Observer-Based Decentralized Control for Uncertain Interconnected Systems of Neutral Type

The observer-based decentralized control problem is investigated for a class of uncertain interconnected systems of neutral type. Using the singular value decomposition approach, a full-order observer is designed to guarantee the asymptotic stability of the error dynamic system. A novel mathematical technique is developed to solve this design problem. Sufficient condition for uncertain interconnected systems of neutral type to be asymptotic stable is established based on the singular value decomposition method. Furthermore, the desired gains of observer and controller are obtained by the explicit expressions in terms of some free parameters. Finally, an illustrative example is used to demonstrate the proposed approach, and the corresponding simulation results are given to elucidate the effectiveness.


Introduction
Nowadays the systems have become more and more large, and for the interconnected systems decentralized control has obvious advantage that overcomes the limitations of the traditional centralized control requiring sufficiently large communication bandwidth to exchange information between the subsystems.Therefore, the decentralized control scheme [1][2][3][4][5][6][7][8][9][10][11] only making use of local information available is very popular among the researchers and the engineers.In [1], an algorithm formulated within the convex optimization framework is proposed to investigate the strict dissipativity of the linear interconnected systems.A decentralized structure of dissipative state-feedback controllers is designed.In [2,3], a decentralized adaptive output-feedback stabilizer and a decentralized  1 adaptive controller are proposed to stabilize a class of large-scale nonlinear systems, respectively.Different from the constant delays involved in the considered system without a priori knowledge of subsystem high-frequencygain signs in [2], the interconnected nonlinearities and unmodeled dynamics are included in the considered systems in [3].Based on the concepts of dynamic graphs and dynamic adjacency matrix, a modeling method of a complex dynamic interconnected system is considered in [4].In [5], a decentralized dynamic output feedback based on linear controller is proposed to robust stabilize a class of nonlinear interconnected systems coupled by nonlinear interconnections that are unknown and quadratically bounded.However, neither constant delay nor time-varying delays are considered in [4,5].
On the other hand, the neutral system is the general form of delay system that contains the same highest order derivatives for the state vector (), at both time  and past time(s)   ≤ .Many models of practical systems can be described by functional differential equation of neutral type [12].Physical examples for neutral system have distributed networks [13], population ecology [14], heat exchangers, robots in contact with rigid environments [15], and so forth.In recent years, the stability analysis and robust control problems of neutral delay systems have been considered extensively (see, e.g., [16][17][18][19][20][21][22][23]).In [16][17][18], the stability problems of neutral systems are investigated.The difference among them is that Balasubramaniam et al. [16] focus on the stability of the neutral systems with both constant and timevarying delays using the method of nonuniformly dividing the whole delay interval into multiple segments, Rakkiyappan et al. [17] study the stability of the neutral systems with interval time-varying delays and nonlinear perturbations using the method of a new Lyapunov functional approach, and Nian et al. [18] deal with the stability of neutral systems with only a constant delay using the method of state matrix decomposition.However, the design problems of the control law have not been considered in them.
Furthermore, Ma et al. [19] develop a control method for neutral systems with a single input and some restrictions on the system matrices using a differential-difference inequality and the transformation technique.Han et al. [20] utilize a discretized Lyapunov-Krasovskii functional approach to investigate the stability of the linear neutral systems with small and nonsmall discrete delays.But its discrete application to control design yields nonlinear conditions, which may not be easily computable.In [21], the robust adaptive stabilization problem is investigated for neutral timedelay systems with uncertainties, and an adaptive scheme is introduced to estimate the bounds on uncertainties.But the matched condition is required for the disturbance vector of the considered system.It is worth pointing out that many researches on the neutral systems are often restricted to the stability analysis without controller or static state feedback control schemes.
Nowadays, observer control is also an attractive topic [24][25][26][27][28].In [24], two controllers based on state feedback and observer output feedback are designed for networked systems with discrete and distributed delays subject to quantization and packet dropout.Also, a compensation scheme is proposed to deal with the effect of random packet dropout through communication network.In [25], Su et al. introduce a new model transformation for considered discrete-time T-S fuzzy systems to realize the design of dynamic outputfeedback controller.Utilizing an approximation for timevarying delay state, a new comparison model is proposed.Furthermore, the  2 - ∞ filtering problem for a class of discrete-time T-S fuzzy systems with time-varying delays is studied in [26].The anticipated full-and reduced-order filter design is cast into a convex optimization problem, which can be efficiently solved by standard numerical algorithms.In [27], an improved fuzzy observer design that has some advantages, such as the less conservatism and the satisfactory multiple performance required by the fault detection, is presented.Moreover, Messaoud et al. [28] propose a new design of functional unknown input observer for nonlinear systems.In particular, necessary and sufficient conditions for the existence of the anticipated observer are presented.
To the best of the authors' knowledge, the observer-based decentralized control problems of uncertain interconnected systems with neutral type have not yet been investigated, which motivates the present study.In this paper, the asymptotic stabilization of a class of uncertain neutral interconnected systems with time-varying delays in state, control input, and interconnections is made.In framework of linear matrix equalities, the design of the observer and controller is formulated.Sufficiently the influence of interconnections on the system performance is taken into account, and interconnections are dealt with by flexible techniques.These strategies allow one to obtain less conservative stabilization conditions.A numerical example and the corresponding simulation results are given to illustrate the effectiveness of the proposed method of decentralized controller based on observer.
The remainder of the paper is organized as follows.The observer-based decentralized control problem formulation is described in Section 2. In Section 3, the desired gains of observer and controller are obtained by the explicit expressions in terms of some free parameters.A numerical example and the corresponding simulation results are presented in Section 4. The conclusion is provided in Section 5.

