Robust H_∞ Control for Fractional Order Systems with Order α (0 < α < 1)

Abstract

In the paper, $H_{\infty }$ and robust $H_{\infty }$ control for fractional order systems (FOS) with order $0<\alpha <1$ are studied. Firstly, necessary and sufficient conditions of $H_{\infty }$ control and state feedback controller design are proposed. Then, robust $H_{\infty }$ control for FOS with uncertainty is studied, and state feedback controller is designed. These conditions are based on linear matrix inequalities (LMI) and can be easily solved by the LMI toolbox. Finally, the effectiveness of these conditions is verified by two numerical examples.

1. Introduction

Fractional calculus was born almost at the same time as integer calculus [1]. However, fractional calculus developed slowly due to simulation difficulties. During the last two decades, with the development of computers, fractional calculus can be better described through simulations, fractional order systems (FOS) have been gradually studied [2,3]. The fundamental reason is that physical systems can be described more accurately with FOS [4,5,6,7]. Therefore, FOS have been extensively developed, especially in the aspect of dynamic process modeling [8]. For the FOS, stability and stabilization are the fundamental problems. In the 1990s, Oustaloup introduced fractional calculus into the control field, and proposed the well-known CRONE control technique [9], which can be used to judge the stability for FOS through characteristic root distribution. Nonlinear FOS have been studied by using the Lyapunov function and Mittag–Leffler function in [10,11]. Although the Matignon’s result can judge the stability of FOS, compared with the linear matrix inequality (LMI) approach, it is difficult to apply. Hence, a lot of researchers have used LMI to describe stability and stabilization of FOS. By using MATLAB’s LMI toolbox, these result based on LMI can be easily solved. Many works use LMI to describe stability and stabilization of FOS with $1<\alpha <2$ in [12,13,14,15]. For FOS with order $0<\alpha <1$, the stable region is non-convex. Therefore, the stability study of $0<\alpha <1$ is more interesting and difficult. Some conditions have been proposed to solve the non-convex problem based on LMI [16,17,18,19,20]. The LMI-based conditions for FOS with order $0<\alpha <2$ have been reported in [21], which can be used to determine the stability for FOS in the uniform form. $H_{\infty }$ control and analysis have been considered in [22,23,24].

Although several contributions of $H_{\infty }$ control for FOS are available in the literature, the results are a little conservative for FOS with $0<\alpha <1$. Moreover, robust $H_{\infty }$ control and analysis have not been reported. Compare with previous results, conjugate matrix or Hermitian matrix is replaced by two real matrices in the paper. These results in the paper are less conservative and much more tractable in simulation. With the above motivations, $H_{\infty }$ and robust $H_{\infty }$ control are studied in the paper. The contributions are as follows.

• The results of $H_{\infty }$ control based on LMI are proposed for FOS with order $0<\alpha <1$. Firstly, the necessary and sufficient condition of stability is proved in Theorem 1. Then, the condition of stabilization with state feedback controller is proved in Theorem 2 of Section 3.
• Robust $H_{\infty }$ control and analysis are studied in the paper. Theorem 3 in Section 4 is proposed and proved to solve the stability of FOS with uncertainty. Then, a condition of stabilization with state feedback controller is proved in Theorem 4.

The paper is organized as follows. FOS models and several definitions are given in Section 2.1, some preliminaries results and lemmas are provided regarding the FOS in the Section 2.2, respectively. In Section 3, $H_{\infty }$ control for FOS with order $0<\alpha <1$ is studied. The necessary and sufficient LMI-based conditions are formulated. In Section 4, the methods of solving robust $H_{\infty }$ control problem of FOS with uncertainty are derived. In Section 5, two examples are shown to verify these conditions proposed in this paper.

Notation 1.

In the paper, $P^T$ is the transpose of P, and $sym\left(P\right)=P+P^T$. For the matrix P, $P>0$ $(P<0)$ denotes positive definite (negative definite). ⊗ denotes the Kronecker’s product. ${\parallel T\left(s\right)\parallel }_{H_{\infty }}$ denotes the $H_{\infty }$ norm of $T\left(s\right)$. The conjugate transpose matrix of A is denoted $A^{*}$, and $\left[\begin{array}{cc}P& Q\\ *& P\end{array}\right]=\left[\begin{array}{cc}P& Q\\ Q^T& P\end{array}\right]$.

