Robust Stability of Discrete-time Singularly Perturbed Systems with Nonlinear Perturbation


 This paper is concerned with the robust stability and stabilization problems of discrete-time singularly perturbed systems (DTSPSs) with nonlinear perturbations. A proper sufficient condition via the fixed-point principle is proposed to guarantee that the given system is in a standard form. Then, based on the singular perturbation approach, a linear matrix inequality (LMI) based sufficient condition is presented such that the original system is standard and input-to-state stable (ISS) simultaneously. Thus, it can be easily verified for it only depends on the solution of an LMI. After that, for the case where the nominal system is unstable, the problem of designing a control law to make the resulting closed-loop system ISS is addressed. To achieve this, a sufficient condition is proposed via LMI techniques for the purpose of implementation. The criteria presented in this paper are independent of the small parameter and the stability bound can be derived effectively by solving an optimal problem. Finally, the effectiveness of the obtained theoretical results is illustrated by two numerical examples.


Introduction
Singularly perturbation problems are usually described by state-space models with a small singular perturbation parameter  , which naturally arise in practical control engineering field, such as aerospace systems, robot control systems, nuclear reactor control systems and power systems. Such a parameter often can lead to the increased order and stiffness of systems. A so-called reduction technique is usually adopted for dealing with this kind of system [1]- [3], which is a basic characterization of singular perturbation and a useful tool. Thus the numerical stiffness problem resulting from the existence of multi-time-scale phenomena can be alleviated. During the past decades, the study of SPSs has received much attention, the survey on the recent developments in SPSs can be found in [4]- [15] and the references therein.
Recently, many efforts have been made in DTSPSs for their extensive applications in control theory and various engineering. As shown in [16], DTSPSs can be described by slow sampling rate model and fast sampling rate model [17]. As with its continuous-time counterpart, the study of DTSPSs still faces the same problems, such as ingredients-order reduction, boundary layer phenomena etc. [18]- [19]. So far, some results have been obtained for robust stability and stabilization of DTSPSs [20]- [22].
Based on the reduction technique, the composite control for DTSPSs is considered in [19], in which a theoretical framework is given for systems with slow and fast modes.
In [22], a unified state-feedback is designed, under which the considered continuousand discrete time SPS is robust stable. Recently, some LMIs approaches have been developed to deal with the robust control problems for DTSPSs [23]- [31]. However, it should be pointed that the most of the aforementioned works were limited to linear cases, in which the system matrices are time-invariant or state-independent uncertainties. Meanwhile, a prescribed scalar   is needed when evaluating the upper bound of singular perturbation, which would bring some inconvenience for operating. The reason is that how to choose a proper   is difficult to ensure the solvability of the proposed LMI. Moreover, the two-time scale decomposition technique is not used in the literature, in which the small parameter 0   is seen as a static scalar. Whether the obtained result for the limit system (i.e. the slow and fast subsystems of the original system) would still remain as 0  → ? Unfortunately, there are no definite answers right now. It will be more perfect if these problems can be solved.
Based on the above analysis, we, in this paper, consider the robust stability and synthesis for DTSPSs with nonlinear perturbation. Up to now, little work on this topic has been made in the literature. As a prerequisite, by resorting to the fixed-point principle, a sufficient condition is given to ensure that the isolate root is possible and unique, thus the given system is standard. Generally speaking, it is a basic requirement for SPSs [1], [19]. Furthermore, by singular perturbation approach, the ISS results for the slow and fast subsystems were established, respectively. The notion of ISS was first introduced by Sontag in 1989 [32], which play an important role in characterizing the effects of external input to a control system. Recently, the relevant results have been generalized to nonlinear time-delay systems [33], discrete-time systems [34], and singularly perturbed systems [35], etc. Based on the ISS notion, a unified LMI sufficient condition is given, under which the existence of the isolate root and ISS property of the original system is guaranteed. Then, a state feedback control is designed to render the resulting closed-loop system ISS. It is important to point out that the proposed method here is substantially different from the existing literature since the nonlinear term is considered here and the coordinate transformation of the original system is not used although the reduction technique is adopted.
Compared with the existing results, the advantages of this paper can be summarized as follow: 1) A more general system is considered, where the nonlinear term only knows their norm upper bounds; 2) A unified LMI-based sufficient condition is presented such that the isolate root and ISS are guaranteed. Thus, it is very easy to verify as it just depends on the solution of the LMI; 3) The method presented in this paper marries the reduced technique and LMI, thus the numerically stiff problem can be alleviated; 4) In [23], it is difficult to solve the controller gain, the reason is that much more complex equation are involved. However, our method overcomes this constraint, which can be solved easily; 5) Although the two-time scale decomposition technique is adopted in this paper, any coordinate transformations for the original system is not involved, while this point is necessary by the traditional method.
The rest of the paper is organized as follows. Section 2 gives the problem formulation. The main results are given in Section 3. Section 4 gives two examples to show the effectiveness of the proposed methods. Finally, the conclusion is drawn in Section 5.
Notation: Throughout the paper, the notations are fairly standard. 0 P  means that matrix P is positive definite; # denotes that the item will not be used in the following.

