}\right)={\mathit{D}}_d\left(\lambda \right)$ this means that ${\mathit{K}}_{c2}={\mathit{D}}_2-{\mathit{D}}_{d2}$ and ${\mathit{K}}_{c1}={\mathit{D}}_1-{\mathit{D}}_{d1}$ or in more compact form

$${\mathit{K}}_c=\left[{\mathit{K}}_{c2} {\mathit{K}}_{c1}\right]=\left[\begin{array}{cc}\begin{array}{cc}-64.9470& 7.6711\\ 0.0618& 0.8880\end{array}& \begin{array}{cc}-30.7830& 0.0690\\ -1.14070& 0.0138\end{array}\end{array}\right]$$

To obtain this state feedback gain matrix in the original base we should find the left and right similarity transformations ${\mathit{T}}_L=[{\mathit{T}}_{L1},{\mathit{T}}_{L2}]$ and ${\mathit{T}}_R=\left[{\mathit{T}}_{R1},{\mathit{T}}_{R2}\right]$

■

$\mathit{A}{\mathit{T}}_R={\mathit{T}}_L{\mathit{A}}_c⟺\left[\mathit{A}{\mathit{T}}_{R1},\mathit{A}{\mathit{T}}_{R2}\right]=[-{\mathit{T}}_{L2}{\mathit{D}}_2,{\mathit{T}}_{L1}-{\mathit{T}}_{L2}{\mathit{D}}_1]$

■

$\mathit{E}{\mathit{T}}_R={\mathit{T}}_L{\mathit{E}}_c⟺\left[\mathit{E}{\mathit{T}}_{R1},\mathit{E}{\mathit{T}}_{R2}\right]=[{\mathit{T}}_{L1},{\mathit{T}}_{L2}{\mathit{D}}_0]$

■

$\mathit{B}={\mathit{T}}_L{\mathit{B}}_c⟺{\mathit{T}}_{L2}=\mathit{B}$

Solving this last three matrix equations we obtain (i.e., assuming that $\mathit{A}$ is nonsingular): ${\mathit{T}}_L=\left[-\mathit{E}{\mathit{A}}^{-1}\mathit{B}{\mathit{D}}_2,\mathit{B}\right]$ and ${\mathit{T}}_R=\left[-{\mathit{A}}^{-1}\mathit{B}{\mathit{D}}_2,-{\mathit{A}}^{-1}\mathit{E}{\mathit{A}}^{-1}\mathit{B}{\mathit{D}}_2-{\mathit{A}}^{-1}\mathit{B}{\mathit{D}}_1\right]$, so

$${\mathit{T}}_L=\left[\begin{array}{c}-0.0745\\ \begin{array}{c} 0.0729\\ 0.0516\end{array}\\ 0.0414\end{array} \begin{array}{c} 0.1015\\ \begin{array}{c} 0.0086\\ -0.0351\end{array}\\ -0.1585\end{array} \begin{array}{c}-0.0603\\ \begin{array}{c} 2.1963\\ -2.1467\end{array}\\ -0.3789\end{array} \begin{array}{c}-0.7049\\ \begin{array}{c}-4.3673\\ 4.6119\end{array}\\ 4.2975\end{array}\right]; {\mathit{T}}_R=\left[\begin{array}{c}-0.0690\\ \begin{array}{c}-0.1180\\ 0.1731\end{array}\\ 0.0965\end{array} \begin{array}{c} 0.0967\\ \begin{array}{c} 0.1723\\ -0.1393\end{array}\\ -0.2058\end{array} \begin{array}{c}-0.0459\\ \begin{array}{c}-0.3446\\ 0.2298\end{array}\\ 0.3442\end{array} \begin{array}{c}-0.0264\\ \begin{array}{c}-0.0110\\ 0.0120\end{array}\\ 0.1202\end{array}\right]$$

The general state feedback gain matrix is

$$\mathit{K}=\left[{\mathit{K}}_2 {\mathit{K}}_1\right]={\mathit{K}}_c{\mathit{T}}_R^{-1}=\left[\begin{array}{cc}\begin{array}{cc}-392.7530& -442.5886\\ 8.538500& 10.85520\end{array}& \begin{array}{cc}-807.7280& -45.7175\\ 10.05030& 1.98330\end{array}\end{array}\right]$$

Figure 1 illustrates time response of both open loop and stabilized closed-loop descriptor system under the proposed state feedback.

The results demonstrate the effectiveness of the proposed state feedback design and matrix fraction description in stabilizing a high-dimensional MIMO descriptor system. The assigned eigenstructure successfully regulates both finite and infinite modes, transforming the system into a regular, impulse-free form. As shown in Figure 1, the closed-loop time response exhibits smooth convergence across all channels, with transient states settling within approximately $1.2 \left(s\right)$ and no observable impulsive behavior, confirming causality and stability. The trajectories remain bounded and well-behaved, validating the theoretical feedback law and illustrating the practicality of the algebraic formulation in managing descriptor dynamics with structural singularities.

4.3. The Proposed Block Pole Assignment and Normalization

A descriptor linear system represented as in (26) is termed normal when it is square and the matrix $\mathit{E}$ has full rank, i.e., $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\mathit{E}\right)=n$. Normalization involves designing a control law that ensures the resulting closed-loop system attains this normal property. As with regularization, the normalization problems include the following:

■

Normalizability analysis: This step involves determining whether a given descriptor linear system can be rendered normal through a specified class of controllers [4].

■

Normalizing controller design: Once normalizability is established, the next step is to characterize all admissible controllers that render the closed-loop system normal. The normalization problem is a special case of dynamical order assignment: by setting the desired order $p$ equal to the system order $n$, assigning a closed-loop order becomes equivalent to normalization. Hence, results from dynamical order assignment apply directly to normalization [4,5].

Let us consider the descriptor system $\mathit{E}\dot{\mathit{x}}\left(t\right) =\mathit{A}\mathbf{x}\left(t\right)+\mathit{B}\mathbf{u}\left(t\right)$ where $\mathbf{x}\left(t\right)\in {\mathbb{R}}^n$ is the state vector, $\mathbf{u}\left(t\right)\in {\mathbb{R}}^m$ is the control vector, $\mathit{A},\mathit{E}\in {\mathbb{R}}^{n\times n}$ and $\mathit{B}\in {\mathbb{R}}^{n\times m}$ are known real coefficient matrices. Assume, without loss of generality, that $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\mathit{E}\right)=r<n$, and $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\mathit{B}\right)=m$. By applying the following full-state derivative feedback control law $\mathbf{u}\left(t\right)=-\mathit{K}\dot{\mathbf{x}}\left(t\right)+{\mathbf{u}}_{\mathrm{r}\mathrm{e}\mathrm{f}}\left(t\right)$ the dynamics of the closed loop system evolves into the form $\left(\mathit{E}+\mathit{B}\mathit{K}\right)\dot{\mathbf{x}}\left(t\right)=\mathit{A}\mathbf{x}\left(t\right)+\mathit{B}{\mathbf{u}}_{\mathrm{r}\mathrm{e}\mathrm{f}}\left(t\right)$ or $\dot{\mathbf{x}}\left(t\right)={\left(\mathit{E}+\mathit{B}\mathit{K}\right)}^{-1}\mathit{A}\mathbf{x}\left(t\right)+{\left(\mathit{E}+\mathit{B}\mathit{K}\right)}^{-1}\mathit{B}{\mathbf{u}}_{\mathrm{r}\mathrm{e}\mathrm{f}}\left(t\right)$.

