Research Article | Open Access

Heli Hu, Dan Zhao, Qingling Zhang, "Singular Value Decomposition-Based Method for Sliding Mode Control and Optimization of Nonlinear Neutral Systems", *Journal of Applied Mathematics*, vol. 2013, Article ID 357531, 11 pages, 2013. https://doi.org/10.1155/2013/357531

# Singular Value Decomposition-Based Method for Sliding Mode Control and Optimization of Nonlinear Neutral Systems

**Academic Editor:**Hak-Keung Lam

#### Abstract

The sliding mode control and optimization are investigated for a class of nonlinear neutral systems with the unmatched nonlinear term. In the framework of Lyapunov stability theory, the existence conditions for the designed sliding surface and the stability bound are derived via twice transformations. The further results are to develop an efficient sliding mode control law with tuned parameters to attract the state trajectories onto the sliding surface in finite time and remain there for all the subsequent time. Finally, some comparisons are made to show the advantages of our proposed method.

#### 1. Introduction

Time delays often arise in the processing state, input, or related variables of dynamic systems. Inparticular, when the state derivative also contains time-delay, the considered systems are called as neutral systems [1]. The outstanding characteristic of neutral systems is the fact that such systems contain the same highest order derivatives for the state vector , at both time and past time(s) . Many engineering systems can be represented as neutral equation [2–8], such as population ecology [9], distributed networks containing lossless transmission lines [10], heat exchangers, and robots in contact with rigid environments [11]. The delay-dependent stability criteria for neutral stochastic systems with time delay in state are studied in [4, 5]. The difference is that the former is about exponential stability; the other is on robust stochastic stability, stabilization, and control. Furthermore, a robust reduced-order filter and a memory state feedback control are developed to establish the improved stability criteria for neutral systems in [6, 7], respectively. In [12], a periodic output feedback is studied in the context of infinite-dimensional linear systems modeled by neutral functional differential equations, and the main work only focuses on stabilization of neutral systems with delayed control. The stability and performance analysis, the reliable stabilization, and the finite-time control for uncertain switched neutral systems are investigated in [13–15], respectively.

On the other hand, as an important robust control approach, sliding mode control strategy has many advantages such as fast response, insensitiveness to parametric uncertainties and external disturbances, and unnecessariness for online identification. Hence, the sliding mode control for dynamic systems has received attention extensively [16–24]. Xia et al. propose a novel approach combining SMC and ESO and utilize the backstepping technique to control the attitude of a nonlinear missile system in [16]. In [17], a stable integral type fractional-order sliding surface is introduced to stabilize and synchronize a class of fractional-order chaotic systems. In [18, 19], Xia et al. investigate the robust sliding control for uncertain continuous systems and discrete-time systems with constant delays, respectively. In [20, 21], the PSMC approach and the PDSMO approach are utilized, respectively, to deal with the fault-tolerant control question for uncertain systems. It is worth pointing out that the nonlinear terms are subject to the matched condition in [20]. The same requirement also appears in [25, 26] for neutral systems. In addition, the matched condition is required for the parametric uncertainty in the input in [27].

To the best of the authors’ knowledge, the sliding control and optimization for uncertain neutral systems have not yet been investigated, which motivates the present study. One contribution of this paper is the optimization for the upper bound of the unmatched nonlinear term. The other contribution lies in the introduction of the scalars and to prevent the unacceptably high gains. In this paper, the state transformations based on singular value decomposition and the descriptor system model are utilized to obtain the existence conditions for the designed sliding surface and the upper bound . Inparticular, the achieved bound secures the quadratic stability of the sliding motion with all satisfying . In addition, an efficient control law is designed to attract the state trajectories onto the sliding surface in finite time and remain there for all the subsequent time. In order to reduce the chattering on the sliding surface, the tuned parameters are introduced into the developed control law. One example under three cases is given to illustrate our proposed method and make some comparisons. Compared with [27], our results have the advantages of the more rapid convergent rate and the less chattering phenomena.