Problem Formulation
Consider the following fast sampling DTSPSs with nonlinear disturbances described by  Then system (1)-(2) can be rewritten as It's easy to verify that the nonlinear term () fx is bounded by Remark 2.1: Condition (3) is used to show that the system considered here is a standard SPSs. The perturbation in (5) has been widely studied; see [8], [36], [37] and the references therein. It is worth noting that the matched condition can be seen as a special case of (5). However, for the discrete case, the robust stability for system (1)-(2) has not been considered.
A common idea for SPS is that the robust stability of the system (1)-(2) is considered by analyzing the corresponding slow and fast subsystems. Before we move on, some definitions and lemmas are given.
has a unique isolate root 21 ( , ) x x w  = , then system (1)-(2) with 0 u = is said to be standard.
It can be seen from the Definition 2.1 that the existence of the isolate root guarantees that the reduced-order model is well defined.
Definition 2.2: [34] Consider the discrete-time nonlinear system: where state () continuous and locally Lipschitz in x and w . The input w is a bounded function for all 0 k  . Then the system (7) is said to be input-to-state stable (ISS) if there exist a class KL function  and a class K function  such that for any initial state

Remark 2.2:
The last inequality (8) guarantees that for any bounded input () w  , the state () x  will be bounded, and as k increases, the state () xk will be ultimately bounded by a class K function of || || w . Furthermore, the inequality also shows that if () wk converges to zero as k →, so does () xk .
where 1  , 2  are class  K functions,  is a class K function, and () Wx is a continuous and positive definite function on n R . Then, the system (7) is input-to-state stable.

A. The Existence of an Isolate Root Analysis
In the next two sections, the zero control input will be considered for system (1)- (2). We, in this subsection, first establish a sufficient condition to guarantee system (1)-(2) standard, which can be stated in the following Lemma. P , 21 P and 22 P satisfying Proof: Consider the difference algebraic equation From (9), one has Noticing that Now, we make a partition for (12) So, 22 22 It can be seen that the matrices M and N are non-singular, and the following decompositions can be given Applying the Schur's complement, one has from LMI (9) that Premultiplying and postmultiplying (13) with T N and N , respectively, it can be seen that By further calculation, it is shown in (14) that the block matrix at the second block row and the second block column is negative definite, that is The above inequality shows that there exists a scalar . Then last inequality is equivalent to which implies that Thus there exists a scalar 0 Furthermore, for which implies that Next, the following transformation of coordinates is introduced by where 1 11 n xR  , 2 12 n xR  . Then the equation (10)- (11) can be described as: For any given 12 x , 2 12 n xR  , we have It can be seen from (16) that for any given 11 ( , ) xw , there exists a unique solution 12 11 ( , ) x x w  = by resorting to the fixed-point principle. Therefore, the isolate root is guaranteed by (15), that is, the system (1) is standard.■ Remark 3.1: it is shown from Lemma 3.1that the isolate root is property of the system itself, which is a basic requirement for the reduced technique. Moreover, it can be seen that 21 is also Lipschitz. Specially, there exist two scalars 1 0   and 2 0   , such that A similar procedure to that of (16) will be adopted in the proof process, thus the detail is ignored here. Now, using Lemma 3.1 and the reduced technique, the original system will be decomposed into the two subsystems.
A slow subsystem can be defined by setting 22 In order to obtain the fast subsystems, we replace ( 1) ( ) x k x k += , subtracting (19) from the equation (2) and setting 0  = yields the following fast subsystem