2. Preliminaries

2.1. FOS Models

In the subsection, the $H_{\infty }$ norm definitions for FOS are introduced.

Consider FOS described by the following form

$$\begin{array}{ccc}& & D^{\alpha }x\left(t\right)=Ax\left(t\right)+Bu\left(t\right)+B_{w}w\left(t\right)\hfill \\ \hfill & & y\left(t\right)=Cx\left(t\right)+Du\left(t\right)\hfill \\ & & z\left(t\right)=C_{z}x\left(t\right)+D_{z}u\left(t\right)+D_{zw}w\left(t\right)\hfill \end{array}$$

(1)

where D denotes Captuto definition and $0<\alpha <1$. $x\left(t\right)\in {\mathbf{R}}^n$, $u\left(t\right)\in {\mathbf{R}}^u$, $w\left(t\right)\in {\mathbf{R}}^w$, $y\left(t\right)\in {\mathbf{R}}^y$ and $z\left(t\right)\in {\mathbf{R}}^z$ are the state vector, control signal, exogenous input, sensed output, and performance output. A, B, $B_w$, C, D, $C_z$, $D_z$, and $D_{zw}$ are real constant matrices. The transfer matrix from $w\left(t\right)$ to $z\left(t\right)$ is $T_{zw}\left(s\right)=C_z{(s^{\alpha }I-A)}^{-1}{B}_w+D_{zw}$.

Some definitions will be introduced as follows.

Definition 1.

Let X be ${\mathit{R}}^m$. Hence, X can be represented as $x\left(t\right)=[x_1\left(t\right),\dots ,x_m\left(t\right)]$, $t\in \mathit{R}$, then it can be obtained that

$$L_m^{\infty }\left(\mathit{R}\right)={\{x\left(t\right):\parallel x\left(t\right)\parallel }_{\infty }=\underset{t\in \mathit{R}}{sup}{\parallel x\left(t\right)\parallel }_{\infty }<\infty \}$$

$$L_m^p\left(\mathit{R}\right)={\{x\left(t\right):\parallel x\left(t\right)\parallel }_p={({\int }_{-\infty }^{\infty }\sum _{i=1}^m\mid x\left(t\right){\mid }^{p}dt)}^{1/p}<\infty ,1<p<\infty \}$$

Suppose $T_{zw}\left(s\right)$ between $w\left(t\right)$ and $z\left(t\right)$ is linear and bounded. Then, we can obtain the following definitions.

Definition 2.

For the linear FOS $T_{zw}:L_m^2\left(\mathit{R}\right)\to L_p^2\left(\mathit{R}\right)$, define ∞-norm as follows

$$\parallel T_{zw}{\left(s\right)\parallel }_{\infty }=\underset{\omega \in \mathit{R}}{sup}{\parallel T_{zw}\left(jw\right)\parallel }_2$$

Definition 3.

Transfer $\parallel T_{zw}{\left(s\right)\parallel }_{H_{\infty }}$ norm is defined by

$$\parallel T_{zw}{\left(s\right)\parallel }_{H_{\infty }}=\underset{\omega \in \mathit{R}}{sup}{\sigma }_{max}\left(T_{zw}\left(j\omega \right)\right)$$

where ${\sigma }_{max}$ is the maximum singular value.

Remark 1.

According to Definitions 1–3, we can obtain $\parallel T_{zw}{\left(s\right)\parallel }_{H_{\infty }}={sup}_{\omega \in \mathit{R}}{\parallel T_{zw}\left(jw\right)\parallel }_2<\gamma$ for a given real scalar γ. Then, we can obtain that $T_{zw}{\left(jw\right)}^{*}{T}_{zw}\left(jw\right)<{\gamma }^2$.

Remark 2.

In the paper, the system gain ${\mathsf{\Gamma }}_{ee}=\frac{{\parallel z\parallel }_2}{{\parallel w\parallel }_2}$ (Energy-to-Energy gain) is equivalent to $\parallel T_{zw}{\left(s\right)\parallel }_{H_{\infty }}$ for FOS.

Problem 1.