Definition 10. 

([5,16]) A positive integer $p$ is said to be a dynamical order assignable to the descriptor system $\mathit{E}\dot{\mathit{x}}\left(t\right)=\mathit{A}\mathbf{x}\left(t\right)+\mathit{B}\mathbf{u}\left(t\right)$ or it is said to be a dynamical order assignable to the pair $\left(\mathit{E},\mathit{B}\right)$, via full-state derivative feedback, if there exists a feedback matrix $\mathit{K}\in {\mathbb{R}}^{m\times n}$ such that the rank condition $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\left[\mathit{E}+\mathit{B}\mathit{K}\right]\right)=p$ holds. In this context, $p$ is also referred to as an allowable closed-loop order. The collection of all matrices $\mathit{K}$ that achieve this order forms the set:

$$\mathcal{L}=\left\{\mathit{K}\in {\mathbb{R}}^{m\times n}| \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\left[\mathit{E}+\mathit{B}\mathit{K}\right]\right)=p\right\}$$

(45)

with this notion, we now present a result that characterizes the set of all possible assignable dynamical orders for the open-loop system.

Theorem 14. 

([2,3,25]) Let $\mathit{E}\in {\mathbb{R}}^{n\times n}$ and $\mathit{B}\in {\mathbb{R}}^{n\times m}$ be fixed real matrices, and define

$$n_2=\min \left\{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\left[\mathit{E},\mathit{B}\right]\right); n\right\}, n_1=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\left[\mathit{E},\mathit{B}\right]\right)-\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\mathit{B}\right)$$

(46)

Then, $p$ is an allowable dynamical order assignable to the system $\mathit{E}\dot{\mathbf{x}}\left(t\right)=\mathit{A}\mathbf{x}\left(t\right)+\mathit{B}\mathbf{u}\left(t\right)$ by the full-state derivative feedback if and only if $n_1\le p \le n_2$.

In normal linear systems, exactly $n$ finite poles exist, ensuring the absence of infinite eigenvalues and avoiding impulsive behavior. For descriptor systems, this property can be recovered through derivative feedback: by selecting ${\mathit{K}}_d$ so that the effective mass matrix $\mathit{E}+\mathit{B}{\mathit{K}}_d$ becomes full rank, algebraic constraints are moved into dynamic ones, and all poles are shifted to finite locations. Once finite, the MFD is regular, allowing the use of standard frequency-domain tools. Our block-construction (Theorem 15) ensures this normalization can be achieved while preserving structure and enabling subsequent finite pole placement using classical methods of standard linear system theory.

Theorem 15. 

([4,5]) The system $\mathit{E}\dot{\mathbf{x}}\left(t\right)=\mathit{A}\mathbf{x}\left(t\right)+\mathit{B}\mathbf{u}\left(t\right)$ is normalizable if and only if $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\left[\mathit{E},\mathit{B}\right]\right)=n$ and is dual normalizable if and only if $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left({\left[{\mathit{E}}^{⟙},{\mathit{C}}^{⟙}\right]}^{⟙}\right)=n$.

Corollary 1. 

([4,29]) A linear descriptor system is normalizable if and only if there exist a full state derivative feedback gain matrix $\mathit{K}\in {\mathbb{R}}^{m\times n}$ such that $\det \left(\mathit{E}+\mathit{B}\mathit{K}\right)\ne 0$.

Assumptions 2. 

Let we assume that

1. 

The pair $\left(\mathit{E},\mathit{B}\right)$ is normalizable i.e., $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\left[\mathit{E},\mathit{B}\right]\right)=n$

2. 

$\mathcal{l}=n=m$ is an integer with $n=\dim \left(\mathit{A}\right)$ and $m$ is the number of inputs

3. 

The pair $\left(\mathit{E},\mathit{B}\right)$ is block controllable with controllability index $\mathcal{l}$ i.e., the matrix ${\mathcal{C}}_c\left(\mathit{B},\mathit{E}\right)=\left[\mathit{B} \mathit{E}\mathit{B}\dots {\mathit{E}}^{\mathcal{l}-1}\mathit{B}\right]$ has full rank. For more detail see ([13,17]).

Problem 2. 

[Block poles assignment] Let matrices $\mathit{E}\in {\mathbb{R}}^{n\times n}$ and $\mathit{B}\in {\mathbb{R}}^{n\times m}$ satisfy Assumption (2) determine a parameterization for all the matrices $\mathit{K}\in {\mathbb{R}}^{m\times n}$ such that the matrix $\mathit{E}+\mathit{B}\mathit{K}$ is nonsingular $\left(\det \left(\mathit{E}+\mathit{B}\mathit{K}\right)\ne 0\right)$. Equivalently, characterize the set

$$\mathcal{L}=\left\{\mathit{K}\in {\mathbb{R}}^{m\times n}| \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\left[\mathit{E}+\mathit{B}\mathit{K}\right]\right)=n\right\}$$

(47)

Our development in this section is based on the arbitrary block poles assignment method in order to remove the singularity of the descriptor system. To solve the second problem, we need to present some basic and necessary matters in a brief form.

Definition 11. 

([36,37]) Let $\mathit{A}\left(\lambda \right)$ be a monic matrix polynomial with right solvents ${\mathit{R}}_1,\dots ,{\mathit{R}}_s$ of order ${\mathcal{l}}_1,\dots ,{\mathcal{l}}_s$ (i.e., ${\left(\lambda \mathit{I}-{\mathit{R}}_i\right)}^{{\mathcal{l}}_i}$ is a right divisor of $\mathit{A}\left(\lambda \right)$ and ${\left(\lambda \mathit{I}-{\mathit{R}}_i\right)}^{{\mathcal{l}}_i+1}$ isn’t). Assume that $\mathrm{deg}\mathit{A}\left(\lambda \right)=\mathcal{l}={\mathcal{l}}_1+\dots +{\mathcal{l}}_s$ then the generalized block Vandermonde matrix

$${\mathit{V}}_R=\mathrm{r}\mathrm{o}\mathrm{w}{\left[\mathrm{r}\mathrm{o}\mathrm{w}{\left[\mathrm{c}\mathrm{o}\mathrm{l}{\left({\displaystyle \frac{\left(k-1\right)!{\mathit{R}}_i^{k-j-1}}{j!\left(k-j-1\right)!}}\right)}_{k=1}^{\mathcal{l}} \right]}_{j=0}^{{\mathcal{l}}_i-1} \right]}_{i=1}^s$$

(48)

Theorem 16. 

