1 Introduction

For past centuries, fractional calculus has been a very interesting topic, but just for mathematicians, to make the subject well understandable for engineers or scientists point of view [1–4]. Only in the last decades, fractional calculus have been caught much attention, because it has been shown that non-integer models can be both theoretically challenging and pertinent for many fields of science and technology such as chemistry [5–7], biology [8, 9], economics [10, 11], psychology [12, 13], mass diffusion, heat conduction, physical and engineering applications [14, 15] etc.

In some real models, output feedback control may not guarantee the stability of the closed loop, because the system output measurements do not provide a complete information on the internal state of the system. For this reason, observer design for estimating the states of a system has received considerable attention in the past [16–18]. Observer-based controllers are generally used to stabilize unstable systems or to improve the system performances. Recently, the research activities for the OBC have been developed for FOS [19–21].

The \(H_{\infty }\) theory was generally restricted to integer-order systems. In the recent years, some works about the extension of the \(H_{\infty }\)-norm computation to FOS have been appeared [22], where the authors defined the pseudo Hamiltonian matrix of a fractional order system, and they proposed two methods to compute FOS \(H_{\infty }\)-norm based on this pseudo Hamiltonian matrix. The first one was a dichotomy algorithm and the second one used LMI formalism. Based on these analysis results, methods to design \(H_{\infty }\) state feedback controllers and \(H_{\infty }\) observer were proposed in [23–26].

The main idea of our study is to design robust \(H_{\infty }\) observer-based controllers for DU-FOS, affected by disturbances that are supposed to have finite energy. After stating OBCs design objectives, our results are given in matrix inequalities. Firstly, the existence conditions of the OBC of such systems are given. In the second section, by using the \(H_{\infty }\)-optimization technique, the stability of the estimation error and stabilization of the original system are given in inequality condition, where all the observer gains and the control law can be computed by solving this inequality condition in two step. The method proposed have two objectives, the first one, satisfies the \(H_{\infty }\) performance index, and the second one is the stabilization of the DU-FOS.

Notation: \(\mathbb {R}^{n}\) and \(\mathbb {R}^{n\times m}\) denote the n dimensional Euclidean space and the set of all \(n\times m\) real matrices, respectively; \(A^{T}\) and \(A^{*}\) denote the transpose and the conjugate transpose of matrix A, respectively; matrix A is symmetric positive definite if and only if \(A^{T}=A\) and \(A>0\); \(A^{+}\) means the generalized inverse of matrix A which satisfies \(AA^{+}A=A\); \(\Vert \Vert _{\infty }\) is the \(H_{\infty }\) norm; I and 0 denote the identity matrix and zero matrix, respectively, of appropriate dimension. \(Sym\{X\}\) is used to denote \(X^{*} + X\). The notation (\(*\)) is the conjugate transpose of the off-diagonal part.

2 Preliminaries

The fractional-order derivative definition introduced by Caputo for a function f(t) can be given as [27]

$$\begin{aligned} ^C_a D^{\alpha }_t f(t)=\dfrac{1}{\varGamma (\alpha -n)}\int _{a}^{t}\dfrac{f^{n}(\tau )}{(t-\tau )^{\alpha -n+1}}d\tau&\quad ,&(n-1) < \alpha < n \end{aligned}$$
(1)

with \(n \in \mathbb {N^*}\) and \(\alpha \in \mathbb {R}^{+}\), where \( \varGamma (.) \) is the Gamma function.

Consider the following linear fractional-order system

$$\begin{aligned} \left\{ \begin{array}{ll} D^\alpha x(t)=A_0 x(t)+ B_0 u(t)\\ y(t)= C_0 x(t) + D_0 u(t)\\ \end{array} \right.&,&0 < \alpha < 2 \end{aligned}$$
(2)

where \(x(t)\in \mathbb {R}^{n}\) is the state vector, \(u(t)\in \mathbb {R}^{m}\) is the control input vector, \(y(t)\in \mathbb {R}^{p}\) is the measured output and \(D^{\alpha }\) is used to denote the fractional derivative of order \(\alpha \). \(A_0\), \(B_0\), \(C_0\) and \(D_0\) are known constant matrices and with appropriate dimensions.