Design the control law for FOS such that:

• Find the gain K to ensure close-loop FOS with $u\left(t\right)=Kx\left(t\right)$ is stable.
• $H_{\infty }$ norm $\parallel T_{zw}{\left(s\right)\parallel }_{H_{\infty }}$ in Definition 3 is minimal for close-loop FOS.

2.2. Preliminaries and Lemmas

In this subsection, some preliminaries of FOS and some lemmas are introduced in the following part.

For the sake of solving Problem 1 in Section 2.1, we need introduce the below lemmas.

Lemma 1

([19]). $D^{\alpha }x\left(t\right)=Ax\left(t\right),0<\alpha <1$ is asymptotically stable if there exist matrices X, Y such that

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

$$sin\left(\frac{\alpha \pi }{2}\right)AX+cos\left(\frac{\alpha \pi }{2}\right)AY+sin\left(\frac{\alpha \pi }{2}\right)X{A}^T-cos\left(\frac{\alpha \pi }{2}\right)Y{A}^T<0$$

Lemma 2

([25]). (Schur complement) For given matrices $S_1=S_1^T$, $S_2$ and $S_3>0$, the following inequality holds

$$S_1+S_2{S}_3^{-1}{S}_2^T<0$$

iff

$$\left[\begin{array}{cc}{S}_1& S_2\\ S_2^T& -S_3\end{array}\right]<0$$

Lemma 3

([26]). Given matrices M and N, and $F\left(\sigma \right)$ which satisfies $F^T\left(\sigma \right)F\left(\sigma \right)\le I$

$$\Omega +MF\left(\sigma \right)N+N^T{F}^T\left(\sigma \right)M^T<0$$

if there exists the scalar $ϵ>0$ such that

$$\Omega +ϵM{M}^T+{ϵ}^{-1}{N}^{T}N<0$$

Lemma 4

([27]). For region D, if there exist the matrices $X\in {\mathit{R}}^{p\times p}$ and $P\in {\mathit{C}}^{p\times p}$ such that

$$D=\{\lambda \in \mathit{C}:X+\lambda P+\overline{\lambda }{P}^{*}<0\}$$

then, D is the LMI region. The following function

$$f_D\left(\lambda \right)=X+\lambda P+\overline{\lambda }{P}^{*}$$

is the characteristic function of region D.

Lemma 5

([27]). For region D, the matrix $Z\in {\mathit{R}}^{n\times n}$ is D-stable if there exists matrix $X\in {\mathit{C}}^{n\times n}$, such that

$$M_D(Z,X)<0$$

where

$$M_D(Z,X)=L\otimes X+M\otimes \left(XZ\right)+M^{*}\otimes \left(Z^T{X}^T\right)$$

3. ${\mathit{H}}_{\infty }$ Control

In the section, problem statement is first introduced. Then, the necessary and sufficient condition is derived for $H_{\infty }$ control of FOS. Moreover, a condition of the state feedback controller is proposed.

Based on system (1), consider the following state space

$$\begin{array}{ccc}& & D^{\alpha }x\left(t\right)=Ax\left(t\right)+Bu\left(t\right)+B_{w}w\left(t\right)\hfill \\ \hfill & & y\left(t\right)=x\left(t\right)\hfill \\ & & z\left(t\right)=C_{z}x\left(t\right)+D_{z}u\left(t\right)+D_{zw}w\left(t\right)\hfill \end{array}$$

(2)

Then, the $H_{\infty }$ stability and stabilization conditions are proved as follows.

Theorem 1.

System (2) with $u\left(t\right)=0$ and ${\parallel G\left(s\right)\parallel }_{H_{\infty }}<\gamma$ is stable if there exist X, Y and a real scalar γ such that

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

(3)

$$\begin{array}{cc}\hfill & \left[\begin{array}{ccc}{\Sigma }_{11}& {\Sigma }_{12}& B_w\\ *& -\gamma I& D_{zw}\\ *& *& -\gamma I\end{array}\right]<0\hfill \end{array}$$

(4)

where ${\Sigma }_{11}=AXsin\left(\frac{\alpha \pi }{2}\right)+X{A}^{T}sin\left(\frac{\alpha \pi }{2}\right)+AYcos\left(\frac{\alpha \pi }{2}\right)-Y{A}^{T}cos\left(\frac{\alpha \pi }{2}\right)$, and ${\Sigma }_{12}=(Xsin\left(\frac{\alpha \pi }{2}\right)-Ycos\left(\frac{\alpha \pi }{2}\right))C_z^T$.