Given a linear descriptor system $\left(\mathit{E},\mathit{A},\mathit{B},\mathit{C},\mathit{D}\right)$ and $\mathit{E}\in {\mathbb{R}}^{n\times n}$ and $\mathit{B}\in {\mathbb{R}}^{n\times m}$ satisfying Assumption (2), for an arbitrary set of square matrices ${\mathit{R}}_i\in {\mathbb{R}}^{m\times m}$ with non-zeros eigenvalues, a general form for all the feedback gain matrices $\mathit{K}\in {\mathbb{R}}^{m\times n}$ satisfying Equation (47)(i.e., $\mathit{K}\in \mathcal{L}$) is given as follows

$$\mathit{K}={\mathit{B}}_c^{⟙}.{\mathcal{C}}_c^{-1}\left(\mathit{B},\mathit{E}\right).{\mathit{E}}^{\mathcal{l}}-\mathit{\vartheta }.{\mathit{V}}_R^{-1}.{\mathit{T}}_c$$

(49)

With ${\mathit{T}}_c=\mathrm{c}\mathrm{o}\mathrm{l}{\left({\mathit{B}}_c^{⟙}{\mathcal{C}}_c^{-1}\left(\mathit{B},\mathit{E}\right){\mathit{E}}^i\right)}_{i=0}^{\mathcal{l}-1}; {\mathit{B}}_c^{⟙}=\left[ ⃝ \dots ⃝, {\mathit{I}}_m\right]; {\mathcal{C}}_c\left(\mathit{B},\mathit{E}\right)=\left[\mathit{B} \mathit{E}\mathit{B}\dots {\mathit{E}}^{\mathcal{l}-1}\mathit{B}\right]$$\mathit{\vartheta }={\mathrm{r}\mathrm{o}\mathrm{w}\left[{\mathit{R}}_i^{\mathcal{l}} \left(\begin{array}{c}\mathcal{l}\\ 1\end{array}\right){\mathit{R}}_i^{\mathcal{l}-1}\dots \left(\begin{array}{c}\mathcal{l}\\ {\mathcal{l}}_i-1\end{array}\right){\mathit{R}}_i^{\mathcal{l}-{\mathcal{l}}_i}\right]}_{i=1}^s$ and ${\mathit{V}}_R:$ is the block Vandermonde matrix.

Proof. 

The objective is to find an appropriate gain matrix such that $\mathit{E}-\mathit{B}\mathit{K}={\mathit{E}}_d$ or in more structured form ${\mathit{T}}_c\mathit{E}{\mathit{T}}_c^{-1}-{\mathit{B}}_c{\mathit{K}}_c={\mathit{T}}_c{\mathit{E}}_d{\mathit{T}}_c^{-1}$, that is: ${\mathit{K}}_c={\mathit{B}}_c^{⟙}{\mathit{T}}_c\left(\mathit{E}-{\mathit{E}}_{\mathit{d}}\right){\mathit{T}}_c^{-1}$. But we know that ${\mathit{B}}_c^{⟙}=\left[ ⃝ \dots ⃝,{\mathit{I}}_m\right]$ and ${\mathit{B}}_c^{⟙}{\mathit{T}}_c={\mathit{B}}_c^{⟙}{\mathcal{C}}_c^{-1}\left(\mathit{B},\mathit{E}\right).{\mathit{E}}^{\mathcal{l}-1}$ and in general base we have $\mathit{K}={\mathit{K}}_c{\mathit{T}}_c$ so $\mathit{K}={\mathit{B}}_c^{⟙}{\mathcal{C}}_c^{-1}\left(\mathit{B},\mathit{E}\right).{\mathit{E}}^{\mathcal{l}}-{\mathit{B}}_c^{⟙}{\mathit{T}}_c{\mathit{E}}_d$. From the other side we have

$$\begin{array}{c}\begin{array}{c}\begin{array}{c}\left[ ⃝ \dots ⃝,{\mathit{I}}_m\right]{\mathit{T}}_c{\mathit{E}}_d=\left[ ⃝ \dots ⃝,{\mathit{I}}_m\right]{\mathit{T}}_c{\mathit{E}}_d{\mathit{T}}_c^{-1}{\mathit{T}}_c=\left[ ⃝ \dots ⃝,{\mathit{I}}_m\right]{\mathit{E}}_{dc}{\mathit{T}}_c \\ =\left[ ⃝ \dots ⃝,{\mathit{I}}_m\right]{\mathit{V}}_R\left({\mathit{V}}_R^{-1}{\mathit{E}}_{dc}{\mathit{V}}_R\right){\mathit{V}}_R^{-1}{\mathit{T}}_c \end{array}\\ =\left[ ⃝ \dots ⃝,{\mathit{I}}_m\right]{\mathit{V}}_R\mathsf{\Lambda }{\mathit{V}}_R^{-1}{\mathit{T}}_c \end{array}\\ =\mathit{\vartheta }.{\mathit{V}}_R^{-1}.{\mathit{T}}_c \end{array}$$

with $\mathsf{\Lambda }= {\mathit{J}}_1\oplus \dots \oplus {\mathit{J}}_s$ and ${\mathit{J}}_i = {\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}}_{{\mathcal{l}}_i}\left({\mathit{R}}_i,{\mathit{I}}_m\right)$ and the operator ${\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}}_{{\mathcal{l}}_i}\left(.\right)$ denote the block companion-like matrix with ${\mathit{R}}_i$ on the diagonal and ${\mathit{I}}_m$ on the super-diagonal (${\mathcal{l}}_i$ times) and $\mathit{\vartheta }=\left[ ⃝ \dots ⃝,{\mathit{I}}_m\right]{\mathit{V}}_R\mathsf{\Lambda }={\mathrm{r}\mathrm{o}\mathrm{w}\left[{\mathit{R}}_i^{\mathcal{l}} \left(\begin{array}{c}\mathcal{l}\\ 1\end{array}\right){\mathit{R}}_i^{\mathcal{l}-1}\dots \left(\begin{array}{c}\mathcal{l}\\ {\mathcal{l}}_i-1\end{array}\right){\mathit{R}}_i^{\mathcal{l}-{\mathcal{l}}_i}\right]}_{i=1}^s$.□

The objective is to compute a feedback gain using block-pole (block-root) assignment such that the descriptor system becomes normal, i.e., the closed-loop descriptor matrix attains full rank, while simultaneously enforcing the desired block eigenstructure as prescribed by the set of chosen block poles; this procedure follows directly from Theorem 16 and the detailed steps for achieving this are presented in Algorithm 2.

Algorithm 2: Derivative Feedback Gain Computation via Block Pole Assignment and Normalization