Problem Formulation
where Time-varying parametric uncertainties Δ  (), Δ   (), Δ  (), and Δ  () are assumed to satisfy where matrices   ,  1 ,    ,  2 , and   are constant matrices of appropriate dimensions and   () is the unknown matrix function satisfying The following assumptions are made on the considered system (1).
Assumption 1. Suppose that the matrix   has full row rank (i.e., rank(  ) =   ).Then the singular value decomposition of   presents as follows: where   ∈ R   ×  is a diagonal matrix with positive elements in a decreasing order, 0 ∈ R   ×(  −  ) is a zero matrix, and   ∈ R   ×  and   ∈ R   ×  are unitary matrices.
Consider the following observer-based decentralized control for system (1): where x () ∈ R   and ŷ () ∈ R   are the state and output vectors of the observer and   ∈ R   ×  and   ∈ R   ×  are the controller gain and observer gain to be designed.System (1) with the observer-based control (6) can be rewritten as where the signal   () =   () − x () is defined as the estimated error of system, and the uncertainty Δ  () satisfies the following quadratic inequality: where Definition 3. Consider system (1) with the observer-based control (6).System (1) is said to be robustly stabilizable by the observer-based control (6), if the closed-loop system (7) satisfying Assumptions 1 and 2 is asymptotically stable.
In addition, Assumption 2 guarantees system (1) is Lipschitzian in the term ẋ  ( −   ()) with Lipschitz constant less than 1 [12].Therefore, by Definition 3 with conditions ( 20) and (24), system ( 7) is asymptotically stable and system (1) is robustly stabilizable by observer-based control (6).This completes the proof.Remark 6.One of the distinctive features of this paper is that the  inequalities are included in (11), and only one is LMI due to the existence of the interconnected matrix variables   =  1    1 ,   ,   =  2    2 , and   among  inequalities.When  = 1, the corresponding equality Ω 1 < 0 in ( 11) is an LMI.Under the solvable condition of Ω 1 < 0, we can apply Schur complement formula to the second inequality Ω 2 < 0 in order to obtain the solvable LMI.The process is repeated until the last inequality.Thus, the controller gain   and observer gain   can be obtained by finding feasible set to Ω  < 0 with feasp in [30].

Illustrative Example
Consider system (1) composed of two three-order subsystems with the following parameters: Applying Matlab toolbox to solving inequality (11), we obtain the following results: Under the following initial condition  from Ω 1 < 0).However, the transformed inequality is an LMI after applying Schur complement to Ω 2 < 0. Hence, the controller gain  2 and observer gain  2 can be obtained by finding feasible set to the transformed inequality of Ω 2 < 0.

Conclusion
The observer-based decentralized control problem of uncertain interconnected systems of neutral type is complex and challenging.In framework of Lyapunov stability theory, a novel mathematical technique to deal with the parametric disturbances is developed to obtain the sufficient conditions of existing anticipated controller and observer.The sufficient conditions are the coupled LMIs and depend on not only the  the Doctorial Science Foundation of Shenyang University (no.20212340).

Figure 5 :Figure 6 :
Figure 5: Output responses of the first subsystem.
Figure 3: The control signals of the first subsystem.For this example,  = 2.When  = 1, we can obtain  11 ,  12 ,  21 ,  12 ,  21 , and  12 by solving LMI Ω 1 < 0 in (11).Furthermore, according to  21 =  11  21  11 and  21 =  12  21  12 , we have  21 and  21 .When  = 2, inequality Ω 2 < 0 in (11) is not LMI because of the existence of the interconnection matrix  12 =  21  12  21 and  12 =  22  12  22 (here,  12 and  12 are constant matrices coming (6) state responses of the closed-loop system are depicted in Figures1-2.The control signals based on the observer are rather smooth in Figures3 and 4. The output signals of the closed-loop system are shown in Figures5 and 6.The state responses and the output signals of the observer system(6)are presented in Figures7-10, respectively.The state responds State of the observer x11 (t) State of the observer x12 (t) State of the observer x13 (t)