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

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 α∗ 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.


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][3][4][5][6][7][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 finitetime  ∞ control for uncertain switched neutral systems are investigated in [13][14][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][17][18][19][20][21][22][23][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   2 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.

Problem Formulation
Consider the following class of nonlinear neutral systems: where () ∈ R  is the state of the system and () ∈ R  is the input vector. ∈ R × ,   ∈ R × ,   ∈ R × , and  ∈ R × are known constant matrices, and ℎ : R  → R  represents a piecewise-continuous nonlinear function satisfying ℎ(0) = 0.It is assumed that the nonlinear term ℎ() can be bounded by a quadratic inequality where  is a constant matrix and  > 0 is a scalar parameter.() is the initial condition.() and () are the time-varying delays.Assume that there exist constants  0 ,  0 ,  0 , , and  satisfying Time-varying parametric uncertainties Δ() and Δ  () are assumed to be of the following form: where , ,   , and   are constant matrices of appropriate dimensions and () and   () are the unknown matrix function satisfying  T ()() ≤  and  T  ()  () ≤  for all  ≥ 0.
Without loss of the generality, assume that rank() = .One can easily get the singular value decomposition of where Σ ∈ R × is a diagonal positive definite matrix; ) , and  ∈ R × are unitary matrices.
The following state transformation is similar to [18,20,28].Choose  = Γ, where Γ = [ ], another form of the system (1) can be obtained where and () = Γ().Furthermore, the system (6) can be rewritten as where ) 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  ∈ R ×(−) is the gain to be designed.
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  ∈ R × , inequality holds for any pair of symmetric positive definite matrix  ∈ R × and positive number  > 0.
Lemma 2 (see [29]).Let Z, X, S, and Y be matrices of appropriate dimensions.Assume that  is symmetric and  T  ≤ , then  +  +  T  T  T < 0 if and only if there exists a scalar  > 0 satisfying

Main Result
has a solution set (positive scalars , where  is a given scalar parameter in (2) and and Ω 99 = −.
Furthermore, for some desired value  satisfying (2), the design of gain  can be formulated as an LMI problem in ,  1 ,  2 ,   2 ,   ,  1 ,  2 ,  3 ,  4 , 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 ‖  ‖ ≤ √    2 .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  =  −1 2 .
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).
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.

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.
and the desired parameter  = 0.1, by solving the optimization problem (13) subject to ( 14), ( 15), ( 16), (17), and (18), we can get following results:  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.

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  + Δ(), where Δ() = (), and does not have the term ℎ(), 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 ℎ(), we chose ℎ() = [0 0 0]  , 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) Thus, the gain  is computed as Under the initial (0) = [1 0 − 1] in [27], when the tuned parameters  and  in (30) [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.

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.

Figure 1 :
Figure 1: State response of the open-loop system (1) in Case 1.

Figure 2 :
Figure 2: State response of the open-loop system (6) in Case 1.