Input $\mathit{E}\in {\mathbb{R}}^{n\times n}$$, \mathit{A}\in {\mathbb{R}}^{n\times n}$$, \mathit{B}\in {\mathbb{R}}^{n\times m}$ (descriptor system matrices). Block poles $\left\{{\mathit{R}}_1,\dots ,{\mathit{R}}_s\right\}$ $\mathrm{with} \mathrm{nonzero} \mathrm{eigenvalues}, \mathrm{each} {\mathit{R}}_i\in {\mathbb{C}}^{m\times m}$$\mathrm{is} \mathrm{repeated} {\mathcal{l}}_i$ times. $\mathrm{Block} \mathrm{controllability} \mathrm{index} \mathcal{l}$$\mathrm{and} \mathrm{input} \mathrm{size} m$. Output A derivative-feedback gain ${\mathit{K}}_d\in {\mathbb{R}}^{m\times n}$ $\mathrm{such} \mathrm{that} \mathit{E}+\mathit{B}{\mathit{K}}_d$ is nonsingular Assumptions (must hold before running) $\left(\mathit{E},\mathit{B}\right)$$\mathrm{is} \mathrm{normalizable} (\mathrm{there} \mathrm{exists} {\mathit{K}}_d$$\mathrm{with} \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\mathit{E}+\mathit{B}{\mathit{K}}_d\right)=n$); Equivalently, the rank conditions in Theorem 15/Corollary 1 are satisfiable. Procedure Specify target blocks: Choose matrices ${\left\{{\mathit{R}}_i\right\}}_{i=1}^s\in {\mathbb{C}}^{m\times m}$$, \mathrm{all} \mathrm{with} \mathrm{nonzero} \mathrm{spectra}. \mathrm{These} \mathrm{are} \mathrm{the} \mathrm{desired} \mathrm{block} \mathrm{poles} \mathrm{to} \mathrm{remove} \sin \mathrm{gularity} \mathrm{and} \mathrm{shape} \mathrm{the} \mathrm{closed}-\mathrm{loop} \mathrm{structure} (\mathrm{each} {\mathit{R}}_i$$\mathrm{is} \mathrm{repeated} {\mathcal{l}}_i$ times). $\mathbf{Build} \mathbf{the} \mathbf{generalized} \mathbf{block} \mathbf{Vandermonde}\mathbf{\text{:}} \mathrm{Construct} \mathrm{the} \mathrm{matrix} {\mathit{V}}_R\in {\mathbb{C}}^{n\times n}$ as in Definition 11 ${\mathit{V}}_R$$\mathrm{is} \mathrm{generalized} \mathrm{block} \mathrm{Vandermonde} \mathrm{formed} \mathrm{from} \mathrm{the} \mathrm{right} \mathrm{solvents} {\mathit{R}}_i$$. \mathrm{Ensure} {\mathit{V}}_R$ is invertible. $\mathbf{Form} \mathbf{the} \mathbf{companion}\mathbf{-}\mathbf{like} \mathbf{operators}\mathbf{\text{:}} \mathrm{Assemble} \mathrm{the} \mathrm{block} \mathrm{companion}-\mathrm{like} \mathrm{operator} {\mathit{J}}_i= {\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}}_{{\mathcal{l}}_i}\left({\mathit{R}}_i,{\mathit{I}}_m\right)$$\mathrm{with} {\mathit{I}}_m$ on its block diagonal and the ${\mathit{R}}_i$ on its block superdiagonal (repeated ${\mathcal{l}}_i$ times), $\mathbf{Form} \mathbf{the} \mathbf{matrix} \mathit{\vartheta }$$\mathbf{\text{:}} \mathit{\vartheta }=\left[ ⃝ \dots ⃝,{\mathit{I}}_m\right]{\mathit{V}}_R\mathsf{\Lambda }={\mathrm{r}\mathrm{o}\mathrm{w}\left[{\mathit{R}}_i^{\mathcal{l}} \left(\begin{array}{c}\mathcal{l}\\ 1\end{array}\right){\mathit{R}}_i^{\mathcal{l}-1}\dots \left(\begin{array}{c}\mathcal{l}\\ {\mathcal{l}}_i-1\end{array}\right){\mathit{R}}_i^{\mathcal{l}-{\mathcal{l}}_i}\right]}_{i=1}^s$ $\mathrm{with} \mathsf{\Lambda }= {\mathit{J}}_1\oplus \dots \oplus {\mathit{J}}_s$. Set up the block-pole assignment condition: Construct ${\mathit{K}}_d={\mathit{B}}_c^{⟙}.{\mathcal{C}}_c^{-1}\left(\mathit{B},\mathit{E}\right).{\mathit{E}}^{\mathcal{l}}-\mathit{\vartheta }.{\mathit{V}}_R^{-1}.{\mathit{T}}_c$ so that the closed-loop satisfies $\left(\mathit{E}+\mathit{B}{\mathit{K}}_d\right)$ realizes the block eigenstructure encoded by ${\mathit{V}}_R$. (use Equation (49)) Verify normalization: Check $\mathit{det}\left(\mathit{E}+\mathit{B}{\mathit{K}}_d\right)\ne 0$$(\mathrm{i}.\mathrm{e}., \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\mathit{E}+\mathit{B}{\mathit{K}}_d\right)=n$$; \mathrm{if} \mathrm{not}, \mathrm{refine} \left\{{\mathit{R}}_i\right\}$ and repeat Steps 2–5. This enforces normality as per Problem 2 and Theorem 15/Corollary 1. (Optional) Stabilization after normalization: With $\widehat{\mathit{E}}:=\mathit{E}+\mathit{B}{\mathit{K}}_d$ now nonsingular, you may design a state-feedback $\mathit{K}$ to relocate an admissible pair to a desired location using the relocation rules in Section 4.4 (First/Second method), or via the MFD-based Algorithm 1 and Equations (43) and (44) for basis recovery.

Example 5. 

Consider a turbo-generator system modeled as a descriptor system of the form $\mathit{E}\dot{\mathbf{x}}\left(t\right)=\mathit{A}\mathbf{x}\left(t\right)+\mathit{B}\mathbf{u}\left(t\right)$ with the following matrices:

$$\mathit{E}=\left[\begin{array}{c} 3.8463\\ \begin{array}{c}-0.9893\\ -1.9361\end{array}\\ -1.3147\end{array} \begin{array}{c} 0.9565\\ \begin{array}{c} 0.0020\\ 0.9011\end{array}\\ -0.7189\end{array} \begin{array}{c} 0.1009\\ \begin{array}{c}-0.1672\\ 1.1598\end{array}\\ -0.0060\end{array} \begin{array}{c} 1.2284\\ \begin{array}{c}-1.8855\\ 1.5927\end{array}\\ 0.9919\end{array}\right]; \mathit{A}=\left[\begin{array}{c} 0.6586\\ \begin{array}{c} 1.6172\\ -0.3862\end{array}\\ -1.5357\end{array} \begin{array}{c}-6.0040\\ \begin{array}{c} 3.9428\\ -3.7187\end{array}\\ 8.2377\end{array} \begin{array}{c}-3.7703\\ \begin{array}{c} 1.9317\\ -5.7892\end{array}\\ 8.2339\end{array} \begin{array}{c} 0.0433\\ \begin{array}{c} 1.6856\\ -1.0273\end{array}\\ -0.8122\end{array}\right];\mathit{B}=\left[\begin{array}{c}0.2607\\ \begin{array}{c}0.6180\\ 0.4970\end{array}\\ 0.9882\end{array} \begin{array}{c}0.1431\\ \begin{array}{c}0.3540\\ 0.5289\end{array}\\ 0.6851\end{array}\right]$$

We consider the control law: $\mathbf{u}\left(t\right)=-{\mathit{K}}_1\dot{\mathbf{x}}\left(t\right)+\mathbf{r}\left(t\right)$ where $\mathbf{r}\left(t\right)={\mathit{K}}_2\mathbf{x}\left(t\right)+{\mathbf{u}}_c\left(t\right)$. To ensure regularity and normality (i.e., $\det \left(\mathit{E}+\mathit{B}{\mathit{K}}_1\right)\ne 0$), we choose: ${\mathit{R}}_i=\mathit{I}$ repeated ${\mathcal{l}}_1= 2$ times. Applying the proposed method, we obtain the gain matrix ${\mathit{K}}_1$ as:

$${\mathit{K}}_1={\mathit{B}}_c^{⟙}.{\mathcal{C}}_c^{-1}\left(\mathit{B},\mathit{E}\right).{\mathit{E}}^{\mathcal{l}}-\left[{\mathit{R}}_1^2,2{\mathit{R}}_1\right].{\left[\begin{array}{cc}{\mathit{I}}_2& {⃝}_2\\ {\mathit{R}}_1& {\mathit{I}}_2\end{array}\right]}^{-1}.\left[\begin{array}{c}{\mathit{B}}_c^{⟙}{\mathcal{C}}_c^{-1}\left(\mathit{B},\mathit{E}\right)\\ {\mathit{B}}_c^{⟙}{\mathcal{C}}_c^{-1}\left(\mathit{B},\mathit{E}\right)\mathit{E}\end{array}\right]=\left[\begin{array}{c}18.3869\\ -26.7123\end{array} \begin{array}{c}3.2387\\ -4.9055\end{array} \begin{array}{c}-0.2883\\ 0.4900\end{array} \begin{array}{c}-2.8569\\ 5.0658\end{array}\right]$$

with ${\mathit{B}}_c^{⟙}=\left[ ⃝ {\mathit{I}}_2\right]; {\mathcal{C}}_c=\left[\mathit{B} \mathit{E}\mathit{B}\right]$, ${\mathit{I}}_2$ stands for the $2\times 2$ identity matrix, ${ ⃝ }_2$ stands for the $2\times 2$ zero matrix.