Proof. 

(Necessity) Firstly, we can obtain the following from Definitions 2 and 3

$$\begin{array}{c}\hfill {\parallel G\left(s\right)\parallel }_{H_{\infty }}=\underset{Re\left(s\right)\ge 0}{sup}\sigma \left(T_{zw}\left(s\right)\right)<\gamma \end{array}$$

(5)

Then, according to Lemma 5, there exist $Z=A+C_z{(I-D_{zw})}^{-1}{B}_w$, $L=0$ and $M=M_1^T{M}_2$ such that

$$\begin{array}{ccc}& & {\gamma }^{2}I>M_D(Z,P)=L\otimes P+M\otimes \left(PZ\right)+M^{*}\otimes \left(Z^T{P}^T\right)\hfill \\ \hfill & & =M_D(A,P)+sym[\left(M_1^T{M}_2\right)\otimes \left(P{C}_z{(I-D_{zw})}^{-1}{B}_w\right)]\hfill \\ \hfill & & =M_D(A,P)+sym[(M_1^T\otimes P{C}_z)\times (M_2\otimes {(I-D_{zw})}^{-1}{B}_w)]\hfill \end{array}$$

(6)

The inequality (6) can be expressed as

$$\begin{array}{ccc}& & v^T{M}_D(A,P)v+2v^T(M_1^T\otimes P{C}_z)\times (M_2\otimes {(I-D_{zw})}^{-1}{B}_w)v<{\gamma }^{2}I\hfill \end{array}$$

(7)

where we can see $w=(M_2\otimes {(I-D_{zw})}^{-1}{B}_w)v$ is the unique solution of the equation $w=q_{w,v}$, where

$$\begin{array}{ccc}& & q_{w,v}=(I_k\otimes D_{zw})w+(M_2\otimes B_w)v,k=rank\left(M\right).\hfill \end{array}$$

(8)

Now, we can know that

$$\begin{array}{cc}& q_{w,v}^T{q}_{w,v}={\left[\begin{array}{c}v\\ w\end{array}\right]}^T(\left[\begin{array}{c}{M}_2\otimes B_w\\ I_k\otimes D_{zw}\end{array}\right]\left[\begin{array}{cc}{M}_2\otimes B_w& I_k\otimes D_{zw}\end{array}\right])\left[\begin{array}{c}v\\ w\end{array}\right]\ge 0\end{array}$$

(9)

Then, according to $q^T\left({\gamma }^{2}I\right)q\ge 0$, (9) can be expressed as

$$\begin{array}{ccc}& & {\left[\begin{array}{c}v\\ w\end{array}\right]}^T\left[\begin{array}{cc}{M}_D(A,P)& M_1^T\otimes P{C}_z\\ *& -{\gamma }^{2}I\end{array}\right]\left[\begin{array}{c}v\\ w\end{array}\right]<0\hfill \\ \hfill \end{array}$$

(10)

Then, Using a standard $\mathbf{S}$-procedure argument [25], inequality (10) and inequality (11) are in turn equivalent to the single LMI constraint

$$\begin{array}{ccc}& & \left[\begin{array}{cc}{M}_D(A,P)& M_1^T\otimes P{C}_z\\ *& -{\gamma }^{2}I\end{array}\right]+\left[\begin{array}{c}{M}_2\otimes B_w\\ I_k\otimes D_{zw}\end{array}\right]\left[\begin{array}{cc}{M}_2\otimes B_w& I_k\otimes D_{zw}\end{array}\right]<0\hfill \end{array}$$

(11)

where, let $M=I$, $M_1^T=M_2=I$ and $P=Xsin\left(\frac{\pi \alpha }{2}\right)+Ycos\left(\frac{\pi \alpha }{2}\right)$. Using Lemma 2, by premultiplying and postmultiplying the matrix $diag\{I,{\gamma }^{\frac{1}{2}}I,{\gamma }^{-\frac{1}{2}}I\}$, we can find that (11) is equivalent to (4). The necessity is proved. □

