Robust Nonfragile Controllers Design for Fractional Order Large-Scale Uncertain Systems with a Commensurate Order 1 < α < 2

The paper concerns the problem of stabilization of large-scale fractional order uncertain systems with a commensurate order 1 < α < 2 under controller gain uncertainties. The uncertainties are of norm-bounded type. Based on the stability criterion of fractional order system, sufficient conditions on the decentralized stabilization of fractional order large-scale uncertain systems in both cases of additive and multiplicative gain perturbations are established by using the complex Lyapunov inequality. Moreover, the decentralized nonfragile controllers are designed. Finally, some numerical examples are given to validate the proposedmethod.


Introduction
In the past decades, a great deal of attention has been paid to the stability and stabilization of large-scale systems [1][2][3][4][5][6][7].This is due to the fact that there exist a large number of large-scale interconnected dynamical systems in many practical physical systems, such as process control systems, computer communication networks, transportation systems, and economic systems.Meanwhile, nonfragile controllers have been nominated by resilient and the fragility of the PID controllers has been analyzed in [8].The controller gain perturbations can commonly be modeled as uncertain gains which are dependent on uncertain parameters in the literature [9,10].The robust nonfragile control problem for uncertain integer order large-scale system has been studied [11][12][13].In recent years, the nonfragile control problem has been an attractive topic in theory analysis and practical implement, because of perturbations often appearing in the controller gain, which may result in either the actuator degradations or the requirements for readjustment of controller gains.The problem of reliable dissipative control within nonfragile control framework has been investigated in [14,15].The nonfragile control idea is how to design a feedback control that will be insensitive to perturbations in gains of feedback control.The robust resilient stabilization problem is to design a nonfragile state feedback controller such that the uncertain fractional order large-scale interconnected closedloop system with a commensurate order 1 <  < 2 is robustly stable for all admissible parameter uncertainties.
On the other hand, pioneering works in stability analysis and stabilization of fractional order control systems can be found in [16][17][18][19][20].The robust stability of fractional order interval systems has been investigated in [21,22].It is well known that Matignon's stability theorem [16] is the basis for stability analysis of the fractional order system by checking the location of eigenvalues in the complex plane.Matignon's theorem is in fact the pioneering works of stability analysis of the fractional order system.Based on Matignon's theorem, the stability criteria of fractional order systems have been proposed in both cases of 1 <  < 2 and 0 <  < 1 in [23,24].The necessary and sufficient LMI conditions for stability analysis of a commensurate fractional order system have been established in [23,24], in which complex Lyapunov inequality holds.However, very few studies provide LMI conditions for the stability analysis of the fractional order large-scale interconnected system in the literature.Our study is mainly motivated by the works [23,24].The important feature is that the proposed method can be implemented to 2 Mathematical Problems in Engineering the fractional order large-scale interconnected system.The objective of the paper is to design a nonfragile controller which is robust to system uncertainties and resilient to controller gain variations for the fractional order large-scale interconnected systems with a commensurate order 1 <  < 2. Here, it should be also pointed out that [25] only focuses on the case of a fractional order 0 <  < 1.This paper is organized as follows.Some preliminaries and the problem statement are given in Section 2. The main results of the sufficient condition of stabilization of the fractional order system under additive gain perturbations are presented in Section 3. Furthermore, the decentralized stabilization state feedback controller are designed.Meanwhile, the LMI results of the sufficient condition of stabilization of the fractional order system under multiplicative gain perturbations are presented in Section 4. The examples are given in Section 5 to illustrate the effectiveness of our LMI-based results for checking the stabilization of the fractional order large-scale interconnected system.Finally, a brief conclusion is drawn in Section 6.
Notations.Throughout the paper, we denote by  the conjugate of the complex number .  denotes the imaginary unit. denotes the identity matrix with appropriate dimensions.block diag denotes the block diagonal matrix.  denotes the -dimensional Euclidean space and  × is the set of all  ×  real matrices.  denotes the transpose of  and  * denotes the Hermitian transpose of .Re() and Im() are corresponding to the real and imaginary parts of the matrix, respectively.