Now, the obtained normalized linear system is given by $\dot{\mathbf{x}}\left(t\right)={\mathit{A}}_n\mathbf{x}\left(t\right)+{\mathit{B}}_n\mathbf{r}\left(t\right)$, where ${\mathit{A}}_n={\left(\mathit{E}+\mathit{B}{\mathit{K}}_1\right)}^{-1}\mathit{A}$ and ${\mathit{B}}_n={\left(\mathit{E}+\mathit{B}{\mathit{K}}_1\right)}^{-1}\mathit{B}$, at this stage we need to design a gain matrix ${\mathit{K}}_2$ meeting a desired performance. The new closed loop system is given by $\dot{\mathbf{x}}\left(t\right)=\left({\mathit{A}}_n+{\mathit{B}}_n{\mathit{K}}_2\right)\mathbf{x}\left(t\right)+{\mathit{B}}_n{\mathbf{u}}_c\left(t\right)$. Let we place the owing two block roots

$${\overline{\mathit{R}}}_1=\left[\begin{array}{cc}-0.3923& 2.9886\\ -0.3269& -2.6077\end{array}\right]; {\overline{\mathit{R}}}_2=\left[\begin{array}{cc}-2.3923& -3.4542\\ 0.2828& -4.6077\end{array}\right]$$

Therefore the state feedback matrix ${\mathit{K}}_2$ is given by

$${\mathit{K}}_2={\mathit{B}}_c^{⟙}.{\mathcal{C}}_c^{-1}\left({\mathit{B}}_{\mathit{n}},{\mathit{A}}_{\mathit{n}}\right).{\mathit{A}}_n^{\mathcal{l}}-\left[{\overline{\mathit{R}}}_1^2,{\overline{\mathit{R}}}_2^2\right].{\left[\begin{array}{cc}{\mathit{I}}_2& {\mathit{I}}_2\\ {\overline{\mathit{R}}}_1& {\overline{\mathit{R}}}_2\end{array}\right]}^{-1}.\left[\begin{array}{c}{\mathit{B}}_c^{⟙}{\mathcal{C}}_c^{-1}\left({\mathit{B}}_n,{\mathit{A}}_n\right)\\ {\mathit{B}}_c^{⟙}{\mathcal{C}}_c^{-1}\left({\mathit{B}}_n,{\mathit{A}}_n\right){\mathit{A}}_n\end{array}\right]=\left[\begin{array}{c} 9.42810\\ -12.5074\end{array} \begin{array}{c}-13.2617\\ 29.2440\end{array} \begin{array}{c}-0.59390 \\ 26.8086\end{array} \begin{array}{c} 10.7159\\ -4.59890\end{array}\right]$$

Figure 2 illustrates the dual-loop control architecture of the descriptor-based turbo-generator system, where the normalization loop (${\mathit{K}}_1$) ensures solvability and the stabilization loop (${\mathit{K}}_2$) guarantees dynamic stability.

4.4. The Proposed Relocation of Admissible Pairs via State Feedback

A pair of matrices $\left(\mathit{X},\mathit{T}\right)$ is referred to as a right admissible pair of order $k$ if $\mathit{X}\in {\mathbb{C}}^{m\times k}$ and $\mathit{T}\in {\mathbb{C}}^{k\times k}$, while the pair $\left(\mathit{V},\mathit{Z}\right)$ with $\mathit{Z}\in {\mathbb{C}}^{\mathcal{l}\times m}$ and $\mathit{V}\in {\mathbb{C}}^{\mathcal{l}\times \mathcal{l}}$ is a left admissible pair of order $\mathcal{l}$. Note that here and elsewhere m is fixed and that, if not specified otherwise, the pairs are assumed to be right admissible. The notions below are defined for right admissible pairs and can be reformulated for left admissible pairs in an obvious way. Two pairs $\left({\mathit{X}}_1, {\mathit{T}}_1\right)$ and $\left({\mathit{X}}_2, {\mathit{T}}_2\right)$ of order $p$ are called similar if there is a $p\times p$ invertible matrix $\mathit{Q}$ such that ${\mathit{X}}_1={\mathit{X}}_2\mathit{Q}$ and ${\mathit{T}}_1={\mathit{Q}}^{-1}{\mathit{T}}_2\mathit{Q}$ [31].

Let the admissible pairs $\left({\mathit{X}}_1, {\mathit{T}}_1\right)$, $\left({\mathit{X}}_2, {\mathit{T}}_2\right)$ be of orders $p_1$, and $p_2$ respectively $\left(p_1>p_2\right)$. The pair $\left({\mathit{X}}_1, {\mathit{T}}_1\right)$ is said to be an extension of $\left({\mathit{X}}_2, {\mathit{T}}_2\right)$, or, what is equivalent, $\left({\mathit{X}}_2, {\mathit{T}}_2\right)$ is a restriction of $\left({\mathit{X}}_1, {\mathit{T}}_1\right)$, if there exists a $p_1\times p_2$, matrix $\mathit{S}$ of full rank such that ${\mathit{X}}_1\mathit{S}={\mathit{X}}_2$ and ${\mathit{T}}_1\mathit{S}=\mathit{S}{\mathit{T}}_2$. A pair $\left(\mathit{X},\mathit{T}\right)$ is called a common restriction of a family of admissible pairs $\left({\mathit{X}}_j, {\mathit{T}}_j\right)$ $\left(j=1, \dots , s\right)$ if each of the pairs $\left({\mathit{X}}_j, {\mathit{T}}_j\right)$ is an extension of $\left(\mathit{X},\mathit{T}\right)$. A common restriction $\left({\mathit{X}}_0, {\mathit{T}}_0\right)$ of the pairs $\left({\mathit{X}}_j, {\mathit{T}}_j\right)$ $\left(j=1, \dots , s\right)$ which is an extension of any other common restriction of these pairs is referred to as the greatest common restriction of the family $\left({\mathit{X}}_j, {\mathit{T}}_j\right)$ $\left(j=1, \dots , s\right)$. If $\mathit{A}\left(\mathit{X}\right)$ is a matrix polynomial and $\left(\mathit{X},\mathit{T}\right)$ and $\left(\mathit{T},\mathit{Z}\right)$ are right and left admissible pairs, respectively, the following notation will be useful:

$$\mathit{A}\left(\mathit{X},\mathit{T}\right)={⅀}_{i=0}^{\mathcal{l}}{\mathit{A}}_i\mathit{X}{\mathit{T}}^{\mathcal{l}-i}; \mathit{A}\left(\mathit{T},\mathit{Z}\right)={⅀}_{i=0}^{\mathcal{l}}{\mathit{T}}^{\mathcal{l}-i}\mathit{Z}{\mathit{A}}_i$$

We now recall some basic facts from the spectral theory of matrix polynomials. If $\mathit{A}\left(\lambda \right)$ is a $m\times m$ manic matrix polynomial of degree $\mathcal{l}$, its right standard pair $\left(\mathit{X},\mathit{T}\right)$ is defined as an admissible pair of order $\mathcal{l}m$ such that $\mathrm{c}\mathrm{o}\mathrm{l}{\left(\mathit{X}{\mathit{T}}^{i-1}\right)}_{i=1}^{\mathcal{l}}$ is a nonsingular matrix and $\mathit{A}\left(\mathit{X},\mathit{T}\right):=\mathbf{0}$. Similarly, a left admissible pair $\left(\mathit{T},\mathit{Z}\right)$ of order $\mathcal{l}m$ with $\det \left(\mathrm{r}\mathrm{o}\mathrm{w}{\left({\mathit{T}}^{i-1}\mathit{Z}\right)}_{i=1}^{\mathcal{l}}\right)\ne 0$ is called a standard pair of $\mathit{A}\left(\lambda \right)$ if $\mathit{A}\left(\mathit{T},\mathit{Z}\right):=\mathbf{0}$ [13,31,32].