Two tools of \(H_{\infty }\) norm computation for FOS were firstly proposed in [22], and the calculation of \(L_{2}\)-gain for FOS was initially presented in [22] and it was extended to give a new formulation in [28]. The following lemmas show these computations.

Lemma 1

([22]) \(H_{\infty }\)-norm of a fractional-order system \(G=(A,B,C,D,\alpha )\) is bounded by a real positive number \(\gamma \) if and only if the eigenvalues of matrix \(A_{\gamma }\) lie in the stable domain defined by \({s \in C : |arg(s)| > \alpha \frac{\pi }{2}}\).

$$\begin{aligned} {A_{\gamma } =\begin{pmatrix} (A+BRD^{T}C) &{} \varXi BRB^{T}\\ C^{T}(I+DRD^{T})C &{} \varXi (A^{T}+C^{T}DRB^{T}) \end{pmatrix}} \end{aligned}$$
(3)

where

$$\begin{aligned} \varXi =e^{-\alpha j\pi } \quad {\text {and}} \quad R=(\gamma ^2 I - D^{T}D)^{-1} \end{aligned}$$

Lemma 2

([28]) For a LTI-FOS \(G=(A_0,B_0,C_0,D_0,\alpha )\), the \(L_{2}\)-gain is bounded by \(\gamma \), if there exists a positive definite Hermitian matrix P, such that

$$\begin{aligned} \varGamma _1 = {\begin{bmatrix}\varXi _1 P A_0 + \varXi _1^{*} A_0^{T}P&PB_0&\varXi _1^{*} C_0^{T}\\ B_0^{T}P&-\gamma ^{2}I&D_0^{T} \\ \varXi _1 C_0&D_0&-I \end{bmatrix} < 0 } \end{aligned}$$
(4)

with

$$\begin{aligned} \varXi _1=e^{(1-\alpha )j\pi /2} {\text {and}} \varXi _1^{*}=e^{-(1-\alpha )j\pi /2} \end{aligned}$$

Lemma 3

([29]) Let D, E and F be real matrices of appropriate dimensions and F satisfies \(F^{T}F\leqslant I\). Then for any scalar \(\epsilon >0\) and vectors \(x, y \in \mathbb {R}^{n}\), we have

$$\begin{aligned} 2x^{T}DFEy\leqslant \epsilon ^{-1}x^{T}DD^{T}x+\epsilon y^{T}E^{T}Ey \end{aligned}$$
(5)

3 Main Results

3.1 \(H_{\infty }\)-norm Computation for FOS with Uncertainties

Let us consider the following linear systems with uncertainties and disturbances:

$$\begin{aligned} \left\{ \begin{array}{ll} D^\alpha x(t)=(A+ \varDelta A)x(t)+ (B_{w} + \varDelta B_{w})w(t)\\ y(t)=(C + \varDelta C)x(t) + (D_{w} + \varDelta D_{w})w(t)\\ \end{array} \right.&,&0 < \alpha < 2 \end{aligned}$$
(6)

The terms \(\varDelta A\), \(\varDelta B_{w}\), \(\varDelta C\), and \(\varDelta D_{w}\) are unknown matrices representing time-varying parameter uncertainties.

Assumption 1

In this paper, we will consider the following structure for the uncertainties

$$\begin{aligned} \begin{bmatrix} \varDelta A&\varDelta B_{w} \\ \varDelta C&\varDelta D_{w} \end{bmatrix} = M F(t) \begin{bmatrix} N_A&N_{B_{w}} \\ N_C&N_{D_{w}} \end{bmatrix} \end{aligned}$$
(7)

where M, \(N_A\), \(N_{B_{w}}\), \(N_C\), and \(N_{D_{w}}\) are known real constant matrices and of appropriate dimensions and F(t) is an unknown real-valued time-varying matrix satisfying

$$\begin{aligned} F^{T}(t)F(t)\le I \end{aligned}$$
(8)

when the elements of F(t) are Lebesgue measurable.

