Generalized Criteria for Admissibility of Singular Fractional Order Systems

Abstract

Unified frameworks for fractional order systems with fractional order $0<\alpha <2$ are worth investigating. The aim of this paper is to provide a unified framework for stability and admissibility for fractional order systems and singular fractional order systems with $0<\alpha <2$, respectively. By virtue of the LMI region and GLMI region, five stability theorems are presented. Two admissibility theorems for singular fractional order systems are extended from Theorem 5, and, in particular, a strict LMI stability criterion involving the least real decision variables without equality constraint by isomorphic mapping and congruent transform. The equivalence between the admissibility Theorems 6 and 7 is derived. The proposed framework contains some other existing results in the case of $1\le \alpha <2$ or $0<\alpha <1$. Compared with published unified frameworks, the proposed framework is truly unified and does not require additional conditional assignment. Finally, without loss of generality, a unified control law is designed to make the singular feedback system admissible based on the criterion in a strict LMI framework and demonstrated by two numerical examples.

1. Introduction

In the development of fractional calculus theory [1] and computer technology, the study of fractional order systems (FOSs) has become an important research topic. Since fractional derivatives can be used to study the behavior of materials and systems with power-law, nonlocal, long-term memory, or fractal properties, FOSs described by fractional order differential equations have better modeling and analysis capabilities in describing complex systems [2,3] and nonlinear phenomena [4,5,6].

The stability analysis of FOSs is an important aspect of fractional order control theory. With further research, it has been found that the stability of fractional order systems is quite different from that of integer order systems. Therefore, new methods for analyzing stability need to be developed to accommodate the special properties of FOSs. Many basic concepts and results of traditional state-space systems have been successfully extended to FOSs. Matignon’s fractional order stability theorem [7] proposed that the stability of fractional order linear time-invariant systems can be checked by the eigenvalues of the system matrix, while it is inconvenient to calculate all the eigenvalues of the matrix to control the system. In the 1990s, with the introduction of the interior point method for solving convex optimization problems [8,9,10], the LMI problem was well solved, and the LMI toolbox was also launched in MATLAB, which greatly promoted the application of LMI in the control field. Based on the importance of LMI in the control theory of integer order systems, many scholars aimed to apply LMI to the control theory of FOSs. Furthermore, many scholars began to explore more effective and LMI-based stability criteria [11,12,13]. The robust stability problem of FOSs with interval uncertainties was studied in [14,15], and an algorithm code in MATLAB was proposed.

The key point in dealing with LMIs is convexity. However, the stability domain of FOSs with fractional order $0<\alpha <1$ is nonconvex. Therefore, the classical stability conditions cannot be directly extended to FOSs. The stability in the case of $1\le \alpha <2$ is studied in [12] and a criterion in the form of LMI is given. Another LMI-based criterion in the case of $1\le \alpha <2$ is given in [13]. These two criteria are consistent with the results of pole placement in convex subregions of integer order systems in the left complex plane given in [16,17]. In addition, Ref. [13] also gives the stability criteria of FOSs with multiple differential orders in the interval of $(0,1)$. However, these criteria are given in the form of matrix inequalities including the terms of $A^{\frac{1}{\alpha }}P$, ${(-(-A))}^{\frac{1}{2-\alpha }}P$ and $e^{\mathrm{j}(1-\alpha )\frac{\pi }{2}}AP$, which make it difficult to solve by using the LMI toolbox when designing the controller for the associated closed-loop system, because of the existing powers of $\frac{1}{\alpha }$, $\frac{1}{2-\alpha }$ at $A+BK$ and complex variable in $e^{\mathrm{j}(1-\alpha )\frac{\pi }{2}}(A+BK)P$. Another criterion for the stability in the case of $0<\alpha <1$ is studied in [15] and the result is in the form of LMI, which requires additional conditions when designing the controller. Ref. [18] first proposed the conditions guaranteeing the admissibility of singular fractional order systems (SFOSs) when the fractional order $\alpha$ belongs to the interval $(0,2)$. The unified LMI criterion of $0<\alpha <2$ is first proposed in [19]. However, the criterion is unified only in the form of mathematical expression, that is, it needs to add a conditional assignment statement associated with the fractional order $\alpha$ when using the LMI toolbox to solve the problem.

Corresponding to the fact that FOSs have attracted extensive attention in the control community, SFOSs have also become an important research object. Singular systems [20] have a more general form than normal systems. Singular systems are described by both differential and algebraic equations, so they are also called differential-algebraic systems, generalized systems, etc. Singular systems, due to the consideration of admissibility, including regularity, impulse-freeness and stability, are more complex to study than normal systems. In [21], the robust stabilization of SFOSs is studied in two cases of $1\le \alpha <2$ and $0<\alpha <1$.

Ref. [22] extended the concept of singular systems from integer order to fractional order and proposed a criterion for the admissibility of SFOSs on the basis of the stability results given in [15]. However, this result requires additional conditions when designing controllers. Another criterion for the admissibility in the case of $0<\alpha <1$ is proposed in [23], which is in the form of complex LMI. Ref. [24] gave three different criteria for the admissibility and stabilization of SFOSs when the differential order is in the interval $0<\alpha <1$, which promoted the research [25,26] on related problems of SFOSs to some extent. The admissibility of SFOSs is still a hot topic in the field of control. Refs. [27,28,29] give LMI-based criteria with complex matrices in the case of $0<\alpha <1$. The criteria for the admissibility with differential order $1\le \alpha <2$ are given in [27], and the scheme for designing the controller is given in the form of bilinear matrix inequality. Based on the result in [19], the necessary and sufficient condition for admissibility and quadratic admissibility of SFOSs with fractional order $0<\alpha <2$ are given in [26,30], respectively.

At present, the stability of FOSs is still a hot topic [31,32,33,34,35]. Although there have been a lot of papers in the field of stability for FOSs, most of the existing results focus on the stability analysis of FOSs separately when the fractional order $\alpha$ belongs to the interval $[1,2)$ and $(0,1)$, and thus can only give different forms of criteria. This means other related studies based on the premise of stability can only be divided into two cases, which makes the research results relatively fragmented. The stability and admissibility problems of FOSs with $0<\alpha <2$ have been studied in [26,30], but their results are still in the form of a piece-based function, that is, it is still divided into two criteria rather than a unified form. In view of the above observation, this paper is concerned with the stability and admissibility in terms of fractional order $0<\alpha <2$ and provides several unified frameworks, respectively, for stability and admissibility regardless of the fractional order interval. Here is the contribution of our work:

• An LMI region and a GLMI region are appiled to study the stability region of FOSs in the case of $0<\alpha <2$, leading to a unified criterion for the stability of FOSs, which can only be discussed and studied separately in most existing results because of the different convexity in the case of $0<\alpha <1$ and $1\le \alpha <2$. Furthermore, since this method is applicable to larger fractional intervals, it is more advantageous than the existing results when investigating systems with varying fractional orders.
• Based on isomorphic mapping and congruent transform to deal with the unified criteria for stability with complex decision variables, the new approach to stability analysis of FOSs with $0<\alpha <2$ is derived, which contains the least decision variables and does not involve a complex matrix. This avoids the problem that the existing results need to solve the complex variables directly, which is difficult to solve by using the LMI toolbox.
• The new way to analyze the admissibility for FOSs with $0<\alpha <2$ eliminates the equality constraints, avoids the non-strict LMIs and has the form of strict LMIs, so it offers an easier method to solve the decision variables than the existing literature. In addition, this method only involves a few real decision variables, so it can be directly used to design the controller without additional conditions and has no conservatism.

Notation 1.

In the paper, $A<0$ and $A\le 0$ denote that the matrix A is negative definite and negative semi-definite, respectively. We denote by $A^{\mathrm{T}}$ the transpose of matrix A, by $A^{*}$ the transpose conjugate of A and by $\mathcal{H}\left\{A\right\}$ the Hermitian expression $A+A^{*}$. We denote by $\Re \left(A\right)$ the real part of A and by $\Im \left(A\right)$ the imaginary of A. For $0<\alpha <2$, $a=\sin \left(\alpha \frac{\pi }{2}\right)$ and $b=\cos \left(\alpha \frac{\pi }{2}\right)$. We denote $\left. [\begin{array}{cc}a& b\\ -b& a\end{array}\right]$ by Θ. $I_t$ represents the $t-$dimension identity matrix. The zero matrix with appropriate dimensions is represented by 0. The symbol * is used to represent the matrices irrelevant in the later analysis. $\lceil ·\rceil$ is the ceiling function to increase · to the next highest whole number and $\lfloor ·\rfloor$ is the floor function to reduce · to the next nearest whole number. Given two arbitrary matrices $F={\left(f_{ij}\right)}_{m\times n}$ and $G={\left(g_{ij}\right)}_{t\times r}$, the Kronecker product is defined as $F\otimes G=\left(f_{ij}G\right)$.

