## Mathematical Modeling, Analysis, and Control of Hybrid Dynamical Systems

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Heli Hu, Dan Zhao, Qingling Zhang, "Observer-Based Decentralized Control for Uncertain Interconnected Systems of Neutral Type", *Mathematical Problems in Engineering*, vol. 2013, Article ID 134935, 12 pages, 2013. https://doi.org/10.1155/2013/134935

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

**Academic Editor:**Rongni Yang

#### Abstract

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.

#### 1. 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 [1–11] 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 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-frequency-gain 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., [16–23]). In [16–18], 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 time-varying 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 time-delay 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 [24–28]. 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 output-feedback controller. Utilizing an approximation for time-varying delay state, a new comparison model is proposed. Furthermore, the 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.

#### 2. Problem Formulation

Consider the following uncertain neutral interconnected systems composed of subsystems: where , , and are the state, control input, and measurement output of the th subsystem, respectively. , , , , , , and are known constant matrices of appropriate dimensions. is the initial condition. , , and are the time-varying delays. Assume that there exist constants , , , , , , and satisfying

Time-varying parametric uncertainties , , , and are assumed to satisfy where matrices , , , , 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 is a diagonal matrix with positive elements in a decreasing order, is a zero matrix, and and are unitary matrices.

*Assumption 2. *The matrix and .

Consider the following observer-based decentralized control for system (1): where and are the state and output vectors of the observer and and 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 is defined as the estimated error of system, , and the uncertainty 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.

Lemma 4 (see [29]). *For a given with rank, assume that is a symmetric matrix; then there exists a matrix such that if and only if**
where and .*

#### 3. Main Result

Theorem 5. *Consider uncertain neutral interconnected systems (1) with (2), (3), and (6). If there exists a solution , , , , , , , , and such that the following inequalities hold:
**
then system (1) is robustly stabilizable by the observer-based control (6) with and , where*

*Proof. *According to Schur complement, inequality (11) is equivalent to the following inequality:

where

By Lemma 4, the conditions and imply the condition . In view of , , , and , the matrix inequality (13) is equivalent to

where

Pre- and postmultiplying matrix in (15) by and , where
we obtain
where

Construct the following Lyapunov functional candidate:

For the system (1), the following structural identity holds:

By some simple derivations, the time derivative of along the trajectories of (7) satisfies the following inequality:
where

According to conditions (18) and (22), one can obtain that, for all ,

In addition, Assumption 2 guarantees system (1) is Lipschitzian in the term 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 , , , and among inequalities. When , the corresponding equality in (11) is an LMI. Under the solvable condition of , we can apply Schur complement formula to the second inequality 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 with feasp in [30].

#### 4. 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 the simulation results are shown in Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. With the observer-based control applied, the state responses of the closed-loop system are depicted in Figures 1–2. The control signals based on the observer are rather smooth in Figures 3 and 4. The output signals of the closed-loop system are shown in Figures 5 and 6. The state responses and the output signals of the observer system (6) are presented in Figures 7–10, respectively. The state responds of the error dynamic system (7) are shown in Figures 11 and 12.

The simulations results indicate that the designed observer can stabilize the error dynamic system and estimate the states of the interconnected systems of neutral type.

*Remark 7. *For this example, . When , we can obtain , , , , , and by solving LMI in (11). Furthermore, according to and , we have and . When , inequality in (11) is not LMI because of the existence of the interconnection matrix and (here, and are constant matrices coming from ). However, the transformed inequality is an LMI after applying Schur complement to . Hence, the controller gain and observer gain can be obtained by finding feasible set to the transformed inequality of .

#### 5. 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 sizes of delays, but also the information of derivatives. The numerical example and the corresponding simulation results elucidate that the results obtained in this paper are effective.

#### Acknowledgments

This research is supported by the Natural Science Foundation of China (no. 61104106), the Science Foundation of Department of Education of Liaoning Province (no. L2012422), and the Doctorial Science Foundation of Shenyang University (no. 20212340).

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#### Copyright

Copyright © 2013 Heli Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.