Lemma 4

The \(L_{2}\)-gain of an uncertain LTI-FOS (6) is bounded by \(\gamma \), if there exists a positive definite Hermitian matrix P and two positive scalars \(\mu _1\) and \(\mu _2\) such that

(9)

with

$$\begin{aligned} \varXi _1=e^{(1-\alpha )j\pi /2}, \varXi _1^{*}=e^{-(1-\alpha )j\pi /2}, \varXi _1 \times \varXi _1^{*}=1 \end{aligned}$$

Proof

By letting \(A_0=A+\varDelta A\), \(B_0=B_{w}+\varDelta B_{w}\), \(C_0=C+\varDelta C\) and \(D_0=D_{w}+\varDelta D_{w}\), the inequality (4) can be written as

$$\begin{aligned} \varGamma _1 = \varOmega _1 + Sym\{ X_1 F(t) Y_1 + X_2 F(t) Y_2 \} < 0 \end{aligned}$$
(10)

with

$$\begin{aligned} \varOmega _1&= \begin{bmatrix}\varXi _1 PA + \varXi _1^{*} A^{T}P&PB&\varXi _1^{*} C^{T}\\ B^{T}P&-\gamma ^{2}I&D^{T} \\ \varXi _1 C&D&-I \end{bmatrix}&X_1&= \begin{bmatrix} PM \\ 0 \\ 0 \end{bmatrix}&X_2&= \begin{bmatrix} 0 \\ 0 \\ M \end{bmatrix} \\ Y_1&= \begin{bmatrix} \varXi _1 N_A&N_{B_{w}}&0 \end{bmatrix}&Y_2&= \begin{bmatrix} \varXi _1 N_C&N_{D_{w}}&0 \end{bmatrix}&\end{aligned}$$

According to Lemma 3, we obtain the following inequality

$$\begin{aligned} \varGamma _1 < \hat{\varGamma }_1 = \varOmega _1 + \mu _1 X_1 X_1^T + \mu _1^{-1} Y_1^T Y_1 + \mu _2 X_2 X_2^T + \mu _2^{-1} Y_2^T Y_2 \end{aligned}$$
(11)

One can see from the above results that the inequality (10) is verified, if there exists two positive scalars \(\mu _1\) and \(\mu _2\) such that

$$\begin{aligned} \varOmega _1 + \mu _1 X_1 X_1^T + \mu _1^{-1} Y_1^T Y_1 + \mu _2 X_2 X_2^T + \mu _2^{-1} Y_2^T Y_2 < 0 \end{aligned}$$
(12)

Then, the inequality (12) is equivalent to

(13)

According to Schur complement, one can see that the equivalence between the above inequality (13) and the the following

$$\begin{aligned} {{\left[ {\begin{matrix} Sym\{\varXi _1 PA \} &{} N_{A}^{T} &{} N_{C}^{T} &{} PM &{} PB_{w} + \frac{1}{\mu _1} N_{A}^{T}N_{B_{w}} + \frac{1}{\mu _2} N_{C}^{T}N_{D_{w}} &{} 0 &{} 0 &{} \varXi _1^{*} C^{T} &{} 0 \\ *&{} - \frac{1}{\mu _1} I &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ *&{} *&{} - \frac{1}{\mu _2} I &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ *&{} *&{} *&{} -\mu _1 I &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ *&{} *&{} *&{} *&{} -\gamma ^{2}I &{} N_{B_{w}}^{T} &{} N_{D_{w}}^{T} &{} \ D_{w}^T &{} 0\\ *&{} *&{} *&{} *&{} *&{} - \frac{1}{\mu _1} I &{} 0 &{} 0 &{} 0 \\ *&{} *&{} *&{} *&{} *&{} *&{} - \frac{1}{\mu _2} I &{} 0 &{} 0 \\ *&{} *&{} *&{} *&{} *&{} *&{} *&{} -I &{} M \\ *&{} *&{} *&{} *&{} *&{} *&{} *&{} *&{} - \mu _2 I \end{matrix}}\right] }} < 0 \end{aligned}$$
(14)

We can easily deduce by permuting some rows and columns, the equivalence between the inequalities (9) and (14).