2. Problem Statement and Preliminaries

In this section, the purpose of this paper is given and some definitions and lemmas are recalled as mathmatical preliminaries.

2.1. System Description

Consider a singular fractional electrical circuit with resistors, supercapacitances, superinductors and voltage sources. There are the following relationships:

$$i_C\left(t\right)=C{D}^{\alpha }{u}_C\left(t\right),$$

$$u_L\left(t\right)=L{D}^{\alpha }{i}_L\left(t\right),$$

where $\alpha$ is the fractional order. C and L are the capacity and the inductance, respectively. $i_C\left(t\right)$ and $i_L\left(t\right)$ are in turn the currents in a supercondensator and a superinductor. $u_C\left(t\right)$ and $u_L\left(t\right)$ are the voltage on the supercondensator and the superinductor. Due to the above physical expression, a singular fractional electrical circuit is essentially an SFOS described by

$$\begin{array}{cc}\hfill E{D}^{\alpha }x\left(t\right)& =Ax\left(t\right)+Bu\left(t\right),\hfill \end{array}$$

(1)

where $A\in {\mathbb{R}}^{n\times n}$, $B\in {\mathbb{R}}^{n\times l}$, $E\in {\mathbb{R}}^{n\times n}$ is supposed to satisfy $0< \mathrm{rank}\left(E\right)=m<n.$$x\left(t\right)$ and $u\left(t\right)$ are n-dimensions system state and l-dimensions control input. The symbol $D^{\alpha }x\left(t\right)$ is used for the Caputo fractional order derivative of $x\left(t\right)$, which is defined in [1]. When $u\left(t\right)=0$, system (1) becomes an unforced SFOS described by

$$E{D}^{\alpha }x\left(t\right)=Ax\left(t\right).$$

(2)

We can always find two matrices L, R such that

$$LER=\left. [\begin{array}{cc}{I}_m& 0\\ 0& 0\end{array}\right],LAR=\left. [\begin{array}{cc}{A}_1& A_2\\ A_3& A_4\end{array}\right].$$

System (2) is regular if $\det (s^{\alpha }E-A)\not\equiv 0$ and is impulse-free if $A_4$ is invertible. Then we denote by $\mathrm{spec}(E,A)=\mathrm{spec}(I_m,A_1-A_2{A}_4^{-1}{A}_3)$ the spectrum of matrix $A_1-A_2{A}_4^{-1}{A}_3$, where $A_1$ is a square matrix in m-dimensions. If $E=I$, system (2) comes down to a normal FOS with the following form:

$$D^{\alpha }x\left(t\right)=Ax\left(t\right),$$

(3)

and then we abbreviate $\mathrm{spec}(I,A)$ to $\mathrm{spec}\left(A\right)$ for ease of expression. The stability of (3) and the admissibility of system (2) both with the fractional order $0<\alpha <2$ are studied in the sequel. For the stability of system (3), Matignon’s fractional order stability theorem presented in [7] states that system (3) with $0<\alpha <2$ is stable iff

$$\left|\arg \left(\mathrm{spec}\left(A\right)\right)\right|>\alpha \frac{\pi }{2}.$$

(4)

However, it is inconvenient to control the system by computing all eigenvalues of the matrix A, so a more suitable unified framework needs to be found. As shown in Figure 1, from the criterion (4), the stability domain for the system (3) is $D_1\cap D_2$ when $1\le \alpha <2$ (${\phi }_1=-{\phi }_2=(\alpha -1)\frac{\pi }{2}$) and $D_1\cup D_2$ when $0<\alpha <1$ (${\phi }_1=-{\phi }_2=(1-\alpha )\frac{\pi }{2}$). In addition, system (2) is called admissible if it is regular, impulse-free and stable. Thus, this paper aims to provide a unified framework for stability and admissibility.

2.2. Definitions and Lemmas

In this subsection, we introduced the definitions of the LMI region, generalized LMI region in the complex plane and their associated lemmas. Note that the stability domain of the FOS (3) is a subregion of the complex plane according to the criterion (4), so these definitions and lemmas are useful tools for the analysis of stability and admissibility of FOSs in this paper.

Definition 1 ([16]).

A convex LMI region is a convex subset $D_R$ of the complex plane defined by

$$D=\{s\in \mathbb{C}:f_{D_R}\left(s\right)<0\},$$

(5)

where $f_{D_R}\left(s\right)=L+sM+\overline{s}{M}^{\mathrm{T}}$, L is a symmetric matrix and $L\in {\mathbb{R}}^{q\times q}$, $M\in {\mathbb{R}}^{q\times q}$.

Lemma 1 ([16]).

All the eigenvalues of A are in the region described by (5) iff there exists $P\in {\mathbb{R}}^{n\times n}>0$ such that $L\otimes P+M\otimes \left(AP\right)+M^{\mathrm{T}}\otimes \left(P{A}^{\mathrm{T}}\right)<0$.

It can be seen that the LMI region defined in Definition 1 is limited to a convex subregion in the complex plane, so it can only be used to study the stability for FOSs with $1\le \alpha <2$. To solve the stability problem in the case of $0<\alpha <1$, the following definition is introduced, which generalizes the region in Definition 1 to regions including but not limited to convexity.

Definition 2 ([36]).

Let $\mathcal{M}$ be a set of $n_M$ Hermitian matrices $M_k$, $k=1,2,\dots ,n_M$, $M_k\in {\mathbb{C}}^{2d\times 2d}$. A generalized LMI region $D_U$ of degree d is defined by

$$D_U=\{s\in \mathbb{C}:\exists \eta ={\left. [\begin{array}{cccc}{\eta }_1& {\eta }_2& \dots & {\eta }_{n_M}\end{array}\right]}^{\mathrm{T}},{\eta }_k\in {\mathbb{R}}^{+*},f_{D_U}<0\},$$

where $f_{D_U}={\sum }_{k=1}^{n_M}{f}_{D_k}$, $f_{D_k}={\eta }_k\left. [\begin{array}{cc}{I}_d& s{I}_d\end{array}\right]M_k{\left. [\begin{array}{cc}{I}_d& \overline{s}{I}_d\end{array}\right]}^{\mathrm{T}}$. When $d=2$, $f_{D_U}$ is an available description deduced from a scalar inequality. In addition, $D_U$ is the union of $n_M$ subregions $D_k$, $k=1,2,\dots ,n_M$.

Lemma 2 ([36]).

Let $D_U$ be a generalized LMI region described in Definition 2. All eigenvalues of the matrix $A\in {\mathbb{C}}^{n\times n}$ are in the region $D_U$ iff there exists a set of Hermitian positive definite matrices $P_k\in {\mathbb{C}}^{n\times n}$, $k=1,2,\dots ,n_M$ such that

$$\left. [\begin{array}{cc}{I}_{dn}& I_d\otimes A\end{array}\right](\sum _{k=1}^{n_M}{M}_k\otimes P_k){\left. [\begin{array}{cc}{I}_{dn}& I_d\otimes A\end{array}\right]}^{*}<0.$$

An LMI in complex variables can be converted to an LMI of larger dimension in real variables through an equivalence. Before giving the framework, it remains to give the following lemma to solve the solution problem caused by complex linear matrix inequalities.

Lemma 3 ([37,38]).

Given $X\in {\mathbb{R}}^{n\times n}$ and $Y\in {\mathbb{R}}^{n\times n}$,

$$X+\mathrm{j}Y>0,$$

iff

$$\left. [\begin{array}{cc}X& -Y\\ Y& X\end{array}\right]>0,$$

or

$$\left. [\begin{array}{cc}X& Y\\ -Y& X\end{array}\right]>0.$$

Proof. 