Preliminaries and Problem Formulation
Let us consider a fractional order large-scale interconnected uncertain system with a commensurate order 1 <  < 2 composed of  fractional order subsystems: where  ∈  is the fractional commensurate order, ,  = such that the resulting fractional order closed-loop system is asymptotically stable, where   ∈ with the composite matrices  and  having the structure Definition 1 (see [26]).For all nonzero real vectors  ∈   ,  ∈  × is real matrix; if the inequality    < 0 holds, then  is said to be negative definite matrix.
is asymptotically stable.

Nonfragile Controller Design of Fractional Order Large-Scale System with Additive Gain Perturbations
In this section, the resilient controller synthesis problem is formulated for the fractional order large-scale interconnected system under additive gain perturbations.Sufficient conditions are firstly derived for the decentralized stabilization of fractional order large-scale interconnected system with norm-bounded uncertainties given by (1).Before proceeding further, we will state the following well-known lemmas.We will use the lemmas and theorems to establish sufficient conditions on decentralized stabilization of fractional order large-scale interconnected system with norm-bounded uncertainties under additive gain perturbations.
Lemma 4 (see [27]).For any matrices  and  with appropriate dimensions and for any  > 0, the following inequality holds: Lemma 5 (see [28]).The fractional order system   ()/  = () with a commensurate order  is asymptotically stable if | arg(spec())| > (/2), where  is the order of fractional order system and spec() is the spectrum of all eigenvalues of .
Proof.The idea is mainly based on the geometric analysis of a fractional system stability domain.Based on Lemma 5, the stability domain for a fractional order 1 <  < 2 is convex.By using Linear Matrix Inequalities (LMI) approach, it is obtained as the above LMI.Therefore, it is equivalent to  +    < 0.

Lemma 7.
A complex Hermitian matrix  satisfies  < 0 if and only if the following real LMI inequality holds: Under commensurate order hypothesis, our finding is summarized in the following theorem.
Theorem 8. Consider the fractional order large-scale interconnected system (1) with a commensurate order where , and  1 and  2 are the real part and imaginary part of matrices   , respectively.Moreover, the stabilization decentralized state-feedback gain matrix can be calculated as follows:   =     .
Proof.Under decentralized state-feedback control law (2), the closed-loop fractional order large-scale interconnected system is obtained as where , = 1, 2, . . ..Based on Lemma 6, the necessary and sufficient condition on the asymptotical stability of the fractional order system with order 1 <  < 2 is that  +    < 0. According to Definition 1, the sufficient condition on the stabilization of fractional order largescale interconnected system satisfies the following quadratic matrix inequality: Consequently, the sufficient condition on the decentralized stabilization of fractional order large-scale interconnected system is that quadratic matrix inequality (11) holds.
Based on Lemmas 3 and 4, by means of enlarging the inequality, it yields Meanwhile, based on Lemmas 3 and 4, by means of enlarging the inequality we have Substituting ( 11), (12), and ( 13) into ( 12) results in the following quadratic matrix inequality: Let   =  −1  and   =     .By premultiplying and postmultiplying  −1  onto ( 16), one has If the quadratic matrix inequality holds, then the fractional order large-scale interconnected system is asymptotically stable.
By applying Schur complement, the above matrix inequality is equivalent to the following complex LMI: where   =     + Based on Lemma 7, the complex LMI ( 18) is transformed into the real LMI.Consider where  1 = Re(  ) and  2 = Im(  ).This completes the proof.
Therefore, the sufficient condition for decentralized robust stabilization of fractional order large-scale interconnected system with norm-bounded uncertainties under additive gain perturbations is derived.Furthermore, this condition is transformed into the solvability problem of linear matrix inequalities.In summary, by solving the LMI (18), we derive the sufficient condition on stabilizability via decentralized state feedback of the fractional order uncertain system with order 1 <  < 2.