One can see from the above results that a sufficient condition for \(L_{2}\)-gain of the LTI-FOS (6) to be bounded by \(\gamma \) is that if there exist a positive definite Hermitian matrix P and two positive scalars \(\mu _1\) and \(\mu _2\) such that LMI (9) must hold, which completes the proof. \(\square \)

3.2 Robust \(H_{\infty }\) Observer-Based Controller Parametrization

Without loss of generality, we consider the DU-FOS represented by the following form:

$$\begin{aligned} \left\{ \begin{array}{ll} D^\alpha x(t)=(A+ \varDelta A)x(t)+ B_{u} u(t) + B_{w} w(t)\\ y(t)=C x(t) \\ \end{array} \right. \end{aligned}$$
(15)

where \(x(t)\in \mathbb {R}^{n}\) is the state vector, \(u(t) \in \mathbb {R}^{m}\) is the input vector, \(y(t)\in \mathbb {R}^{q} \) the measurement output vector and \(w(t) \in \mathbb {R}^{p}\) is the disturbance input vector. A, \(B_u\), \(B_{w}\) and C are known matrices of appropriate dimensions. The term \(\varDelta A\) is an unknown matrix uncertainty, assumed to respect the Assumption 1.

Assumption 2

We assume that \(w(t) \in L_2\), where the \(L_2\)-norm is defined as

$$\begin{aligned} ||w||_{L_2}&= \left( \int _{0}^{\infty } w(t)^T w(t) \mathrm {d} t\right) ^{\frac{1}{2}} \end{aligned}$$
(16)

To allow the stabilization of system using measurement feedback, we need to reconstruct the state variable. For this, we consider a robust \(H_\infty \) observer with a linear feedback control law of the form

$$\begin{aligned} \left\{ \begin{array}{rl} D^\alpha \eta (t) &{} = N \eta (t) + Hu(t) + J y(t) \\ \hat{x}(t) &{} = \eta (t)+ Ey(t) \\ u(t) &{} = K_u \hat{x}(t) \end{array} \right. \end{aligned}$$
(17)

where \(\eta (t)\in \mathbb {R}^{n}\) is the state vector of robust \(H_\infty \) observer, \(\hat{x}(t)\in \mathbb {R}^{n}\) is the estimate of x(t) and \(u(t)\in \mathbb {R}^{m}\) is the feedback control law. The matrices N, J, H and E are observer unknown matrices of appropriate dimensions which must be determined, such that \( \hat{x}(t)\) converges asymptotically to x(t) for \(w(t)=0\) and \(\frac{||e||_2}{||w||_2}<\gamma \), for \(w(t)\ne 0\). The matrix \(K_u \in \mathbb {R}^{m \times n}\) is the controller gain to be determined.

Before designing the robust \(H_{\infty }\) observer, the estimation error is defined as

$$\begin{aligned} e(t)= x(t)- \hat{x}(t) \end{aligned}$$
(18)

and has the fractional-order dynamic

$$\begin{aligned} D^\alpha e(t)= D^\alpha x(t) - D^\alpha \hat{x}(t) \end{aligned}$$
(19a)

or equivalently

$$\begin{aligned} D^\alpha e(t) = N e(t) + (RA - NR - JC&+ R \varDelta A)x(t)\nonumber \\&\qquad + (RB_u - H)u(t) + RB_w w(t) \end{aligned}$$
(19b)

where \(R=I_{n}-EC\).