First, we aim to demonstrate the equivalence of $X+\mathrm{j}Y>0$ and $X-\mathrm{j}Y>0$. By the definition of a Hermitian positive definite matrix, $X+\mathrm{j}Y>0$ is equivalent to $z^{*}(X+\mathrm{j}Y)z>0$ for all non-zero vectors $z\in {\mathbb{C}}^n$. Since z is arbitrary, we also have ${\overline{z}}^{*}(X+\mathrm{j}Y)\overline{z}>0$. By the property that the conjugate of a positive real number is itself, it follows that $\overline{{\overline{z}}^{*}(X+\mathrm{j}Y)\overline{z}}>0$, which implies $z^{*}(X-\mathrm{j}Y)z>0$ for all non-zero vectors $z\in {\mathbb{C}}^n$, and hence $X-\mathrm{j}Y>0$. Similarly, it can be shown that $X-\mathrm{j}Y>0$ implies $X+\mathrm{j}Y>0$. Thus, we have

$$\left. [\begin{array}{cc}X-\mathrm{j}Y& 0\\ 0& X+\mathrm{j}Y\end{array}\right]>0.$$

(6)

Pre- and post- multiplying (6) by

$$\left. [\begin{array}{cc}1& 1\\ \mathrm{j}& -\mathrm{j}\end{array}\right]\otimes \left(\frac{I_n}{\sqrt{2}}\right),$$

and its conjugate transpose, respectively, from the congruence transformation property, (6) turns to

$$\left. [\begin{array}{cc}X& -Y\\ Y& X\end{array}\right]>0.$$

(7)

Using the Schur complement on (6), the following inequality

$$\left. [\begin{array}{cc}X& Y\\ -Y& X\end{array}\right]>0,$$

is derived. And vice versa, (6) can be derived from (7) in the same way. This completes the proof.    □

3. Main Results

The unified criteria for stability and admissibility of FOSs are sequentially introduced in this section.

3.1. Stability of FOSs

This subsection provides five criteria for the stability of FOSs with $0<\alpha <2$ in a unified framework.

Theorem 1.

System (3) with $0<\alpha <2$ is stable iff there exist $X_1,X_2\in {\mathbb{C}}^{n\times n}$ satisfying $X_1>0$, $X_2>0$ and

$$\mathcal{H}\{(a+\mathrm{j}b)A{X}_1+\lceil 1-\alpha \rceil (a-\mathrm{j}b)A{X}_2\}<0.$$

(8)

Proof. 

Note that the convexity of the stability domain of $0<\alpha <2$ changes with different value ranges of $\alpha$. Therefore, it is appropriate to discuss two cases: $1\le \alpha <2$ (its stability region is convex) and $0<\alpha <1$ (whose stability region is nonconvex).

Case $1\le \alpha <2$: In this case, the stable region is $D_1\cap D_2$, shown in Figure 1. Note that the stability region is an intersection of $D_1$ and $D_2$, where

$$D_1=\{s\in \mathbb{C}:f_{D_1}<0\},$$

$$D_2=\{s\in \mathbb{C}:f_{D_2}<0\},$$

where $f_{D_1}$, $f_{D_2}$ are defined in Definition 2, $n_M=1$ both in $f_{D_1}$ and $f_{D_2}$, and the matrix $M_1$, $M_2$ in $f_{D_1}$, $f_{D_2}$ in turn are

$$\left. [\begin{array}{cc}0& a+\mathrm{j}b\\ a-\mathrm{j}b& 0\end{array}\right],\left. [\begin{array}{cc}0& a-\mathrm{j}b\\ a+\mathrm{j}b& 0\end{array}\right].$$

(9)

As for some $s\in \mathrm{spec}\left(A\right)$, $\overline{s}\in \mathrm{spec}\left(A\right)$, and note that $D_1$ and $D_2$ are symmetric with respect to the real axis, either one of $f_{D_1}$ and $f_{D_2}$ is equivalent to the stability region of A. From Lemma 2, all eigenvalues of the matrix A are in the region $D_1$ iff there exists a Hermitian positive definite matrix $X_1$ satisfying

$$(a+\mathrm{j}b)A{X}_1+(a-\mathrm{j}b)X_1{A}^{\mathrm{T}}<0,$$

which is identical to (8) with $\lceil 1-\alpha \rceil =0$. This completes the proof for the case of $1\le \alpha <2$.

Case $0<\alpha <1$: In this case, the stability region becomes $D_1\cup D_2$, where $D_1$ and $D_2$ are defined in Definition 2 with $M_1$, $M_2$ in $f_{D_1}$, $f_{D_2}$ defined by (9). From Lemma 2, all eigenvalues of the matrix A are in the region $D_U=D_1\cup D_2$ iff there exist two Hermitian positive definite matrices $X_1$, $X_2$ satisfying

$$(a+\mathrm{j}b)A{X}_1+(a-\mathrm{j}b)X_1{A}^{\mathrm{T}}+(a-\mathrm{j}b)A{X}_2+(a+\mathrm{j}b)X_2{A}^{\mathrm{T}}<0,$$

(10)

which is identical to (8) with $\lceil 1-\alpha \rceil =1$. This completes the proof for the case of $0<\alpha <1$.

Combining the above two cases, this completes the proof.    □

Theorem 2.

System (3) with $0<\alpha <2$ is stable iff there exist matrices $X_1,X_2\in {\mathbb{C}}^{n\times n}$ satisfying $X_1>0$, $X_2>0$ and

$$\mathcal{H}\{\Theta \otimes \left(A{X}_1\right)+\lceil 1-\alpha \rceil ({\Theta }^{\mathrm{T}}\otimes \left(A{X}_2\right))\}<0.$$

(11)

Proof. 

Along the same lines as in the proof of Theorem 1 with $M_1$ defined as

$$\left. [\begin{array}{cc}0& \Theta \\ {\Theta }^{\mathrm{T}}& 0\end{array}\right],$$

in the case of $1\le \alpha <2$ and

$$M_1=\left. [\begin{array}{cc}0& \Theta \\ {\Theta }^{\mathrm{T}}& 0\end{array}\right],M_2=\left. [\begin{array}{cc}0& {\Theta }^{\mathrm{T}}\\ \Theta & 0\end{array}\right],$$

in the case of $0<\alpha <1$. Now it only remains to show that the region $D_1$ described by $f_{D_1}$ and $D_2$ described by $f_{D_2}$ is identical to the region $D_1$, $D_2$ shown in Figure 1, with $M_k$, $k=1,2$ in $f_{D_1}$, $f_{D_2}$ defined as above. Note that the equation $f_{D_1}<0$ with $M_1$ is

$$\left. [\begin{array}{cc}2a\Re \left(s\right)& 2b\Im \left(s\right)\\ -2b\Im \left(s\right)& 2a\Re \left(s\right)\end{array}\right]<0.$$

(12)

Then the following region can be equivalently derived by Lemma 3

$$\{s\in \mathbb{C}:a\Re (s)+\mathrm{j}b\Im (s)<0\}.$$

(13)

Clearly the region described by expression (12) is the region $D_1$. Similarly, the region characterized by $f_{D_{U_2}}<0$ can be equivalently rewritten as

$$\{s\in \mathbb{C}:a\Re (s)-\mathrm{j}b\Im (s)<0\},$$

(14)

which is identical to the region $D_2$. This completes the proof.    □

The two LMI-based criteria stated in Theorems 1 and 2 involve complex variables that are untoward when solved with the LMI toolbox. In order to avoid using complex matrices, the following theorems are presented.

Theorem 3.

System (3) with $0<\alpha <2$ is stable iff there exist matrices $X_1,X_2,Y_1,Y_2\in$${\mathbb{R}}^{n\times n}$ satisfying

$$\left. [\begin{array}{cc}{X}_1& Y_1\\ -Y_1& X_1\end{array}\right]>0,\left. [\begin{array}{cc}{X}_2& Y_2\\ -Y_2& X_2\end{array}\right]>0,$$

(15)

$$\mathcal{H}\{(\Theta \otimes A)\left. [\begin{array}{cc}{X}_1& Y_1\\ -Y_1& X_1\end{array}\right]+\lceil 1-\alpha \rceil ({\Theta }^{\mathrm{T}}\otimes A)\left. [\begin{array}{cc}{X}_2& Y_2\\ -Y_2& X_2\end{array}\right]\}<0.$$

(16)

Proof. 