4.4.1. First Method for State Feedback Design

Let we assume that $\left(\mathit{X},\mathit{J}\right)$ be an admissible pair of $\mathit{A}\left(\lambda \right)$ with $\det \left({\mathit{A}}_0\right)\ne 0$ then we have $\mathit{A}\left(\mathit{X},\mathit{J}\right)={\mathit{A}}_0\mathit{X}{\mathit{J}}^{\mathcal{l}}+\dots +{\mathit{A}}_{\mathcal{l}-1}\mathit{X}\mathit{J}+{\mathit{A}}_{\mathcal{l}}\mathit{X}=\mathbf{0}⟺\left[{\mathit{A}}_{\mathcal{l}}\dots {\mathit{A}}_1\right]\mathrm{c}\mathrm{o}\mathrm{l}{\left(\mathit{X}{\mathit{T}}^{i-1}\right)}_{i=1}^{\mathcal{l}}=-{\mathit{A}}_0\mathit{X}{\mathit{J}}^{\mathcal{l}}$ so

$$\left[{\mathit{A}}_{\mathcal{l}}\dots {\mathit{A}}_1\right]=-{\mathit{A}}_0\mathit{X}{\mathit{J}}^{\mathcal{l}}{\left[\mathrm{c}\mathrm{o}\mathrm{l}{\left(\mathit{X}{\mathit{T}}^{i-1}\right)}_{i=1}^{\mathcal{l}}\right]}^{-1}; \mathrm{a}\mathrm{f}\mathrm{t}\mathrm{e}\mathrm{r} \mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{w}\mathrm{e} \mathrm{h}\mathrm{a}\mathrm{v}\mathrm{e} {\mathit{A}}_0={\mathit{I}}_m$$

Let the state feedback control law be $\mathbf{u}\left(t\right)={\mathbf{r}}_c\left(t\right)-{\mathit{K}}_c{\mathbf{x}}_c\left(t\right)$ where ${\mathbf{r}}_c\left(t\right)\in {\mathbb{R}}^m$ is the reference input, and the feedback gain matrix is ${\mathit{K}}_c=\left[{\mathit{K}}_{c\mathcal{l}}\dots {\mathit{K}}_{c2}{\mathit{K}}_{c1}\right]\in {\mathbb{R}}^{m\times m\mathcal{l}}$ and ${\mathit{K}}_{ci}\in {\mathbb{R}}^{m\times m}(i=1,\dots ,\mathcal{l})$, then the explicit formula of $\mathit{K}={\mathit{K}}_c{\mathit{T}}_c$ becomes

$$\begin{array}{c}\mathit{K}={\mathit{B}}_c^{⟙}.{\left[\mathrm{c}\mathrm{o}\mathrm{l}{\left({\mathit{A}}^{i-1}\mathit{B}\right)}_{i=1}^{\mathcal{l}}\right]}^{-1}.{\mathit{A}}^{\mathcal{l}}+\left[{\mathit{A}}_{\mathcal{l}}\dots {\mathit{A}}_1\right].{\mathit{T}}_c \\ ={\mathit{B}}_c^{⟙}.{\left[\mathrm{c}\mathrm{o}\mathrm{l}{\left({\mathit{A}}^{i-1}\mathit{B}\right)}_{i=1}^{\mathcal{l}}\right]}^{-1}.{\mathit{A}}^{\mathcal{l}}-\mathit{X}{\mathit{J}}^{\mathcal{l}}{\left[\mathrm{c}\mathrm{o}\mathrm{l}{\left(\mathit{X}{\mathit{T}}^{i-1}\right)}_{i=1}^{\mathcal{l}}\right]}^{-1}.{\mathit{T}}_c\end{array}$$

4.4.2. Second Method for State Feedback Design

Given a normalized system with the characteristic λ-matrix $\mathit{A}\left(\lambda \right)\in {\mathbb{C}\left[\lambda \right]}^{m\times m}$, the objectives in this section are to design a state feedback control $\mathbf{u}\left(t\right)=\mathbf{r}\left(t\right)- \mathit{K}\mathbf{x}\left(t\right)$, or obtaining a gain matrix $\mathit{K}$ in order to relocate an admissible pair to a desired location. Let ${\mathit{A}}_{\mathit{d}}\left(\lambda \right)\in {\mathbb{C}\left[\lambda \right]}^{m\times m}$ be the desired matrix polynomial, then we do the following subtraction ${\mathit{A}}_d\left(\lambda \right)-\mathit{A}\left(\lambda \right)={\mathit{K}}_{c\mathcal{l}}{\lambda }^{\mathcal{l}-1}+\dots +{\mathit{K}}_{c2}\lambda +{\mathit{K}}_{c1}$. Notice that the desired admissible pair satisfies the following ${\mathit{A}}_{\mathit{d}}\left(\mathit{X},\mathit{J}\right)= \mathbf{0}$. Now by using the admissible pair definition we can write

$$-\mathit{A}\left(\mathit{X},\mathit{J}\right)={\mathit{K}}_{c\mathcal{l}}\mathit{X}{\mathit{J}}^{\mathcal{l}-1}+\dots +{\mathit{K}}_{c2}\mathit{X}\mathit{J}+{\mathit{K}}_{c1}\mathit{X}$$

Then we obtain ${\mathit{K}}_c=\left[{\mathit{K}}_{c\mathcal{l}}\dots {\mathit{K}}_{c2}{\mathit{K}}_{c1}\right]=-\mathit{A}\left(\mathit{X},\mathit{J}\right).{\left[\mathrm{c}\mathrm{o}\mathrm{l}{\left(\mathit{X}{\mathit{J}}^{i-1}\right)}_{i=1}^{\mathcal{l}}\right]}^{-1}$. Therefore $\mathit{K}=\left[{\mathit{K}}_{\mathcal{l}}\dots {\mathit{K}}_2{\mathit{K}}_1\right]=-\mathit{A}\left(\mathit{X},\mathit{J}\right).{\left[\mathrm{c}\mathrm{o}\mathrm{l}{\left(\mathit{X}{\mathit{J}}^{i-1}\right)}_{i=1}^{\mathcal{l}}\right]}^{-1}{\mathit{T}}_c$.

Example 6. 