We can easily deduce that the input control law stabilizing the system (15) has the following form

$$\begin{aligned} u(t)&= K_u x(t) - K_u e(t) \end{aligned}$$
(20)

The system (15) can be rewritten as

$$\begin{aligned} D^\alpha x(t) = (A +B_u K_u + \varDelta A) x(t) - B_u K_u e(t) + B_{w} w(t) \end{aligned}$$
(21)

System (15) and estimation error (19b) can be combined in the following augmented system

$$\begin{aligned} D^\alpha \begin{bmatrix} x(t) \\ e(t) \end{bmatrix}&= \begin{pmatrix} \begin{bmatrix} A +B_u K_u &{} -B_u K_u \\ \varTheta _{1} &{} N \end{bmatrix} + \begin{bmatrix} \varDelta A &{} 0 \\ R \varDelta A &{} 0 \end{bmatrix} \end{pmatrix} \begin{bmatrix} x(t) \\ e(t) \end{bmatrix} + \begin{bmatrix} B_{w} \\ R B_{w} \end{bmatrix} w(t) + \begin{bmatrix} 0 \\ \varTheta _{2} \end{bmatrix} u(t) \end{aligned}$$
(22)

where

$$\begin{aligned} \varTheta _{1}&= NR + JC - RA \\ \varTheta _{2}&= H - RB_u \end{aligned}$$

Problem 1

Get, if possible, an OBC (17), i.e. determine all the observer and controller gain matrices N, J, H, E and \(K_u\) of appropriate dimensions such that the uncertain system (15) is stabilized for all initial states values.

Proposition 1

System (17) is a robust \(H_{\infty }\) observer-based controller of the system (15), with respect to the Assumption 2 and for any finite x(0) and \(\hat{x}(0)\) if

  1. (i)

    The \(L_{2}\)-gain of the augmented system (22) is bounded by \(\gamma > 0\).

  2. (ii)

    \(NR + JC - RA=0\)

  3. (iii)

    \(H = RB_u\)

where \(R=I_n-EC\).

By using the definition of R, the expression of \(\varTheta _{1}\) can be rewritten as

$$\begin{aligned} N + ECA + KC&= A \end{aligned}$$
(23a)

where \(K = J - NE\).

Now, equation (23a) can be written as

$$\begin{aligned} \begin{bmatrix} N&K&E\end{bmatrix}\mathcal {M}_1 = \mathcal {M}_2 \end{aligned}$$
(24)

where

$$\begin{aligned} \mathcal {M}_1 = \begin{bmatrix} I_n \\ C \\ CA \end{bmatrix}&and&\mathcal {M}_2 = \begin{bmatrix} A \end{bmatrix} \end{aligned}$$
(25)

The necessary and sufficient condition for the existence of the solution of (24) can be given by the following lemma.

Lemma 5

([30]) There exists a solution to (24) if and only if

$$\begin{aligned} rank \begin{bmatrix} \mathcal {M}_1 \\ \mathcal {M}_2 \end{bmatrix} = rank \begin{bmatrix} \mathcal {M}_1 \end{bmatrix} \; . \end{aligned}$$
(26)

If (26) is satisfied, the general solution of (24) is given by

$$\begin{aligned} \begin{bmatrix} N&K&E \end{bmatrix} = \mathcal {M}_2 \mathcal {M}_1^{+} - Z( I - \mathcal {M}_1 \mathcal {M}_1^{+}) \end{aligned}$$
(27)

where \(\mathcal {M}_1^{+}\) is a generalized inverse matrix of \(\mathcal {M}_1\) [30] (i.e. \(\mathcal {M}_1=\mathcal {M}_1 \mathcal {M}_1^{+} \mathcal {M}_1\)) and Z is an arbitrary matrix of appropriate dimension.