We divide the discussion of $\alpha$ values in the interval $0<\alpha <2$ into two cases: $1\le \alpha <2$ and $0<\alpha <1$.

Case $1\le \alpha <2$: Following the proof of Theorem 1, the stability domain can be described by $f_{D_{U_1}}$ with

$$M_1=\left. [\begin{array}{cc}0& a+\mathrm{j}b\\ a-\mathrm{j}b& 0\end{array}\right].$$

From (15), we obtain $P:=X+\mathrm{j}Y>0$. It follows from Lemma 2 that system (3) is stable iff there exists a feasible solution $P=X+\mathrm{j}Y\in {\mathbb{C}}^{n\times n}$ such that (15) and

$$\begin{array}{cc}\hfill & (a+\mathrm{j}b)\otimes \left(A(X+\mathrm{j}Y)\right)+(a-\mathrm{j}b)\otimes \left((X-\mathrm{j}Y)A^{\mathrm{T}}\right)<0.\hfill \end{array}$$

(17)

From Lemma 3, the inequality (17) is equivalent to

$$\left. [\begin{array}{cc}aAX+aX{A}^{\mathrm{T}}-bAY+bY{A}^{\mathrm{T}}& aAY+bAX+aX{A}^{\mathrm{T}}-bX{A}^{\mathrm{T}}\\ -(aAY+bAX+aX{A}^{\mathrm{T}}-bX{A}^{\mathrm{T}})& aAX+aX{A}^{\mathrm{T}}-bAY+bY{A}^{\mathrm{T}}\end{array}\right]<0,$$

(18)

which can be arranged as

$$\mathcal{H}\left\{(\Theta \otimes A)\left. [\begin{array}{cc}X& Y\\ -Y& X\end{array}\right]\right\}<0.$$

(19)

Note that the inequality (19) is identical to (16) when $1\le \alpha <2$.

Case $0<\alpha <1$: From (15) and Lemma 3, we obtain

$$P_1:=X_1+\mathrm{j}{Y}_1>0,P_2:=X_2+\mathrm{j}{Y}_2>0.$$

(20)

Following the proof of Theorem 1, substituting (20) into (10) and merging the real and imaginary part terms separately, we obtain (10) is identical to

$$\begin{array}{c}\hfill \mathcal{H}\{aA{X}_1-bA{Y}_1+aA{X}_2+bA{Y}_2+\mathrm{j}(aA{Y}_1+bA{X}_1+aA{Y}_2-bA{X}_2)\}<0.\end{array}$$

(21)

Using Lemma 3 on (21), it is derived that

$$\mathcal{H}\left\{\left. [\begin{array}{cc}aA{X}_1-bA{Y}_1+aA{X}_2+bA{Y}_2& aA{Y}_1+bA{X}_1+aA{Y}_2-bA{X}_2\\ -aA{Y}_1-bA{X}_1-aA{Y}_2+bA{X}_2& aA{X}_1-bA{Y}_1+aA{X}_2+bA{Y}_2\end{array}\right]\right\}<0,$$

which can be rewritten as

$$\mathcal{H}\{(\Theta \otimes A)\left. [\begin{array}{cc}{X}_1& Y_1\\ -Y_1& X_1\end{array}\right]+({\Theta }^{\mathrm{T}}\otimes A)\left. [\begin{array}{cc}{X}_2& Y_2\\ -Y_2& X_2\end{array}\right]\}<0.$$

(22)

Both of these cases cover all situations where $0<\alpha <2$. This completes the proof.    □

It is noted that the criterion in Theorem 3 requires additional conditions when it is used to design a controller, which is conservative. Therefore, by reducing the matrix variables, the following theorems are presented.

Theorem 4.

System (3) with $0<\alpha <2$ is stable iff there exist $X,Y\in {\mathbb{R}}^{n\times n}$ satisfying

$$\left. [\begin{array}{cc}X& Y\\ -Y& X\end{array}\right]>0,$$

(23)

$$\mathcal{H}\{(\Theta \otimes A)\left. [\begin{array}{cc}X& Y\\ -Y& X\end{array}\right]+\lceil 1-\alpha \rceil ({\Theta }^{\mathrm{T}}\otimes A)\left. [\begin{array}{cc}X& -Y\\ Y& X\end{array}\right]\}<0.$$

(24)

Proof. 

It only remains to prove that the criterion in Theorem 4 is equivalent to the one in Theorem 3.

(Necessity). Pre- and post- multiplying (16) by $\left. [\begin{array}{cc}0& I\\ I& 0\end{array}\right]$ and using congruent transformation, we obtain

$$\mathcal{H}\{({\Theta }^{\mathrm{T}}\otimes A)\left. [\begin{array}{cc}{X}_1& -Y_1\\ Y_1& X_1\end{array}\right]+\lceil 1-\alpha \rceil (\Theta \otimes A)\left. [\begin{array}{cc}{X}_2& -Y_2\\ Y_2& X_2\end{array}\right]\}<0.$$

(25)

Summing Equations (16) and (25) yields

$$\mathcal{H}\{(\Theta \otimes A)\left. [\begin{array}{cc}{X}_1+X_2& Y_1-Y_2\\ Y_2-Y_1& X_1+X_2\end{array}\right]+\lceil 1-\alpha \rceil ({\Theta }^{\mathrm{T}}\otimes A)\left. [\begin{array}{cc}{X}_1+X_2& Y_2-Y_1\\ Y_1-Y_2& X_1+X_2\end{array}\right]\}<0.$$

(26)

Let $X=X_1+X_2$ and $Y=Y_1-Y_2$; Equation (26) becomes Equation (24). From Lemma 3, Equation (15) is equivalent to

$$\left. [\begin{array}{cc}{X}_1& Y_1\\ -Y_1& X_1\end{array}\right]>0,\left. [\begin{array}{cc}{X}_2& -Y_2\\ Y_2& X_2\end{array}\right]>0.$$

Equation (23) is derived by summing

$$\left. [\begin{array}{cc}{X}_1& Y_1\\ -Y_1& X_1\end{array}\right],\left. [\begin{array}{cc}{X}_2& -Y_2\\ Y_2& X_2\end{array}\right].$$

(Sufficiency). Suppose there exist X, Y satisfying (23) and (24). Letting $X_1=X_2=\frac{1}{2}X$, $Y_1=Y_2=\frac{1}{2}Y$, it is easy to obtain (15) and

$$\mathcal{H}\{({\Theta }^{\mathrm{T}}\otimes A)\left. [\begin{array}{cc}2X_1& -2Y_1\\ 2Y_1& 2X_1\end{array}\right]+\lceil 1-\alpha \rceil (\Theta \otimes A)\left. [\begin{array}{cc}2X_2& 2Y_2\\ -2Y_2& 2X_2\end{array}\right]\}<0,$$

which is equivalent to Equation (16). This completes the proof.    □

It is noted that the above unified form criteria are useful tools for testing stability, but when designing and solving the controller gain using them, additional conditions are required, which is conservative to a certain extent. Therefore, the following theorem for stability is presented to reduce the conservatism in controller design.

Theorem 5.

System (3) with $0<\alpha <2$ is stable iff there exist matrices $X,Y\in {\mathbb{R}}^{n\times n}$ satisfying

$$\left. [\begin{array}{cc}X& Y\\ \lfloor \alpha -1\rfloor Y& X\end{array}\right]>0,$$

(27)

$$\mathcal{H}\{(\Theta \otimes A)\left. [\begin{array}{cc}X& Y\\ \lfloor \alpha -1\rfloor Y& X\end{array}\right]+\lceil 1-\alpha \rceil ({\Theta }^{\mathrm{T}}\otimes A)\left. [\begin{array}{cc}X& \lfloor \alpha -1\rfloor Y\\ Y& X\end{array}\right]\}<0.$$

(28)

Proof. 

Similar to the proof of Theorems 1 and 3, the proof also proceeds by two cases.

Case $0<\alpha <1$: In this case, Equations (27) and (28) become Equations (23) and (24) in Theorem 4, respectively.

Case $1\le \alpha <2$: As presented in [16], the stability domain is an LMI region that can be characterised by

$$\{s\in \mathbb{C}:s\Theta +s^{*}{\Theta }^{\mathrm{T}}<0\}.$$

(29)

From Lemma 2, it can be derived that system (3) is stable iff there exists $X\in {\mathbb{R}}^{n\times n}$ satisfying

$$\Theta \otimes \left(AX\right)+{\Theta }^{\mathrm{T}}\otimes \left(X{A}^{\mathrm{T}}\right)<0,$$

which is obviously equivalent to (28) with $1\le \alpha <2$.