(Sufficiency) From inequality (3), we can obtain that

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

For inequality (4), premultiplying and postmultiplying the matrix $diag\{I,{\gamma }^{\frac{1}{2}}I,{\gamma }^{-\frac{1}{2}}I\}$, we can obtain

$$\begin{array}{cc}& \left[\begin{array}{ccc}{\Sigma }_{11}& {\Sigma }_{12}& B_w\\ *& -{\gamma }^{2}I& D_{zw}\\ *& *& -I\end{array}\right]<0\end{array}$$

(12)

where ${\Sigma }_{11}=AXsin\left(\frac{\pi \alpha }{2}\right)+X{A}^{T}sin\left(\frac{\pi \alpha }{2}\right)+AYcos\left(\frac{\pi \alpha }{2}\right)-Y{A}^{T}cos\left(\frac{\pi \alpha }{2}\right)$, and ${\Sigma }_{12}=(Xsin\left(\frac{\pi \alpha }{2}\right)-Ycos\left(\frac{\pi \alpha }{2}\right))C_z^T$.

Then according to Lemma 2, inequality (12) can be represented as

$$\begin{array}{cc}& \left[\begin{array}{cc}{\Sigma }_{11}& {\Sigma }_{12}\\ *& -{\gamma }^{2}I\end{array}\right]+\left[\begin{array}{c}{B}_w\\ D_{zw}\end{array}\right]\left[\begin{array}{cc}{B}_w^T& D_{zw}^T\end{array}\right]<0\end{array}$$

(13)

Let $L=0$, $M=M_1^T{M}_2=I$, $M_1^T=M_2=I$ and $P=Xsin\left(\frac{\pi \alpha }{2}\right)+Ycos\left(\frac{\pi \alpha }{2}\right)$. We can obtain

$$\begin{array}{ccc}& & \left[\begin{array}{cc}{M}_D(A,P)& M_1^T\otimes P{C}_z\\ *& -{\gamma }^{2}I\end{array}\right]+\left[\begin{array}{c}{M}_2\otimes B_w\\ I_k\otimes D_{zw}\end{array}\right]\left[\begin{array}{cc}{M}_2\otimes B_w& I_k\otimes D_{zw}\end{array}\right]<0\hfill \end{array}$$

(14)

where we can know that the system (2) is stable and inequality (12) can be expressed as inequality (11). Then using inequalities (6)–(10), the proof of sufficiency is completed. Hence, Theorem 1 is proved completely.

Remark 3.

Compare with the existing results in [23,24], two real matrices are used to replace the conjugate matrix or Hermitian matrix; Theorem 1 in the paper can be solved easily. Moreover, when $\alpha \to 1$, Theorem 1 can be used to solve $H_{\infty }$ control for integer order systems. Hence, Theorem 1 is not conservative, and seen as generalization of the results for integer systems.

Next, state feedback $H_{\infty }$ control should be considered in the below theorem.

Theorem 2.

System (2) with $u\left(t\right)=Kx\left(t\right)$ and ${\parallel G\left(s\right)\parallel }_{H_{\infty }}<\gamma$ is stable if there exist three matrices X, Y, Z and a real scalar γ such that

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

(15)

$$\begin{array}{cc}\hfill & \left[\begin{array}{ccc}{\Pi }_{11}& {\Pi }_{12}& B_w\\ *& -\gamma I& D_{zw}\\ *& *& -\gamma I\end{array}\right]<0\hfill \end{array}$$

(16)

where ${\Pi }_{11}=AXsin\left(\frac{\alpha \pi }{2}\right)+X{A}^{T}sin\left(\frac{\alpha \pi }{2}\right)+AYcos\left(\frac{\alpha \pi }{2}\right)-Y{A}^{T}cos\left(\frac{\alpha \pi }{2}\right)+BZ+Z^T{B}^T$, ${\Pi }_{12}=X{C}_z^{T}sin\left(\frac{\alpha \pi }{2}\right)-Y{C}_z^{T}cos\left(\frac{\alpha \pi }{2}\right)+Z^T{D}_z^T$, and $K=Z{(Xsin\left(\frac{\alpha \pi }{2}\right)+Ycos\left(\frac{\alpha \pi }{2}\right))}^{-1}$.