From (28a), we obtain

$$\begin{aligned} N&= \mathbb {A}_N-Z\mathbb {B}_N \end{aligned}$$
(28a)
$$\begin{aligned} K&= \mathbb {A}_K-Z\mathbb {B}_K \end{aligned}$$
(28b)
$$\begin{aligned} E&= \mathbb {A}_E-Z\mathbb {B}_E \end{aligned}$$
(28c)

where

$$\begin{aligned} \mathbb {A}_N&= (\mathcal {M}_2 \mathcal {M}_1^{+})\begin{bmatrix} I&0&0 \end{bmatrix}^{T}&\mathbb {B}_N&= ( I - \mathcal {M}_1 \mathcal {M}_1^{+})\begin{bmatrix} I&0&0 \end{bmatrix}^{T} \\ \mathbb {A}_K&= (\mathcal {M}_2 \mathcal {M}_1^{+})\begin{bmatrix} 0&I&0 \end{bmatrix}^{T}&\mathbb {B}_K&= ( I - \mathcal {M}_1 \mathcal {M}_1^{+})\begin{bmatrix} 0&I&0 \end{bmatrix}^{T}\\ \mathbb {A}_E&= (\mathcal {M}_2 \mathcal {M}_1^{+})\begin{bmatrix} 0&0&I \end{bmatrix}^{T}&\mathbb {B}_E&= ( I - \mathcal {M}_1 \mathcal {M}_1^{+})\begin{bmatrix} 0&0&I \end{bmatrix}^{T} \end{aligned}$$

Matrices J and H are obtained from

$$\begin{aligned} \left\{ \begin{array}{ll} J = K + NE \\ H = (I_n-EC)B \end{array} \right. \end{aligned}$$
(29)

By using this results, all the parameters of the Robust \(H_{\infty }\) fractional-order observer (17) can be computed if matrix parameter Z is known.

3.3 Robust \(H_{\infty }\) Observer-Based Controller Design

Now, if conditions (ii and iii) in Proposition 1 are satisfied, then the augmented system (22) can be expressed as

$$\begin{aligned} D^\alpha \begin{bmatrix} x(t) \\ e(t) \end{bmatrix}&= \begin{pmatrix} \begin{bmatrix} A +B_u K_u &{} -B_u K_u \\ 0 &{} N \end{bmatrix} + \begin{bmatrix} \varDelta A &{} 0 \\ R \varDelta A &{} 0 \end{bmatrix} \end{pmatrix} \begin{bmatrix} x(t) \\ e(t) \end{bmatrix} + \begin{bmatrix} B_{w} \\ R B_{w} \end{bmatrix} w(t) \end{aligned}$$
(30)

or equivalently

$$\begin{aligned} D^\alpha \widetilde{X}(t)&= \begin{pmatrix} \widetilde{A} + \varDelta \widetilde{A} \end{pmatrix} \widetilde{X}(t) + \widetilde{B} w(t) \end{aligned}$$
(31)

with

$$\begin{aligned} \widetilde{X}(t)&= \begin{bmatrix} x(t) \\ e(t) \end{bmatrix}&\widetilde{A}&=\begin{bmatrix} A +B_u K_u&-B_u K_u \\ 0&N \end{bmatrix}&\widetilde{B}&=\begin{bmatrix} B_{w} \\ R B_{w} \end{bmatrix} \\ \varDelta \widetilde{A}&=\widetilde{M} F(t) \widetilde{N}_{A}&\widetilde{M}&=\begin{bmatrix} M \\ RM \end{bmatrix}&\widetilde{N}_{A}&=\begin{bmatrix} N_{A}&0 \end{bmatrix} \end{aligned}$$