This completes the proof.    □

Clearly the criterion provided in Theorem 5 is in the form of real LMIs, which is suitable to be solved using the LMI toolbox in MATLAB.

3.2. Admissibility of SFOSs

A similar unified approach can be used to construct unified criteria for the admissibility of SFOSs in the following theorems. Since the criterion in Theorem 5 is most suitable for designing controllers for normal FOSs in the proposed framework, only the result in Theorem 5 is extended to admissibility and stabilization for SFOSs in this subsection.

Theorem 6.

System (2) with $0<\alpha <2$ is admissible iff there exist matrices $X,Y\in {\mathbb{R}}^{n\times n}$ and $Q\in {\mathbb{R}}^{(n-m)\times n}$ satisfying (28) and

$$\left. [\begin{array}{cc}EX& EY\\ \lfloor \alpha -1\rfloor EY& EX\end{array}\right]=\left. [\begin{array}{cc}{X}^{\mathrm{T}}{E}^{\mathrm{T}}& \lfloor \alpha -1\rfloor Y^{\mathrm{T}}{E}^{\mathrm{T}}\\ Y^{\mathrm{T}}{E}^{\mathrm{T}}& X^{\mathrm{T}}{E}^{\mathrm{T}}\end{array}\right]\ge 0,$$

(30)

Proof. 

(Sufficiency). For system (2), choose two invertible matrices $L,R$ such that

$$LER=\left. [\begin{array}{cc}{I}_m& 0\\ 0& 0\end{array}\right],LAR=\left. [\begin{array}{cc}{A}_1& A_2\\ A_3& A_4\end{array}\right].$$

(31)

Let

$$R^{-1}X{L}^{\mathrm{T}}=\left. [\begin{array}{cc}{X}_{11}& X_{12}\\ X_{13}& X_{14}\end{array}\right],R^{-1}Y{L}^{\mathrm{T}}=\left. [\begin{array}{cc}{Y}_{11}& Y_{12}\\ Y_{13}& Y_{14}\end{array}\right].$$

(32)

Considering (32), pre- and post-multiplying (28), (30), respectively, by

$$\left. [\begin{array}{cc}L& 0\\ 0& L\end{array}\right],$$

and its transpose, using contract transformation gives that the following two equations hold:

$$\begin{gathered} \mathcal{H}\{(\Theta \otimes \left. [\begin{array}{cc}{A}_1& A_2\\ A_3& A_4\end{array}\right])\left(\left. [\begin{array}{cc}{R}^{-1}& 0\\ 0& R^{-1}\end{array}\right]\left. [\begin{array}{cc}X& Y\\ \lfloor \alpha -1\rfloor Y& X\end{array}\right]\left. [\begin{array}{cc}{L}^{\mathrm{T}}& 0\\ 0& L^{\mathrm{T}}\end{array}\right]\right) \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\lceil 1-\alpha \rceil ({\Theta }^{\mathrm{T}}\otimes \left. [\begin{array}{cc}{A}_1& A_2\\ A_3& A_4\end{array}\right])\left({\left. [\begin{array}{cc}R& 0\\ 0& R\end{array}\right]}^{-1}\left. [\begin{array}{cc}X& \lfloor \alpha -1\rfloor Y\\ Y& X\end{array}\right]\left. [\begin{array}{cc}{L}^{\mathrm{T}}& 0\\ 0& L^{\mathrm{T}}\end{array}\right]\right)\}<0, \end{gathered}$$

(33)

$$\left. [\begin{array}{cccc}{X}_{11}& X_{12}& Y_{11}& Y_{12}\\ 0& 0& 0& 0\\ \lfloor \alpha -1\rfloor Y_{11}& \lfloor \alpha -1\rfloor Y_{12}& X_{11}& X_{12}\\ 0& 0& 0& 0\end{array}\right]=\left. [\begin{array}{cccc}{X}_{11}^{\mathrm{T}}& 0& \lfloor \alpha -1\rfloor Y_{11}^{\mathrm{T}}& 0\\ X_{12}^{\mathrm{T}}& 0& \lfloor \alpha -1\rfloor Y_{12}^{\mathrm{T}}& 0\\ Y_{11}^{\mathrm{T}}& 0& X_{11}^{\mathrm{T}}& 0\\ Y_{12}^{\mathrm{T}}& 0& X_{12}^{\mathrm{T}}& 0\end{array}\right]\ge 0.$$

(34)

Letting

$$V=\left. [\begin{array}{cccc}I& -A_2{A}_4^{-1}& 0& 0\\ 0& I& 0& 0\\ 0& 0& I& -A_2{A}_4^{-1}\\ 0& 0& 0& I\end{array}\right],$$

(35)

considering (32) and the constraints among the relevant variables in (34), pre- and post- multiplying (33) by V and $V^{\mathrm{T}}$ and using contract transformation, we have

$$\begin{gathered} \mathcal{H}\{\left. [\begin{array}{cccc}\tilde{A}(a{X}_{11}+\lfloor \alpha -1\rfloor b{Y}_{11})& 0& \tilde{A}(a{Y}_{11}+b{X}_{11})& 0\\ *& A_4(a{X}_{14}+\lfloor \alpha -1\rfloor b{Y}_{14})& *& A_4(a{Y}_{14}+b{X}_{14})\\ \tilde{A}(-b{X}_{11}+\lfloor \alpha -1\rfloor a{Y}_{11})& 0& \tilde{A}(-b{Y}_{11}+a{X}_{11})& 0\\ *& A_4(-b{X}_{14}+\lfloor \alpha -1\rfloor a{Y}_{14})& *& A_4(-b{Y}_{14}+a{X}_{14})\end{array}\right] \\ +\lceil 1-\alpha \rceil \left. [\begin{array}{cccc}\tilde{A}(a{X}_{11}-b{Y}_{11})& 0& \tilde{A}(\lfloor \alpha -1\rfloor a{Y}_{11}+b{X}_{11})& 0\\ *& A_4(a{X}_{14}-b{Y}_{14})& *& A_4(\lfloor \alpha -1\rfloor a{Y}_{14}+b{X}_{14})\\ \tilde{A}(b{X}_{11}+a{Y}_{11})& 0& \tilde{A}(\lfloor \alpha -1\rfloor b{Y}_{11}+a{X}_{11})& 0\\ *& A_4(b{X}_{14}+a{Y}_{14})& *& A_4(\lfloor \alpha -1\rfloor b{Y}_{14}+a{X}_{14})\end{array}\right]\}<0, \end{gathered}$$

(36)

where $\tilde{A}=A_1-A_2{A}_4^{-1}{A}_3$. Two results can be obviously obtained from (36):

(i)  $\mathcal{H}\left\{A_4(\lfloor \alpha -2\rfloor b{Y}_{14}+2a{X}_{14})\right\}<0$, which implies $A_4$ is nonsingular.

(ii) The following inequality holds.

$$\begin{gathered} \mathcal{H}\left. [\begin{array}{cc}\tilde{A}(a{X}_{11}+\lfloor \alpha -1\rfloor b{Y}_{11})& \tilde{A}(a{Y}_{11}+b{X}_{11})\\ \tilde{A}(-b{X}_{11}+\lfloor \alpha -1\rfloor a{Y}_{11})& \tilde{A}(-b{Y}_{11}+a{X}_{11})\end{array}\right] \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\lceil 1-\alpha \rceil \left. [\begin{array}{cc}\tilde{A}(a{X}_{11}-b{Y}_{11})& \tilde{A}(\lfloor \alpha -1\rfloor a{Y}_{11}+b{X}_{11})\\ \tilde{A}(b{X}_{11}+a{Y}_{11})& \tilde{A}(\lfloor \alpha -1\rfloor b{Y}_{11}+a{X}_{11})\end{array}\right]<0. \end{gathered}$$

(37)

It is easy to see that the inequality (37) is identical to

$$\begin{gathered} \mathcal{H}\left\{\right(\Theta \otimes (A_1-A_2{A}_4^{-1}{A}_3))\times \left. [\begin{array}{cc}{X}_{11}& Y_{11}\\ \lfloor \alpha -1\rfloor Y_{11}& X_{11}\end{array}\right] \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\lceil 1-\alpha \rceil ({\Theta }^{\mathrm{T}}\otimes (A_1-A_2{A}_4^{-1}{A}_3))\times \left. [\begin{array}{cc}{X}_{11}& \lfloor \alpha -1\rfloor Y_{11}\\ Y_{11}& X_{11}\end{array}\right]\}<0. \end{gathered}$$

(38)

From Theorem 5, the inequality (38) means

$$\begin{array}{c}\hfill \left|\arg \left(\mathrm{spec}(A_1-A_2{A}_4^{-1}{A}_3)\right)\right|>\alpha \frac{\pi }{2}.\end{array}$$

From (i), (ii), it follows that system (2) is admissible.