The sliding mode control problem formulation is described in Section 2. In Section 3, the sliding surface design and the reaching motion control design are developed. An example under three cases and some compared results are provided in Section 4. Finally, conclusion is given in Section 5.

#### 2. Problem Formulation

Consider the following class of nonlinear neutral systems: where is the state of the system and is the input vector. , , , and are known constant matrices, and represents a piecewise-continuous nonlinear function satisfying . It is assumed that the nonlinear term can be bounded by a quadratic inequality where is a constant matrix and is a scalar parameter. 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 be of the following form: where , , , and are constant matrices of appropriate dimensions and and are the unknown matrix function satisfying and for all .

Without loss of the generality, assume that . One can easily get the singular value decomposition of where is a diagonal positive definite matrix; , , and are unitary matrices.

The following state transformation is similar to [18, 20, 28]. Choose , where , another form of the system (1) can be obtained where , , , , , , , and .

Furthermore, the system (6) can be rewritten as where , , , , ,, and and the others can be obtained easily according to the same way.

On account of the second equation of (7) that represents the sliding motion dynamics of (6), one can construct the following sliding surface: where is the gain to be designed.

Based on (8) and the second equation of (7), one can get the following sliding motion:

In view of (2), (8), and in (9), one can obtain that where .

Our purpose is to design a sliding surface and a reaching motion control law such that the following holds:(i)sliding motion (9) is quadratically stable with upper bound ; that is, for any given , find out the upper bound so that for all satisfying , sliding motion (9) is quadratically stable;(ii)the system (7) is asymptotically stable under the reaching control law .

Lemma 1 (see [4]). *For any constant matrix , inequality
**
holds for any pair of symmetric positive definite matrix and positive number .*

Lemma 2 (see [29]). *Let Z, X, S, and Y be matrices of appropriate dimensions. Assume that is symmetric and , then if and only if there exists a scalar satisfying
*

#### 3. Main Result

##### 3.1. Sliding Surface Design for Nonlinear Neutral Systems

Theorem 3. *The sliding motion (9) is quadratically stable with upper bound if the following optimization problem
**
subject to
**
has a solution set (positive scalars , and , positive definite matrices , and , and matrix ). Moreover, the gain of the sliding surface (8) and the upper bound are obtained as follows:
**
where is a given scalar parameter in (2) and , and .*

*Proof. *First, transform the sliding motion (9) to the equivalent descriptor system form as follows:
where

Choose , , , and , and construct the following Lyapunov functional

Obviously, for all . Applying (3), (4), Lemma 1, and the decomposition techniques of matrix to the derivative of along the trajectories of the sliding motion (20), yield
where
.

Pre- and postmultiplying in (23) by and , where , we obtain the following matrix:
where , and .

By Schur complement lemma and Lemma 2, the condition in (14) is equivalent to in (25). Obviously, means . From the condition in (23), there exists a constant , such that

According to Schur complement lemma, LMI (15) is equivalent to the following inequality:
that is,

Thus, one can obtain that

This implies that the sliding motion (9) is Lipschitzian in the term with Lipschitz constant less than 1 [29].

By conditions (22) and (26) and , one can conclude that the sliding motion (9) with (3) and (4) is quadratically stable.

Furthermore, for some desired value satisfying (2), the design of gain can be formulated as an LMI problem in , , , , , , , , , and in (13). Also, one can obtain the upper bound which guarantees the quadratic stability for the sliding motion (9) with all satisfying in (2) if the LMI optimization is feasible. This completes the proof.

*Remark 4. *It should be pointed out that the norm of the gain matrix is implicitly bounded by (17) and (18), which imply that . This is necessary in order to prevent the unacceptably high gains.

*Remark 5. *It is obvious that (14), (15), (16), (17), and (18) are a group of LMIs with respect to solution variables; various efficient convex algorithms can be used to ascertain the LMI solutions. In this paper, we utilize Matlab’s LMI Toolbox [30] to solve the convex optimization problem to obtain directly the gain as .