Lemma 6

System (17) is a robust \(H_{\infty }\) observer-based controller of the uncertain disturbed system (15) with disturbance attenuation given by \(\gamma \), if there exist two positive definite hermitian matrices \(P_1\) and \(P_2\), two matrices \(X_1\) and \(X_2\), and a positive scalar \(\mu _1\), such that the following matrix inequality holds

$$\begin{aligned} \begin{bmatrix} \varOmega _{11}&\varOmega _{12} \\ *&\varOmega _{22} \end{bmatrix} < 0 \end{aligned}$$
(32)

where

$$\begin{aligned} \varOmega _{11}&= \begin{bmatrix} Sym\{\varXi _1 (AP_{1} + B_u X_1) \}&-B_u K_u \\ *&Sym\{\varXi _1( P_2 \mathbb {A}_N - X_2 \mathbb {B}_N )\} \end{bmatrix} \\ \varOmega _{12}&= \begin{bmatrix} \begin{bmatrix} M \\ P_2 \mathbb {A}_R M + X_2 \mathbb {B}_R M \end{bmatrix}&\begin{bmatrix} P_1^{-1} N_A^T \\ 0 \end{bmatrix}&\begin{bmatrix} B_w \\ P_2 \mathbb {A}_R B_w + X_2 \mathbb {B}_R B_w \end{bmatrix}&\varXi _1^{*} I\end{bmatrix} \\ \varOmega _{22}&=\begin{bmatrix} - \mu _1 I&0&0&0\\ *&- \frac{1}{\mu _1} I&0&0 \\ *&*&- \gamma ^2 I&0 \\ *&*&*&- I \end{bmatrix} \end{aligned}$$

with

$$\begin{aligned} \mathbb {A}_R&=I_n-\mathbb {A}_2C&X_1&=K_uP_1^{-1} \\ \mathbb {B}_R&=\mathbb {B}_2 C&X_2&=P_2 Z \end{aligned}$$

Proof

The detailed proof of this result is omitted for space limitation. However, the proof can obtained by replacing all system matrices A, \(B_{w}\), \(N_{A}\) and M in the inequality (9) by their expression given in the augmented system (30) \(\widetilde{A}\), \(\widetilde{B}_{w}\), \(\widetilde{N}_{A}\) and \(\widetilde{M}\), respectively. Then, the inequality (32) is obtained by pre and post multiplication of the above inequality with the following matrices, respectively \(\begin{bmatrix} P_{01}^{-1}&0 \\ 0&I \end{bmatrix}\) and \(\begin{bmatrix} P_{01}^{-1}&0 \\ 0&I \end{bmatrix}^T\), with \(P_{01}^{-1}=P_{1}\). Which complete the proof. \(\square \)

We can remark that the robust \(H_{\infty }\) observer based control problem given by the inequality (32) is a non convex problem. The product between the two decision matrices \(P_1\) and \(K_u\), and the presence of matrices \(P_1\) and its inverse \(P_1^{-1}\) leading to a bilinear matrix inequality (BMI) structure. Then, the inequality (32) can not be solved for (\(P_1\), \(P_2\), \(X_1\), \(X_2\), \(K_u\)) in the same time. According the Schur lemma, all diagonal component must satisfy the inequality. Therefore, we propose to resolve this problem in two step. Firstly, we start by solving the first component in \(\varOmega _{11}\). After obtaining \(P_1\) and \(X_1\), replacing them into the inequality (32) by their value leads to a feasible LMI.

4 Numerical Example

In this section, the performance of the proposed robust \(H_{\infty }\) observer-based stabilizing controller is presented via a numerical example.

Consider the disturbed uncertain fractional-order system described by

$$\begin{aligned} {\small \left\{ \begin{array}{rl} D^{1.5} x(t) &{} =\begin{pmatrix}\begin{bmatrix}0&{}10\\ 15&{}-20\end{bmatrix}+\varDelta A\end{pmatrix}x(t) + \begin{bmatrix}0\\ 0.5\end{bmatrix}u(t) + \begin{bmatrix}1\\ 0.25\end{bmatrix}w(t)\\ y(t) &{} = \begin{bmatrix}1&{}0\end{bmatrix}x(t) \end{array} \right. } \end{aligned}$$
(33)

The uncertainty matrix \(\varDelta A\) is given as

$$\begin{aligned} \varDelta A&= MF(t)N_A=\begin{bmatrix}0.1&-0.5\\0.25&0.4\end{bmatrix} F(t) \begin{bmatrix}-0.1&0\\0&0.5\end{bmatrix} \end{aligned}$$
(34)

with \(F(t)= \mathrm{diag}(0.15\sin (25t),0.15\sin (25t))\).