(Necessity). Supposing that system (2) is admissible, for two arbitrarily selected invertible matrices $L_1$, $R_1$ given by

$$L_{1}E{R}_1=\left. [\begin{array}{cc}{I}_m& 0\\ 0& J_{n-m}\end{array}\right],L_{1}A{R}_1=\left. [\begin{array}{cc}{\overline{A}}_1& 0\\ 0& I_{n-m}\end{array}\right],$$

it is implied that $J_{n-m}=0$ and $|\arg \left(\mathrm{spec}\left({\overline{A}}_1\right)\right)|>\alpha \frac{\pi }{2}$ hold. From Theorem 5, there exist $X_{11},Y_{11}\in {\mathbb{R}}^{m\times m}$, such that

$$\left. [\begin{array}{cc}{X}_{11}& Y_{11}\\ \lfloor \alpha -1\rfloor Y_{11}& X_{11}\end{array}\right]>0,$$

$$\mathcal{H}\{(\Theta \otimes {\overline{A}}_1)\left. [\begin{array}{cc}{X}_{11}& Y_{11}\\ \lfloor \alpha -1\rfloor Y_{11}& X_{11}\end{array}\right]+\lceil 1-\alpha \rceil ({\Theta }^{\mathrm{T}}\otimes {\overline{A}}_1)\left. [\begin{array}{cc}{X}_{11}& \lfloor \alpha -1\rfloor Y_{11}\\ Y_{11}& X_{11}\end{array}\right]\}<0.$$

Letting

$$X=R_1\left. [\begin{array}{cc}{X}_{11}& 0\\ 0& -I_{n-m}\end{array}\right]L_1^{-\mathrm{T}},Y=R_1\left. [\begin{array}{cc}{Y}_{11}& 0\\ 0& 0\end{array}\right]L_1^{-\mathrm{T}},$$

denoting by $\mathsf{\Omega }$ the left side of (28), to verify the matrix variables constructed above satisfy (28) and (30), pre- and post- multiplying $\mathsf{\Omega }$ and

$$\left. [\begin{array}{cc}E& 0\\ 0& E\end{array}\right]\left. [\begin{array}{cc}X& Y\\ \lfloor \alpha -1\rfloor Y& X\end{array}\right],$$

respectively by

$$\left. [\begin{array}{cc}{L}_1& 0\\ 0& L_1\end{array}\right],$$

and its transpose, the following two equations hold:

$$\begin{gathered} \left. [\begin{array}{cc}{L}_1& 0\\ 0& L_1\end{array}\right]\mathsf{\Omega }{\left. [\begin{array}{cc}{L}_1& 0\\ 0& L_1\end{array}\right]}^{\mathrm{T}}=\mathcal{H}\{\left. [\begin{array}{cccc}{\overline{A}}_1(a{X}_{11}+\lfloor \alpha -1\rfloor b{Y}_{11})& 0& {\overline{A}}_1(a{Y}_{11}+b{X}_{11})& 0\\ 0& -a{I}_{n-m}& 0& -b{I}_{n-m}\\ {\overline{A}}_1(-b{X}_{11}+\lfloor \alpha -1\rfloor a{Y}_{11})& 0& {\overline{A}}_1(-b{Y}_{11}+a{X}_{11})& 0\\ 0& b{I}_{n-m}& 0& -a{I}_{n-m}\end{array}\right] \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\lceil 1-\alpha \rceil \left. [\begin{array}{cccc}{\overline{A}}_1(a{X}_{11}+\lfloor \alpha -1\rfloor b{Y}_{11})& 0& {\overline{A}}_1(a{Y}_{11}+b{X}_{11})& 0\\ 0& -a{I}_{n-m}& 0& -b{I}_{n-m}\\ {\overline{A}}_1(-b{X}_{11}+\lfloor \alpha -1\rfloor a{Y}_{11})& 0& {\overline{A}}_1(-b{Y}_{11}+a{X}_{11})& 0\\ 0& b{I}_{n-m}& 0& -a{I}_{n-m}\end{array}\right]\}<0, \end{gathered}$$

(39)

$$\left. [\begin{array}{cc}{L}_1& 0\\ 0& L_1\end{array}\right]\left. [\begin{array}{cc}E& 0\\ 0& E\end{array}\right]\left. [\begin{array}{cc}X& Y\\ -Y& X\end{array}\right]\left. [\begin{array}{cc}{L}_1^{\mathrm{T}}& 0\\ 0& L_1^{\mathrm{T}}\end{array}\right]=\left. [\begin{array}{cccc}{X}_{11}& 0& Y_{11}& 0\\ 0& 0& 0& 0\\ \lfloor \alpha -1\rfloor Y_{11}& 0& X_{11}& 0\\ 0& 0& 0& 0\end{array}\right]\ge 0.$$

(40)

From (39) and (40), it can be easily deduced that $X,Y$ satisfy (28) and (30). This completes the proof.    □

The criterion provided in Theorem 6 contains equality constraints, which are troublesome when solved using the LMI toolbox. Therefore, the following theorem provides a criterion that does not contain equality constraints and is in a form of strict LMI.

Theorem 7.

System (2) with $0<\alpha <2$ is admissible iff there exist matrices $X,Y\in {\mathbb{R}}^{n\times n}$ and $Q\in {\mathbb{R}}^{(n-m)\times n}$ satisfying (27) and

$$\mathcal{H}\{(\Theta \otimes A)\left. [\begin{array}{cc}X{E}^{\mathrm{T}}+SQ& Y{E}^{\mathrm{T}}\\ \lfloor \alpha -1\rfloor Y{E}^{\mathrm{T}}& X{E}^{\mathrm{T}}+SQ\end{array}\right]+\lceil 1-\alpha \rceil ({\Theta }^{\mathrm{T}}\otimes A)\left. [\begin{array}{cc}X{E}^{\mathrm{T}}+SQ& \lfloor \alpha -1\rfloor Y{E}^{\mathrm{T}}\\ Y{E}^{\mathrm{T}}& X{E}^{\mathrm{T}}+SQ\end{array}\right]\}<0,$$

(41)

where $S\in {\mathbb{R}}^{n\times (n-m)}$ is any matrix satisfying full column rank and $ES=0$.

Proof. 

(Sufficiency). Suppose that there exist matrices $X,Y\in {\mathbb{R}}^{n\times n}$ and $Q\in {\mathbb{R}}^{(n-m)\times n}$ satisfying (27) and (41). Denoting $X{E}^{\mathrm{T}}+\frac{1}{2}SQ$ by $\overline{X}$ and $Y{E}^{\mathrm{T}} \mathrm{by} \overline{Y},$ Equations (27) and (41) show that $\overline{X}$, $\overline{Y}$ satisfy (28), (30). From Theorem 6, it can be concluded that the system (2) is admissible.