*Remark 6. *If the convex optimization problem is solvable, the bound guarantees the quadratic stability for the sliding motion (9) with all satisfying in (2).

##### 3.2. Reaching Motion Control Design for Nonlinear Neutral Systems

Theorem 7. *Suppose that the optimization problem (13) has solutions: positive scalars , , , , and , positive definite matrices , and , and matrix , and the sliding surface is given by (8). Then, the trajectory of the closed-loop system (7) can be driven onto the sliding surface in finite time with the control
**
where
**, , and are positive constants.*

*Proof. *From the sliding surface (8), one can compute the derivative of as follows:
Replacing in (32) with in (30) yields that
That is, for each one in , we have

In view of
thus one can conclude that
which means that the trajectory of the system (7) with the control law (30) can be driven onto the sliding surface in finite time and remain there in the subsequent time. This completes the proof.

#### 4. Numerical Example

In this section, three examples are presented to illustrate the design approach of sliding surface and reaching motion control and show their advantages.

Consider parts of parameters in [27]

##### 4.1. Case 1

For the system (1) in our paper, when and the desired parameter , by solving the optimization problem (13) subject to (14), (15), (16), (17), and (18), we can get following results: Thus, the gain and the upper bound are computed as

Under the following initial condition when the tuned parameters and in (30) are chosen as the simulation results are shown in Figures 1–6 based on the above parameters. From Figures 1 and 2, one can see that the nonlinear neutral systems (1) and (6) without control are divergent. From Figures 3 and 4, one can see that the nonlinear neutral system (6) with control (30) is indeed well stabilized. The control signal of the system (6) and the sliding surface (8) are rather smooth in Figures 5 and 6.

##### 4.2. Case 2

In order to compare with [27], we need guarantee no difference for the considered system firstly. Because in [27] the system (1) contains the term , where , and does not have the term , and also the delays are constant, for the system (1) in our paper, matrix is needed between and , where . When is contained in our system (1), the theoretical results should make some modification; that is, the reaching motion control should have the following form: where denotes the pseudoinverse of the argument .

In addition, to remove the nonlinear term , we chose , which means that the parameter in (2) does not need to be optimized.

As for the constant delays, their derivatives, and , are equal to zero.

Utilizing the parameters in [27] to solve LMIs (14) and (15), one can get following results: Thus, the gain is computed as

Under the initial in [27], when the tuned parameters and in (30) are chosen as the simulation results are presented in Figures 7, 8, 9, and 10. Seen from Figures 7–10 in our paper, the convergent time is about 10 seconds. However, the convergent time is about 16 seconds from Figures 2–4 in [27]. Compared with Figures 2–4 in [27], one can conclude that the convergent rate of our results is faster.

##### 4.3. Case 3

Under the initial in [27], when the tuned parameters and in (30) are chosen as and other parameters are similar to [27], the simulation results are presented in Figures 11, 12, 13, and 14.

Seen from Figures 11–14 in our paper, the convergent time is about 20 seconds. However, the convergent time is about 30 seconds from Figures 5–7 in [27]. Also, the amplitudes of the corresponding control signal are 0.45 and 3.5, respectively. Therefore, one can conclude that the convergent rate of our results is faster and the amplitude of control signal of our results is smaller.

#### 5. Conclusion

The sliding mode control and optimization problem of nonlinear neutral systems with time-varying delays are complex and challenging. In the framework of the Lyapunov stability theory, based on the methods of singular value decomposition and descriptor system model transformation, the sliding surface and the reaching control law are designed, which can be obtained by solving the optimization problem. Also, the upper bound that guarantees the quadratic stability of sliding motion (9) for all satisfying is derived. The numerical example has shown the validity of the present design and the advantages of the schemes over the existing results in the literature.

In future research, it is expected to investigate the optimization of reaching motion control for a class of nonlinear neutral systems with time-varying delays.

#### Acknowledgments

This research is supported by Natural Science Foundation of China (no. 61104106), Science Foundation of Department of Education of Liaoning Province (no. L2012422), and Start-up Fund for Doctors 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.