The inequality condition (32) can be solved by using a two-step procedure. The obtained results for \(\gamma = 0.41\) are given by

$$\begin{aligned}&P_1 = \begin{bmatrix} 375.02&-27.3 \\ -27.3&18.9\end{bmatrix}, X_1 = \begin{bmatrix} -12342&970.71 \end{bmatrix} P_2 = \begin{bmatrix} 20449&-0.025 \\ -0.025&26.4\end{bmatrix},\\&\qquad \qquad \qquad X_2 = \begin{bmatrix} 14460&2024.2&-14460&-202.37 \\ 198.19&568.19&-198.18&-56.54 \end{bmatrix} \end{aligned}$$

Finally, the dynamic of the estimate \( \hat{x}(t)\) and the controller law are given by the following observer

$$\begin{aligned} \left\{ \begin{array}{rl} D^{1.5} \eta (t) &{}= \begin{bmatrix} -0.7 &{} -3 e-06 \\ -0.01 &{} -21.7 \end{bmatrix} \eta (t) + \begin{bmatrix} 0 \\ 0.5 \end{bmatrix}u(t) \\ &{} + \begin{bmatrix} -7e-07 \\ 11.2\end{bmatrix} y(t) \\ \hat{x}(t) &{} = \eta (t)+ \begin{bmatrix} 1 &{} 0.2 \end{bmatrix}y(t) \\ u(t) &{} = \begin{bmatrix} -32.6 &{} 4.3 \end{bmatrix} \hat{x}(t)\end{array} \right. \end{aligned}$$
(35)

Figure 1 show the open loop response of the unstable system. In addition, (Figs. 2, 3, 4 and 5) show the performances in the time domain of the proposed robust \(H_{\infty }\) OBC to ensure the stability of the DU-FOS in the closed loop. The actual states are shown with their estimates, and the estimation errors obtained by using the proposed method. It is clear that the estimate state \(\hat{x}(t)\) converges to the actual state x(t).

One can see in (Figs. 2, 3, 4 and 5) that disturbance w(t) is activated in the time interval between five and ten seconds. In this time interval, the estimate \(\hat{x}(t)\) tracks the actual state x(t) with a small error. This is in agreement with the small value of the disturbance attenuation criterion given by \(\gamma =0.41\).

Fig. 1
figure 1

States trajectories \(x_1(t)\) and \(x_2(t)\) of the open loop unstable systems (33)

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Fig. 2
figure 2

Evolution of the vector \(x_1(t)\) and its estimate \(\hat{x}_1(t)\)

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Fig. 3
figure 3

Evolution of the vector \(x_2(t)\) and its estimate \(\hat{x}_2(t)\)

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Fig. 4
figure 4

Evolution of the estimation error for state \(x_1(t)\)

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Fig. 5
figure 5

Evolution of the estimation error for state \(x_2(t)\)

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5 Conclusion

In this paper it has been shown how that a robust \(H_{\infty }\) observer can be used to design a controller such that the robust stabilization of the DU-FOS is ensured. The existence conditions of an observer and the robust stabilization satisfying the performance requirement of closed loop in the presence of uncertainties and disturbances are investigated. When the system is subject to uncertainties and bounded disturbances, the robust \(H_{\infty }\) OBC must ensure the stabilization of the closed loop, and minimize the effect of disturbances on the estimation error and the system. By adopting an \(H_{\infty }\)-norm approach for FOS, and on the basis of the algebraic constraints derived from the analysis of the estimation error dynamics, sufficient conditions formed in inequality are given to satisfy the two above requirements. Finally, a numerical example is provided to show the effectiveness of the proposed method.