Proof. 

Let $A+BK=\widehat{A}$, $C_z+D_{z}K=\widehat{C}$; Theorem 2 is directly derived by using Theorem 1 and Lemma 2. Moreover, the state feedback matrix $K=Z{(Xsin\left(\theta \right)+Ycos\left(\theta \right))}^{-1}$ can be obtained. □

Remark 4.

As a result, based on Theorem 2, Problem 1 proposed in Section 2.1 can be solved.

4. Robust ${\mathit{H}}_{\infty }$ Control

Based on system (2), consider the following FOS with uncertainty

$$\begin{array}{ccc}& & D^{\alpha }x\left(t\right)=(A+{\Delta }_A)x\left(t\right)+Bu\left(t\right)+B_{w}w\left(t\right)\hfill \\ & & y\left(t\right)=x\left(t\right)\hfill \\ & & z_z\left(t\right)=C_{z}x\left(t\right)+D_{z}u\left(t\right)+D_{zw}w\left(t\right)\hfill \end{array}$$

(17)

where ${\Delta }_A$ is bounded disturbance. Hence, ${\Delta }_A=MF\left(\sigma \right)N$, and $F\left(\sigma \right)$ satisfies $F^T\left(\sigma \right)F\left(\sigma \right)\le I$.

Problem 2.

Design the control law for FOS with uncertainty such that:

• Find the gain K to ensure close-loop FOS with $u\left(t\right)=Kx\left(t\right)$ and uncertainty is quadratically stable.
• The $H_{\infty }$ norm $\parallel T_{zw}{\left(s\right)\parallel }_{H_{\infty }}$ in Definition 3 is minimal for close-loop FOS with uncertainty.

Then, robust $H_{\infty }$ stability and stabilization conditions are proved as follows.

Theorem 3.

System (17) with $u\left(t\right)=0$ and ${\parallel G\left(s\right)\parallel }_{H_{\infty }}<\gamma$ is quadratically stable if there exist matrices X, Y and two real scalars γ, ϵ such that

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

(18)

$$\begin{array}{cc}\hfill & \left[\begin{array}{cccc}{\Omega }_{11}& N(Xsin\left(\frac{\alpha \pi }{2}\right)+Ycos\left(\frac{\alpha \pi }{2}\right))& {\Omega }_{12}& B_w\\ *& -ϵI& 0& 0\\ *& *& -\gamma I& D_{zw}\\ *& *& *& -\gamma I\end{array}\right]<0\hfill \end{array}$$

(19)

where ${\Omega }_{11}=AXsin\left(\frac{\alpha \pi }{2}\right)+X{A}^{T}sin\left(\frac{\alpha \pi }{2}\right)+AYcos\left(\frac{\alpha \pi }{2}\right)-Y{A}^{T}cos\left(\frac{\alpha \pi }{2}\right)+ϵM{M}^T$, and ${\Omega }_{12}=(Xsin\left(\frac{\alpha \pi }{2}\right)-Ycos\left(\frac{\alpha \pi }{2}\right))C_z^T$.

Proof. 

From (19), according to Lemma 2, we can obtain

$$\begin{array}{cc}\hfill & \left[\begin{array}{ccc}{\Psi }_{11}& {\Psi }_{12}& B_w\\ *& -\gamma I& D_{zw}\\ *& *& -\gamma I\end{array}\right]<0\hfill \end{array}$$

(20)

where

$$\begin{array}{cc}\hfill {\Psi }_{11}=& AXsin\left(\frac{\alpha \pi }{2}\right)+X{A}^{T}sin\left(\frac{\alpha \pi }{2}\right)+AYcos\left(\frac{\alpha \pi }{2}\right)-Y{A}^{T}cos\left(\frac{\alpha \pi }{2}\right)+ϵM{M}^T+\\ \hfill & +\frac{1}{ϵ}{(Xsin\left(\frac{\alpha \pi }{2}\right)+Ycos\left(\frac{\alpha \pi }{2}\right))}^T{N}^{T}N(Xsin\left(\frac{\alpha \pi }{2}\right)+Ycos\left(\frac{\alpha \pi }{2}\right))\end{array}$$

and ${\Psi }_{12}=(Xsin\left(\frac{\alpha \pi }{2}\right)-Ycos\left(\frac{\alpha \pi }{2}\right))C_z^T$.