Nonfragile Controller Design of Fractional Order Large-Scale System with Multiplicative Gain Perturbations
In this section, the nonfragile controller design problem is formulated for the fractional order large-scale interconnected system under multiplicative gain perturbations.Sufficient conditions are established for the decentralized stabilization of fractional order large-scale interconnected system with norm-bounded uncertainties under multiplicative gain perturbations.We are in a position to present our main result. where , and  1 and  2 are the real part and imaginary part of matrices   , respectively.Moreover, the stabilization decentralized state-feedback gain matrix is given by   =     .
Proof.By the means of decentralized state-feedback control law (2), the closed-loop fractional order large-scale interconnected system is obtained as where ,  = 1, 2, . . ., Δ  =         .Based on Lemma 6, the necessary and sufficient condition on the asymptotical stability of the fractional order system with order 1 <  < 2 is that  +    < 0.
According to Definition 1, the sufficient condition on the stabilization of fractional order large-scale interconnected system satisfies the following quadratic matrix inequality: Consequently, the sufficient condition on the decentralized stabilization of fractional order large-scale interconnected system is that quadratic matrix inequality (23) holds.
Based on Lemmas 3 and 4, by means of enlarging the inequality we have Likewise, it is obtained that Substituting ( 24), (25), and ( 26) into (24) results in the following quadratic matrix inequality, and we have Let   =  −1  and   =     .By premultiplying and postmultiplying  −1  onto (27), one has If the quadratic matrix inequality holds, then the fractional order large-scale interconnected system is asymptotically stable.By applying Schur complement, the above matrix inequality is equivalent to the following complex LMI: where   =     + where  1 = Re(  ),  2 = Im(  ).This completes the proof.
Therefore, the sufficient condition for decentralized robust stabilization of fractional order large-scale interconnected system with norm-bounded uncertainties under multiplicative gain perturbations is obtained.Furthermore, this condition is transformed into the solvability problem of linear matrix inequalities.In summary, by solving the LMI (27), we derive the sufficient conditions on stabilizability via decentralized state feedback of the uncertain fractional order system under multiplicative gain perturbations.

Numerical Examples
In this section, to verify and demonstrate the effectiveness of the proposed method, two numerical examples are investigated.The fractional order large-scale interconnected uncertain system under controller gain perturbations is stabilized by the decentralized state feedback controllers.The Adams-type predictor-corrector method [29] Meanwhile, the following additive gain perturbations are considered: The time responses of system are shown in Figure 1.It is observed that its four states all converge to zero.It can be concluded that fractional order large-scale interconnected system with additive gain perturbations can be stabilized by the nonfragile controller.(40) The time responses of system are shown in Figure 2. It is shown that its four states all converge to zero.It can be concluded that fractional order large-scale interconnected system with multiplicative gain perturbations can be stabilized by the nonfragile controller.

Conclusions
In this paper, sufficient conditions have been derived on the stabilization of fractional order large-scale interconnected uncertain system with a commensurate order 1 <  < 2 under two kinds of controller gain perturbations, that is, additive and multiplicative gain perturbations.The proposed method is based on the stability criterion of fractional order system by using the complex Lyapunov inequality.Moreover, the nonfragile controllers are designed.Simulation results have demonstrated the effectiveness of the proposed method.

Figure 2 :
Figure 2: Example 2: time responses of the closed-loop fractional order large-scale interconnected uncertain system.
1, 2, . . ., and   () ∈    and   () ∈    are the state and input of the th fractional order subsystem, respectively.Assume that the nominal systems   ∈    ×  ,   ∈    ×  , and   ∈    ×  are constant and of appropriate dimensions and the pair (  ,  ) is controllable.
Δ  =         represent the additive and multiplicative gain perturbations, respectively.In this note, the uncertainty is bounded as follows.The parameter uncertainties considered here are norm-bounded and are of the forms Δ  =       ,      ≤ ; Δ  =       ,      ≤ ; Δ  =       ,      ≤ ; Δ  =         ,      ≤ , where the elements are Lebesgue measurable and   ,   ,   ,   ,   ,   ,   , and   are known real matrices of appropriate dimensions which characterize the structure of the uncertainty.The overall system is described by the composite fractional order large-scale state equations ×  is the state feedback gain matrix to be designed and Δ  =       and +     +      +       =        +             +     +       .In practice, the feedback matrix   has no imaginary part.So let Im(  ) = 0; then   = Re(  ).According to the relationship   =     , the output matrix   has no imaginary part; that is, Im(  ) = 0; then   = Re(  ).Substituting  = cos  +  sin  into   gives   = cos     + cos      + cos     + cos      +        +        +             + cos     + cos       +  (sin     − sin      + sin     − sin + sin     − sin       ) .
is used for the numerical solution of fractional differential equations during the simulation.