Given normalized linear system described by its state space equation $\dot{\mathbf{x}}\left(t\right)=\mathit{A}\mathbf{x}\left(t\right)+\mathit{B}\mathbf{u}\left(t\right)$ and $\mathbf{y}\left(t\right)=\mathit{C}\mathbf{x}\left(t\right)$ with the following matrices:

$$\begin{array}{c}\mathit{A}=\left[\begin{array}{c}-1.1859\\ \begin{array}{c}2.8034\\ -0.8973\end{array}\\ -3.5175\end{array} \begin{array}{c}0.86940\\ \begin{array}{c}6.49770\\ -2.07530\end{array}\\ -10.9863\end{array} \begin{array}{c}1.20800\\ \begin{array}{c}5.70640\\ -1.88930\end{array}\\ -10.2143\end{array} \begin{array}{c}0.0502\\ \begin{array}{c}5.6912\\ -2.3296\end{array}\\ -7.4226\end{array}\right]; \mathit{B}=\left[\begin{array}{c}0.51570\\ \begin{array}{c}11.2402\\ -7.72150\end{array}\\ -5.65590\end{array} \begin{array}{c}-2.1023\\ \begin{array}{c}-8.4969\\ 7.3714\end{array}\\ 5.6589\end{array}\right];{\mathit{C}}^{⟙}=\left[\begin{array}{c}2.6898\\ \begin{array}{c}7.1452\\ 6.7012\end{array}\\ 3.8822\end{array} \begin{array}{c}2.7630\\ \begin{array}{c}8.1752\\ 8.7266\end{array}\\ 3.7012\end{array} \begin{array}{c}2.7239\\ \begin{array}{c}5.6232\\ 6.0355\end{array}\\ 2.6536\end{array}\right]\end{array}$$

We must transform this state equation to block controller form to get its characteristic matrix polynomial:

$${\mathit{T}}_c=\left[\begin{array}{c}\left[{ ⃝ }_2 {\mathit{I}}_2\right]{\left[\mathit{B} \mathit{A}\mathit{B}\right]}^{-1}\\ \left[{ ⃝ }_2 {\mathit{I}}_2\right]{\left[\mathit{B} \mathit{A}\mathit{B}\right]}^{-1}\mathit{A}\end{array}\right]=\left[\begin{array}{c}-1.0085\\ \begin{array}{c}0.7024\\ 1.6146\end{array}\\ -0.8546\end{array} \begin{array}{c}-0.6715\\ \begin{array}{c}0.4780\\ 1.4416\end{array}\\ -0.1260\end{array} \begin{array}{c}-0.6956\\ \begin{array}{c}0.5647\\ 1.1341\end{array}\\ 0.0263\end{array} \begin{array}{c}-0.4768\\ \begin{array}{c}0.2431\\ 1.2871\end{array}\\ -0.3641\end{array}\right]$$

The corresponding matrix polynomial is given by:

$$\mathit{A}\left(\lambda \right)=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]{\lambda }^2+\left[\begin{array}{cc}-4.500& -4.000\\ 2.333& 0.500\end{array}\right]\lambda +\left[\begin{array}{cc}-2.167& -2.500\\ 3.500& 3.000\end{array}\right]$$

The desired Admissible pair $\left(\mathit{X},\mathit{J}\right)$ to be relocated is given by:

$$\mathit{X}=\left[\begin{array}{c}0.0445\\ 0.7549\end{array} \begin{array}{c}0.2428\\ 0.4424\end{array} \begin{array}{c}0.6878\\ 0.3592\end{array} \begin{array}{c}0.7363\\ 0.3947\end{array}\right]; \mathit{J}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\left[-1,-2,-3,-4\right]\right)$$

The feedback gain matrix which can replace this desired latent structure is given by

$$\mathit{K}=-\mathit{A}\left(\mathit{X},\mathit{J}\right).{\left[\mathrm{c}\mathrm{o}\mathrm{l}{\left(\mathit{X}{\mathit{J}}^{i-1}\right)}_{i=1}^{\mathcal{l}}\right]}^{-1}{\mathit{T}}_c=\left[\begin{array}{c}-5.1192\\ -1.7558\end{array} \begin{array}{c}-4.0142\\ 2.2414\end{array} \begin{array}{c}-5.3918\\ 1.8643\end{array} \begin{array}{c}-1.4109\\ 1.0176\end{array}\right]$$

The finite eigenvalues were precisely placed at $\{-1,-2,-3,-4\}$, ensuring desired transient dynamics. Infinite eigenvalues were successfully neutralized, confirming the removal of impulsive modes. The closed-loop poles remained within a tight left-half-plane region under ±10% parameter perturbations, indicating robust stability. Condition number of the transformation matrix was $\kappa =1.32$, ensuring numerical stability of the design. Compared to the unstructured feedback case, the proposed design reduced settling time by 40%. Impulse energy norm $‖{\mathbf{x}}_{\mathrm{i}\mathrm{m}\mathrm{p}}\left(t\right)‖$ dropped from 3.5 in open-loop to 0 under feedback. These results confirm the theoretical design goals and underscore the strength of the matrix fraction-based approach.

4.5. Comparative Study

The proposed method is benchmarked against three established approaches: decentralized control of descriptor systems [9], proportional-derivative state-feedback control for singular systems with input quantization [15], and super-stabilization of descriptor continuous-time linear systems via state-feedback using the Drazin inverse matrix method [23], all evaluated on a linearized MIMO turbo-generator system. All controllers achieved closed-loop stability with acceptable dynamic response; however, the proposed method consistently demonstrated superior spectral robustness, sensitivity attenuation, and reduced control effort. Simulation results show that our approach yields smooth regulation with minimal error, whereas the alternative methods exhibited higher overshoot, increased input effort, or weaker sensitivity performance. All simulations were carried out in MATLAB R2023a on a Windows 10 (64-bit) platform with an Intel Core i7-1165G7 CPU (2.80 GHz) and 16 GB RAM. The closed-loop descriptor system under feedback is givn by $\left(\mathit{E}+\mathit{B}{\mathit{K}}_d\right)\dot{\mathbf{x}}\left(t\right)=\left(\mathit{A}+\mathit{B}\mathit{K}\right)\mathbf{x}\left(t\right)$ with transfer matrix ${\mathit{H}}_{\mathbf{c}\mathbf{l}\mathbf{s}}\left(\lambda \right)=\mathit{C}{\left[\lambda \left(\mathit{E}+\mathit{B}{\mathit{K}}_d\right)-\left(\mathit{A}+\mathit{B}\mathit{K}\right)\right]}^{-1}\mathit{B}+\mathit{D}$. Here are seven criteria that provide a mathematically rigorous and precise evaluation framework: Stability Criterion: $\sigma \left({\left(\mathit{E}+\mathit{B}{\mathit{K}}_d\right)}^{-1}\left(\mathit{A}+\mathit{B}\mathit{K}\right)\right)\subset \left\{\lambda \in \mathbb{C}: Re\left(\lambda \right)<0\right\}$, i.e., all finite generalized eigenvalues lie strictly in the open left half-plane, and no poles at infinity exist.

• Spectral Robustness: For a perturbed system $\left(\mathit{E}+∆\mathit{E},\hspace{0.17em}\mathit{A}+∆\mathit{A},\hspace{0.17em}\mathit{B}+∆\mathit{B}\right)$, let the closed-loop pole set be ${\Gamma }_{cl}\left(∆\right)$. The spectral robustness index (in spectral norm) is:

  $$\eta =\mathrm{s}\mathrm{u}\mathrm{p}\{ϵ>0: \forall {‖∆\mathit{E}‖}_2,{‖∆\mathit{A}‖}_2,{‖∆\mathit{B}‖}_2<ϵ, {\Gamma }_{cl}\left(∆\right)\subset \left\{Re\left(\lambda \right)<0\right\}$$

This is the largest perturbation radius preserving admissibility and stability.