Then, from Lemma 3 and (20), we can obtain the following inequality

$$\begin{array}{cc}\hfill & \left[\begin{array}{ccc}{\overline{\Psi }}_{11}& {\Psi }_{12}& B_w\\ *& -\gamma I& D_{zw}\\ *& *& -\gamma I\end{array}\right]<0\hfill \end{array}$$

(21)

where

$$\begin{array}{cc}\hfill {\overline{\Psi }}_{11}=& AXsin\left(\frac{\alpha \pi }{2}\right)+X{A}^{T}sin\left(\frac{\alpha \pi }{2}\right)+AYcos\left(\frac{\alpha \pi }{2}\right)-Y{A}^{T}cos\left(\frac{\alpha \pi }{2}\right)+\\ \hfill & {\Delta }_{A}Xsin\left(\frac{\alpha \pi }{2}\right)+X{\Delta }_A^{T}sin\left(\frac{\alpha \pi }{2}\right)+{\Delta }_{A}Ycos\left(\frac{\alpha \pi }{2}\right)-Y{\Delta }_A^{T}cos\left(\frac{\alpha \pi }{2}\right)\end{array}$$

Let $\overline{A}=A+{\Delta }_A$, obtain

$$\begin{array}{cc}\hfill {\overline{\Psi }}_{11}=& \overline{A}Xsin\left(\frac{\alpha \pi }{2}\right)+X{\overline{A}}^{T}sin\left(\frac{\alpha \pi }{2}\right)+\overline{A}Ycos\left(\frac{\alpha \pi }{2}\right)-Y{\overline{A}}^{T}cos\left(\frac{\alpha \pi }{2}\right)\end{array}$$

Therefore, Theorem 3 can be proved on the basis of Theorem 1. □

Next, robust $H_{\infty }$ control with state feedback should be considered in the following theorem.

Theorem 4.

System (17) with $u\left(t\right)=Kx\left(t\right)$ and ${\parallel G\left(s\right)\parallel }_{H_{\infty }}<\gamma$ is quadratically stable if there exist matrices X, Y, Z and two real scalars γ, ϵ such that

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

(22)

$$\begin{array}{cc}\hfill & \left[\begin{array}{cccc}{\Xi }_{11}& N(Xsin\left(\frac{\alpha \pi }{2}\right)+Ycos\left(\frac{\alpha \pi }{2}\right))& {\Xi }_{12}& B_w\\ *& -ϵI& 0& 0\\ *& *& -\gamma I& D_{zw}\\ *& *& *& -\gamma I\end{array}\right]<0\hfill \end{array}$$

(23)

where ${\Xi }_{11}=AXsin\left(\frac{\alpha \pi }{2}\right)+X{A}^{T}sin\left(\frac{\alpha \pi }{2}\right)+AYcos\left(\frac{\alpha \pi }{2}\right)-Y{A}^{T}cos\left(\frac{\alpha \pi }{2}\right)+BZ+Z^T{B}^T+ϵM{M}^T$, ${\Xi }_{12}=X{C}_z^{T}sin\left(\frac{\alpha \pi }{2}\right)-Y{C}_z^{T}cos\left(\frac{\alpha \pi }{2}\right)+Z^T{D}_z^T$, and $K=Z{(Xsin\left(\frac{\alpha \pi }{2}\right)+Ycos\left(\frac{\alpha \pi }{2}\right))}^{-1}$.

Proof. 

According to Theorem 2 and Theorem 3, we can prove Theorem 4 easily, omitted. □

Remark 5.

According to Theorem 4, Problem 2 can be solved completely. Moreover, in the case of $0<\alpha <1$, above theorems of $H_{\infty }$ and robust $H_{\infty }$ control can be easily used to simulate with LMI toolbox.

5. Simulation Examples

In the following, two numerical examples are shown to verify the effectiveness of the conditions proposed in Section 3 and Section 4.