(Necessity). Supposing that the system (2) is admissible, there exist arbitrarily chosen invertible $L_1$, $R_1$ satisfying (31) and

$$\left|\arg \left(\mathrm{spec}\left({\overline{A}}_1\right)\right)\right|>\alpha \frac{\pi }{2}.$$

That is, there exist $X_1$, $Y_1\in {\mathbb{R}}^{m\times m}$, satisfying the two following inequalities:

$$\left. [\begin{array}{cc}{X}_1& Y_1\\ \lfloor \alpha -1\rfloor Y_1& X_1\end{array}\right]>0,$$

$$\mathcal{H}\{(\Theta \otimes {\overline{A}}_1)\left. [\begin{array}{cc}{X}_1& Y_1\\ \lfloor \alpha -1\rfloor Y_1& X_1\end{array}\right]+\lceil 1-\alpha \rceil ({\Theta }^{\mathrm{T}}\otimes \overline{A})\left. [\begin{array}{cc}{X}_1& \lfloor \alpha -1\rfloor Y_1\\ Y_1& X_1\end{array}\right]\}<0.$$

Let

$$X=R_1\left. [\begin{array}{cc}{X}_1& 0\\ 0& I_{n-m}\end{array}\right]R_1^{\mathrm{T}},Y=R_1\left. [\begin{array}{cc}{Y}_1& 0\\ 0& 0_{n-m}\end{array}\right]R_1^{\mathrm{T}},$$

$$S=R_1\left. [\begin{array}{c}0\\ I_{n-m}\end{array}\right]H,Q=H^{-1}\left. [\begin{array}{cc}0& -I_{n-m}\end{array}\right]L_1^{-\mathrm{T}},$$

where H is any invertible matrix. Then it is derived that (27) and the following inequality

$$\begin{gathered} \hspace{0.17em}\mathcal{H}\{(I_2\otimes L_1^{-1})((\Theta \otimes \left. [\begin{array}{cc}{\overline{A}}_1& 0\\ 0& I_{n-m}\end{array}\right])\times \left. [\begin{array}{cccc}{X}_1& 0& Y_1& 0\\ 0& -\frac{1}{2}{I}_{n-m}& 0& 0\\ \lfloor \alpha -1\rfloor Y_1& 0& X_1& 0\\ 0& 0& 0& -\frac{1}{2}{I}_{n-m}\end{array}\right] \\ +\lceil 1-\alpha \rceil ({\Theta }^{\mathrm{T}}\otimes \left. [\begin{array}{cc}{\overline{A}}_1& 0\\ 0& I_{n-m}\end{array}\right])\times \left. [\begin{array}{cccc}{X}_1& 0& \lfloor \alpha -1\rfloor Y_1& 0\\ 0& -\frac{1}{2}{I}_{n-m}& 0& 0\\ Y_1& 0& X_1& 0\\ 0& 0& 0& -\frac{1}{2}{I}_{n-m}\end{array}\right])(I_2\otimes L_1^{-\mathrm{T}})\}<0, \end{gathered}$$

(42)

hold. Note that Equation (42) is identical to (41). This completes the proof.    □

Without loss of generality, consider the SFOS (1) and the following state feedback controller:

$$u\left(t\right)=Kx\left(t\right),K\in {\mathbb{R}}^{l\times n}.$$

(43)

Using this controller (43) on system (1), the following system can be obtained:

$$E{D}^{\alpha }x\left(t\right)=(A+BK)x\left(t\right).$$

(44)

Then the controller can be designed by the following theorem.

Theorem 8.

There exists a state feedback controller (43) such that the closed-loop SFOS (44) is admissible iff there exist $X,Y\in {\mathbb{R}}^{n\times n}$, $Q\in {\mathbb{R}}^{(n-m)\times n}$ and $Z\in {\mathbb{R}}^{l\times n}$ satisfying (27) and

$$\begin{gathered} \mathcal{H}\{(\Theta \otimes A)(\left. [\begin{array}{cc}X& Y\\ \lfloor \alpha -1\rfloor Y& X\end{array}\right]{\overline{E}}^{\mathrm{T}}+\overline{S}\overline{Q})+\lceil 1-\alpha \rceil ({\Theta }^{\mathrm{T}}\otimes A)(\left. [\begin{array}{cc}X& \lfloor \alpha -1\rfloor Y\\ Y& X\end{array}\right]{\overline{E}}^{\mathrm{T}}+\overline{S}\overline{Q}) \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\lceil \alpha -1\rceil (\Theta \otimes B)(I_2\otimes Z)\lceil 1-\alpha \rceil (I_2\otimes BZ)\}<0, \end{gathered}$$

(45)

where $\overline{E}=I_2\otimes E$, $\overline{S}=I_2\otimes S$, $S\in {\mathbb{R}}^{n\times (n-m)}$, $ES=0$, $\overline{Q}=I_2\otimes Q$. Then, one can choose a stabilizing state feedback controller with gain matrix

$$\begin{array}{c}\hfill K=Z{(\lceil \alpha -1\rceil (X{E}^{\mathrm{T}}+SQ)+2\lceil 1-\alpha \rceil (aX{E}^{\mathrm{T}}-bY{E}^{\mathrm{T}}+aSQ))}^{-1}.\end{array}$$

(46)

The practical code for solving the LMIs (27) and (45) using the LMI toolbox in MATLAB is presented in Appendix A, and it can be seen that the proposed framework does not require additional conditional statements such as: if $1\le \alpha <2$, execute statement 1; elif $0<\alpha <1$, execute statement 2. Thus, it is a truly unified framework. According to

Table 1. Comparison of existing methods and ours.

Ref.	$\mathit{\alpha }$ Range	Var. Kind	FOS or SFOS	Nonconservative Stabilization	As a Special Case of Ours	Unified
[11]	$(0,1)$	Complex	FOS	No	Th. 1	N/A
[12]	$[1,2)$	Real	FOS	No	N/A	N/A
[11]	$[1,2)$	Real	FOS	Yes	Th. 5	N/A
[15]	$(0,1)$	Real	FOS	No	Th. 3	N/A
[19]	$(0,2)$	Real	FOS	Yes	N/A	No
[27]	$[1,2)$	Real	SFOS	Yes	Th. 5	N/A
[24]	$(0,1)$	Real	SFOS	Yes	Th. 7	N/A
[26]	$(0,2)$	Real	SFOS	Yes	N/A	No
[30]	$(0,2)$	Real	SFOS	Yes	N/A	No
Ours (Th. 5)	$(0,2)$	Real	FOS	Yes	N/A	Yes
Ours (Th. 7)	$(0,2)$	Real	SFOS	Yes	N/A	Yes

, the proposed framework in this paper has advantages in the unity of the structure, the generality of the criteria, the non-conservatism of the controller design and the ease for solving.

4. Numerical Examples

Two numerical examples are used to verify the effectiveness of the proposed scheme.

Example 1.

Consider the FOS (3) with the parameters $\alpha =0.5$ and

$$A=\left. [\begin{array}{cccc}1& -1.5& 0& 0\\ 1.5& 1& 0& 0\\ 1& 0& 0.6& -1\\ 0& 1& 1& 0.6\end{array}\right].$$

(47)

It is easy to see that system (3) is stable because all eigenvalues ${\lambda }_i(i=1,2,3,4)$ of the system matrix A satisfy

$$\left|\arg \left({\lambda }_i\right)\right|>\frac{\pi }{4}.$$

Using the LMI toolbox to solve (27) and (28) in Theorem 5, the following feasible solutions can be obtained:

$$X=\left. [\begin{array}{cccc}25.0228& 0& 14.666& -11.5702\\ 0& 25.0228& 11.5702& 14.666\\ 14.666& 11.5702& 94.5643& 0\\ -11.5702& 14.666& 0& 94.5643\end{array}\right],$$

$$Y=\left. [\begin{array}{cccc}0& 21.2230& 7.0284& 15.6533\\ -21.2230& 0& -15.6533& 7.0284\\ -7.0284& 15.6533& 0& 76.8874\\ -15.6533& -7.0284& -76.8874& 0\end{array}\right].$$