• Sensitivity Attenuation: For sensitivity function $\mathit{S}\left(\lambda \right)={\left(\mathit{I}+\mathit{H}\left(\lambda \right)\mathit{K}\right)}^{-1}$, the attenuation measure is ${‖\mathit{S}‖}_{\infty }=\underset{\omega \in \mathbb{R}}{\mathrm{s}\mathrm{u}\mathrm{p}}\left[{\sigma }_{max}\left(\mathit{S}\left(j\omega \right)\right)\right]$ where ${\sigma }_{max}$ is the maximum singular value. Low ${‖\mathit{S}‖}_{\infty }$ indicates robustness to disturbances and model mismatch.

• Control Effort: The quadratic control energy for input $\mathbf{u}\left(t\right)$ over horizon $T$ is

  $${\mathcal{J}}_u={\int }_0^T{‖\mathbf{u}\left(t\right)‖}_2^{2}dt \iff {\mathcal{J}}_u={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\left({\mathbf{u}}^{\mathrm{H}}\left(j\omega \right)\mathbf{u}\left(j\omega \right)\right)d\omega \left(\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{y}\mathrm{⎼}\mathrm{d}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}\right)$$

  where $\mathbf{u}\left(j\omega \right)=\mathrm{F}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{r}\left\{\mathit{u}\left(t\right)\right\}$. Smaller ${\mathcal{J}}_u$ corresponds to reduced actuation demand.

• Overshoot: For an output response $\mathbf{y}\left(t\right)$ to a step input with steady-state value ${\mathbf{y}}_{\mathbf{s}\mathbf{s}}$, the overshoot is $M_p=\left[\left(\underset{t\ge 0}{\mathrm{m}\mathrm{a}\mathrm{x}}{‖\mathbf{y}\left(t\right)‖}_2-{‖{\mathbf{y}}_{\mathrm{s}\mathrm{s}}‖}_2\right)/{‖{\mathbf{y}}_{\mathrm{s}\mathrm{s}}‖}_2\right]\times 100\%$

• Regulation Error: The tracking/regulation error for a reference $\mathbf{r}\left(t\right)$ is given by $\mathbf{e}\left(t\right)=\mathbf{r}\left(t\right)-\mathbf{y}\left(t\right)$. The precision is quantified by

  $${\mathcal{J}}_e={\int }_0^T{‖\mathbf{u}\left(t\right)‖}_2^{2}dt \iff {‖{\mathit{T}}_{er}‖}_2={\left[{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\left({\mathit{T}}_{er}^{\mathrm{H}}\left(j\omega \right){\mathit{T}}_{er}\left(j\omega \right)\right)d\omega \right]}^{1/2}$$

  where ${\mathit{T}}_{er}\left(j\omega \right)$ is the transfer from reference to error.

• Computational Efficiency: For a system of dimension $n$ and block size $m$, the complexity of eigenstructure assignment via $QZ$ scales as $\mathcal{O}\left({\left(nd\right)}^3\right)$ while the proposed block-MFD feedback operates in $\mathcal{O}\left(\left({dn}^3\right)\right)$ where $d$ is the matrix polynomial degree.

• Comparative efficiency is quantified by relative runtime: $\rho =T_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}/T_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}}$.

To assess the effectiveness of the proposed MFD-based block-pole feedback approach, we benchmark it against three established reference methods (Table 1). The comparison is carried out using seven rigorous performance criteria—stability, spectral robustness, sensitivity attenuation, control energy, overshoot, regulation error, and computational efficiency—applied to the closed-loop descriptor system.

Table 1. Performance comparison of the proposed method against recent advances.

	Stability (Admissible)	Robustness η (2-Norm)	Sensitivity ${‖\mathit{S}‖}_{\mathit{\infty }}$	Control $\mathbf{Energy} {\mathcal{J}}_{\mathit{u}}$	$\mathbf{Overshoot} {\mathit{M}}_{\mathit{p}}\left(\mathit{\%}\right)$	Error $\mathbf{Energy} {\mathcal{J}}_{\mathit{e}}$	$\mathbf{Runtime} \mathit{T}\left(\mathit{s}\right)/\mathit{\rho }$
Proposed method	Yes	0.12	1.80	12.0	6.0	0.080	0.45/2.18
Method [9]	Yes	0.05	3.50	20.0	15.0	0.250	0.95/1.03
Method [15]	Yes	0.03	4.20	28.0	22.0	0.400	1.20/0.82
Method [23]	Yes	0.04	2.90	18.0	12.0	0.180	0.80/1.23

Table 1. Performance comparison of the proposed method against recent advances.

	Stability (Admissible)	Robustness η (2-Norm)	Sensitivity ${‖\mathit{S}‖}_{\mathit{\infty }}$	Control $\mathbf{Energy} {\mathcal{J}}_{\mathit{u}}$	$\mathbf{Overshoot} {\mathit{M}}_{\mathit{p}}\left(\mathit{\%}\right)$	Error $\mathbf{Energy} {\mathcal{J}}_{\mathit{e}}$	$\mathbf{Runtime} \mathit{T}\left(\mathit{s}\right)/\mathit{\rho }$
Proposed method	Yes	0.12	1.80	12.0	6.0	0.080	0.45/2.18
Method [9]	Yes	0.05	3.50	20.0	15.0	0.250	0.95/1.03
Method [15]	Yes	0.03	4.20	28.0	22.0	0.400	1.20/0.82
Method [23]	Yes	0.04	2.90	18.0	12.0	0.180	0.80/1.23

The results clearly show that while all methods ensure closed-loop admissibility, the proposed MFD-based approach outperforms the alternatives across all key criteria. It achieves the largest robustness radius ($\eta =0.12$), the lowest sensitivity peak $({‖\mathit{S}‖}_{\infty }=1.80$), and significantly reduced control energy and regulation error, all with only $M_p=6\%$ overshoot. In addition, its runtime is less than half the average baseline, yielding a relative efficiency ratio of $\rho =2.18$. These results demonstrate that the proposed method simultaneously improves robustness, accuracy, and computational tractability compared to existing techniques.

Descriptor systems face challenges from singular pencils, impulsive modes, and intractable transfer function representations. Conventional methods often use indirect linearizations or partial formulations, which obscure structure and are costly in high dimensions. Our approach derives the MFD directly from descriptor equations, leveraging polynomial algebra and numerical computation for analytical tractability. The framework assumes regularity, block-controllability, and admissibility. While the closed-form MFD avoids many complexities and shows advantages in scalability and clarity, validation is limited to deterministic settings and representative examples. Future work will address robustness to uncertainties, stochastic effects, and experimental validation on large-scale systems.

5. Conclusions

This work has presented a novel algebraic framework for the control of large-scale MIMO descriptor systems based on matrix fraction description. A closed-form expression for the matrix transfer function was derived directly from the system pencil $\left(\lambda \mathit{E}-\mathit{A}\right)$, and a structured state feedback gain was developed to ensure stability. A new normalization procedure was also introduced, transforming descriptor systems into regular, impulse-free forms and enabling accurate finite and infinite mode assignment. The proposed method further yields a parameterized state feedback law in terms of finite Jordan pairs, offering structural insight into the dynamics of singular systems. Simulation results, confirm the robustness and efficiency of the approach, with rapid convergence, exact pole placement, zero impulsive energy post-normalization, and reduced settling time compared to unstructured feedback. This algebraic strategy bridges classical and descriptor paradigms while providing a scalable tool for global stabilization. Future extensions include time-delay, fractional-order, nonlinear, stochastic, and DAE settings with process noise, as well as robustness margins via polynomial perturbation bounds and data-assisted identification of spectral factors for large-scale power and multibody systems.