5.1. Example 1

For system (2), consider order $\alpha =\frac{1}{3}$, and

$$A=\left[\begin{array}{cc}0& 1\\ 1& -1\end{array}\right],B=\left[\begin{array}{c}1\\ 1\end{array}\right],B_w=\left[\begin{array}{c}1\\ 0\end{array}\right]$$

$$C_z=\left[\begin{array}{cc}1& 0\end{array}\right],D_z=\left[\begin{array}{c}2\\ 1\end{array}\right],D_{zw}=1$$

The open loop transfer function is

$$T_{zw}=\frac{s^{\frac{2}{3}}+2s^{\frac{1}{3}}}{s^{\frac{2}{3}}+s^{\frac{1}{3}}-1}$$

Figure 1 shows bode diagram, and system is not stable from the state diagram in Example 1.

According to Theorem 2, we can solve the numerical example 1 and obtain

$$X=\left[\begin{array}{cc}58.3347& -2.2373\\ -2.2373& 63.4722\end{array}\right],Y=\left[\begin{array}{cc}0& 1.2917\\ -1.2917& 0\end{array}\right]$$

$$Z=\left[\begin{array}{cc}-52.2858& -22.7870\end{array}\right],\gamma =120.9782$$

Then we can obtain

$$K=Z(Xsin\left(\frac{\alpha \pi }{2}\right)+Ycos{\left(\frac{\alpha \pi }{2}\right)}^{-1}=\left[\begin{array}{cc}-1.8477& -0.7180\end{array}\right]$$

Eigenvalues of $(A+BK)$ are $\{-1.7829+0.4846i\}$ and $\{-1.7829-0.4846i\}$, i.e., the closed loop is stable. Figure 2 also shows that the system in Example 1 is stable with state feedback.

5.2. Example 2

Consider system (17) with fractional order $\alpha =0.5$, and

$$A=\left[\begin{array}{ccc}-1& 2& 2\\ -3& 2& 1\\ 1& -1& 2\end{array}\right],B=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],B_w=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$$

$$C_z=\left[\begin{array}{ccc}1& 0& 0\end{array}\right],D_z=\left[\begin{array}{c}2\\ 1\end{array}\right],D_{zw}=1$$

$$M=\left[\begin{array}{ccc}0.2& 0.2& 0.3\\ 0.1& 0.4& 0.5\\ 0.2& 0.3& 0.4\end{array}\right],N=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],F\left(\sigma \right)=\left[\begin{array}{ccc}sin\left(0.1\pi \right)& 0& 0\\ 0& cos\left(0.1\pi \right)& 0\\ 0& 0& sin\left(0.1\pi \right)\end{array}\right]$$

According to Theorem 4, we can solve the numerical example 2 and obtain

$$X=\left[\begin{array}{ccc}39.3411& 21.4029& -27.3651\\ 21.4029& 43.3116& 3.4823\\ -27.3651& 3.4823& 11.1525\end{array}\right],Y=\left[\begin{array}{ccc}0& 27.3651& -3.4823\\ -27.3651& 0& -11.1525\\ 3.4823& 11.1525& 0\end{array}\right]$$

$$Z=\left[\begin{array}{ccc}69.1734& 47.4414& -28.6476\end{array}\right]$$

$$ϵ=-193.5910,\gamma =99.1739$$

Then we can obtain

$$K=Z{(Xsin\left(\frac{\alpha \pi }{2}\right)+Ycos\left(\frac{\alpha \pi }{2}\right))}^{-1}=\left[\begin{array}{ccc}5.7410& -7.3458& 7.1940\end{array}\right]$$

Figure 3 shows that system (17) with the above K is quadratically stable.

6. Conclusions

In the paper, necessary and sufficient methods of $H_{\infty }$ control for FOS with order $0<\alpha <1$ are proposed. Moreover, robust $H_{\infty }$ control methods for FOS with uncertainty are derived. Comparing these methods to previous results, two real matrices are used to replace conjugate matrix or or Hermitian matrix. These conditions are less conservative, and easier to solve. These proposed methods in the paper are similar to integer order methods of $H_{\infty }$ and robust $H_{\infty }$ control. Moreover, when $\alpha \to 1$, the results of integer systems can be extended by our methods. Hence, our methods are more general and applicable. Finally, two examples are presented to illustrate of our methods.

In the future, observer-based $H_{\infty }$ control and dynamic output feedback $H_{\infty }$ control for FOS will be our work.