Further, for system (3) with system matrix A given in (47) and the fractional order $\alpha =0.7$, it is unstable because there exists an eigenvalue ${\lambda }_i$ of system matrix A that does not satisfy

$$\left|\arg \left({\lambda }_i\right)\right|>0.35\pi .$$

According to Theorem 5, a state feedback controller $u=Kx\left(t\right)$ can be designed to make the closed-loop system (1) with $E=I$ stable, where the control matrix can be arbitrarily chosen as

$$B={\left. [\begin{array}{cccc}1& 1& 1& 1\end{array}\right]}^{\mathrm{T}}.$$

Using the LMI toolbox to solve the LMIs (27) and

$$\mathcal{H}\{(\Theta \otimes A)(\left. [\begin{array}{cc}X& Y\\ \lfloor \alpha -1\rfloor Y& X\end{array}\right])+\lceil 1-\alpha \rceil ({\Theta }^{\mathrm{T}}\otimes A)(\left. [\begin{array}{cc}X& \lfloor \alpha -1\rfloor Y\\ Y& X\end{array}\right])+\lceil \alpha -1\rceil (\Theta \otimes B)(I_2\otimes Z)+\lceil 1-\alpha \rceil (I_2\otimes BZ)\}<0,$$

the following feasible solutions can be obtained:

$$X=\left. [\begin{array}{cccc}80.1117& 6.5456& 13.5362& 14.6681\\ 6.5456& 31.3419& 12.4094& -16.9618\\ 13.5362& 12.4094& 42.7925& 15.3142\\ 14.6681& -16.9618& 15.3142& 47.8344\end{array}\right],$$

$$Y=\left. [\begin{array}{cccc}0& 28.0858& 1.8549& -23.5956\\ -28.0858& 0& 0.9793& -0.9894\\ -1.8549& -0.9793& 0& 14.5734\\ 23.5956& 0.9894& -14.5734& 0\end{array}\right],$$

$$Z=\left. [\begin{array}{cccc}-105.2726& -50.9853& -42.4700& -53.9358\end{array}\right].$$

Then, the controller gain K can be designed by

$$K=Z{(\lceil \alpha -1\rceil X+2\lceil 1-\alpha \rceil (aX-bY))}^{-1}=\left. [\begin{array}{cccc}-0.3080& -2.0298& 0.7833& -1.2884\end{array}\right].$$

Using the LMI toolbox to solve the LMIs provided by Theorem 3 in [15], the following feasible solutions are obtained:

$$X=\left. [\begin{array}{cccc}-3.4790& -2.1393& -2.3830& -1.8168\end{array}\right],$$

$$Q=\left. [\begin{array}{cccc}4.0730& 0.5549& 2.2858& 1.5569\\ 0.5549& 1.0757& 0.3187& 0.05248\\ 2.2858& 0.3187& 1.6994& 1.0874\\ 1.5569& 0.05248& 1.0874& 0.80109\end{array}\right],$$

$$K=X{Q}^{-1}=\left. [\begin{array}{cccc}0.8217& -2.9167& 2.9709& -7.7067\end{array}\right].$$

The positions where the eigenvalues of the system matrix $A+BK$, where K is obtained by the above two methods, fall on the complex plane are shown in Figure 2. It can be seen that the controller designed by Theorem 5 can configure the eigenvalues ${\lambda }_i(A+BK)$ in the right stability region, while the method presented in [15] can only configure the eigenvalues ${\lambda }_i(A+BK)$ in the left semi-complex plane.

Remark 1.

When using the stability analysis method provided by Theorem 1 in [15], four decision variables need to be solved, so the solution calculation speed is slow. In addition, the LMI provided in Theorem 12 of Ref. [13] involves complex decision variables, so it is difficult when solving with the LMI toolbox.

Remark 2.

The LMIs provided in Theorems 9 and 10 of [13] involve the terms $\mathcal{H}\left\{{(A+BK)}^{\frac{1}{\alpha }}X\right\}$ and $\mathcal{H}\{-{(-A-BK)}^{\frac{1}{2-\alpha }}X\}$, respectively, so both of them cannot be used directly to design controllers. The stabilization analysis method presented in [15] provides a sufficient but unnecessary criterion, which is conservative to a certain extent. Using the LMI toolbox to solve the LMIs of Theorem 3 in [15] with $A=\left. [\begin{array}{ccccccc}1& 2& 0;-2& 1& 0;1& 0& 1\end{array}\right]$ and $B=\left. [\begin{array}{c}0;0;1\end{array}\right]$, one obtains the following information, which means Theorem 3 in [15] is invalid.

Result: best value of t: $6.205424\times {10}^{13}$

f-radius saturation: 0.000% of R = ${10}^9$

Marginal infeasibility: these LMI constraints may be feasible but are not strictly feasible.

Example 2.

Consider the SFOS (1) with the fractional order $\alpha =0.7and1.5$, respectively, and other parameters as

$$E=\left. [\begin{array}{cccc}1.2& 0.2& 0.2& -0.8\\ 0.2& 1.2& 0.2& -0.8\\ 0.2& 0.2& 1.2& -0.8\\ 0.4& 0.4& 0.4& -0.6\end{array}\right],A=\left. [\begin{array}{cccc}-0.6& -2.6& 2.4& 0.4\\ 1& -3& 2& 0\\ 0& -2& 3& -1\\ -0.4& -2.4& 2.6& 0.6\end{array}\right],$$

$$B={\left. [\begin{array}{cccc}1& 1& 1& 1\end{array}\right]}^{\mathrm{T}}.$$

It is easy to determine

$$\det (s^{0.7}E-A)=2s^{0.7}-s^{0.21}+4\not\equiv 0,$$

$$\det (s^{1.5}E-A)=2s^{1.5}-s^{4.5}+4\not\equiv 0,$$

$$\mathrm{deg}\left(\det \right(sE-A\left)\right)=\mathrm{rank}\left(E\right)=3,$$

$$\mathrm{spec}(E,A)=\{-1+\mathrm{j},-1-\mathrm{j},2\}.$$

Thus, the system (1) is regular and impulse-free but unstable in the case of $\alpha =0.7and1.5$. That is, the system (1) is not admissible. By solving (45), we obtain the following feasible solutions:

Case $\alpha =0.7$:

$$X=\left. [\begin{array}{cccc}14.6934& 6.8766& 5.0576& -1.0284\\ 6.8766& 25.8541& -2.6300& -2.7650\\ 5.0576& -2.6300& 18.0273& 2.0579\\ -1.0284& -2.7650& 2.0579& 25.4384\end{array}\right],$$

$$Y=\left. [\begin{array}{cccc}0& 0.2877& 1.2145& -0.7511\\ -0.2877& 0& 1.9940& -0.8532\\ -1.2145& -1.9940& 0& 1.6042\\ 0.7511& 0.8532& -1.6042& 0\end{array}\right],$$

$$Q=\left. [\begin{array}{cccc}-377.0389& -380.1311& -376.0045& -351.8424\end{array}\right],$$

$$Z=\left. [\begin{array}{cccc}55.2985& 138.0754& 110.3800& -204.6825\end{array}\right],$$

$$K=\left. [\begin{array}{cccc}3.5125& 10.7956& 21.8878& -21.1369\end{array}\right].$$

Case $\alpha =1.5$:

$$X=\left. [\begin{array}{cccc}0.9927& -0.1731& 0.5139& 0.2102\\ -0.1731& 2.1292& -0.9060& 0.3519\\ 0.5139& -0.9060& 1.2105& 0.4678\\ 0.2102& 0.3519& 0.4678& 1.2390\end{array}\right],$$

$$Y=0,$$

$$Q=\left. [\begin{array}{cccc}-2.2345& -1.4041& -2.8467& 0.4759\end{array}\right],$$

$$Z=\left. [\begin{array}{cccc}-1.8134& 6.6435& -5.4609& -0.26022\end{array}\right],$$

$$K=\left. [\begin{array}{cccc}0.4144& 1.1978& -4.2450& 1.3810\end{array}\right].$$

As shown in Figure 3 and Figure 4, the system (1) is admissible under the designed control law.

5. Conclusions

In this paper, necessary and sufficient conditions for stability and admissibility are presented for fractional order systems and singular fractional order systems with $0<\alpha <2$, respectively. Theorems 1–4 provide LMI-based criteria containing complex matrices and real matrices, respectively. In particular, in Theorem 5, a real LMI-based framework, is proposed, which can be easily used to design controllers. Theorem 6 provides a unified critera for testing admissibility, but it contains an equality constraint. A unified criterion for admissibility without equality constraints and non-strictness is presented in Theorem 7, and is used to design a controller for an SFOS. Compared with the unified framework for stability in [19] and admissibility in [26,30], the proposed framework does not need to design the form of elements in linear matrix inequalities in different cases ($1\le \alpha <2$ or $0<\alpha <1$) and is thus a truly unified framework. Strict linear matrix inequalities with few real variables are developed to test the stability and admissibility of the system. The analytical framework is more applicable than other results, and can be directly used to design the controller. In the future work, the control and stabilization problems of $H_{\infty }$ control and robust control problems for singular fractional order systems with the fractional order $0<\alpha <2$ can be extended based on this result, so that the control synthesis problems of fractional order systems can be unified.