Remark 3.2:
This hybrid form has been discussed in depth discussed in [19].
Meanwhile, the evolution of the slow subsystems has also been proven to be justified in the theory.
It is known that the ISS property of reduced order system does not imply the one of the original system. Therefore, we will focus on how the ISS property of the original system (1)-(2) can be derived from the subsystems.

B. Input-to-State Stability Analysis
Using the two-time scale decomposition technique, a sufficient condition will be proposed such that the full-order system (1) Proof: Consider the Lyapunov function:  Hence, the conditions of Lemma 2.1 are satisfied, and we can conclude that there exist a class KL function  and a class K function  such that for any initial That is, the fast subsystem (20)  1) system (1)-(2) is a standard singularly perturbed systems; 2) system (1)-(2) is made ISS with respect to disturbance w for any Proof: 1) The proof of Lemma 3.1 has shown that system (1)-(2) is in the standard form, which completes the proof of part 1).
2) We now show the ISS property of system (1)- (2). Under the condition of Define a Lyapunov function candidate for system (1)-(2) as follows Then, for any a constant 1 0   , one has where  and  are defined in (26) and (30), respectively. Then system (1) problem [39] min  subject to (33).

T T T A P P A A P A
where  and 3 P are given in Theorem 3.4.

Remark 3.4: Using the frequency domain approach, Li and Li in [20] considered
the stability bound problem. The major drawback is that evaluating the exact value of stability bound by plotting the generalized Nyquist plot is extremely difficult unless the fast subsystem is a scalar. Further, based on the critical stability criteria, a new method was proposed for obtaining the stability bound in [16]. However, the method is still complex to operate. In contrast, Corollary 3.1 provides an LMI based sufficient condition for stability bound of system (34), which can be solved easily by LMI Toolbox. In [25]- [27], it is notice that a prescribed stability bound *  is required when using LMI technique. This leads to some inconvenience for implemention, because how to choose a proper   is difficult in order to ensure the solvability of the resulting linear matrix inequality. However, from Corollary 3.1, we can see that this case has been avoided effectively by our method.

C. The Input-to-State Stability of Closed-loop Systems
It should be worth mentioning that Theorem 3.3 requires that the nominal system be stable. However, in practice, this condition may be unsatisfactory. In this case, a feedback control would be necessary to guarantee that the ISS property can be achieved. Instead of being stable, the system is assumed to be stabilizable. Therefore , a state feedback transformation is given by where 12 () K K K = is a constant matrix, such that the resulting closed-loop system is ISS with w .
Substituting the feedback transformation (36) into (1)-(2), the resulting closed-loop system is given as Applying Theorem 3.3 to (35), we have the following result.  21 X , Y such that where H O H = .
Then the resulting closed-loop system (37) which is equivalent to   [23], it is easy to show that our method is simpler although the system considered here is more complex. In [23], solving a nonlinear matrix inequality is required in order to get the stabilizing controller. Unfortunately, so far there is no effective algorithm for solving it. Thus, the method in [23] is rather conservative, and it will become more evident when dimension of the system is larger.
Moreover, we found that no effective computation method for the upper bound is proposed in [23].

Numerical Examples
In this section, two illustrative examples are given to verify the feasibility of the obtained results.  the numerical computation, it has been clearly shown that solving control gain matrix is simple. However, it should be pointed out that the derived sufficient condition in [23] for solving the control gain matrix involves much more complex nonlinear matrix inequalities, which is numerically inefficient. In addition, with the best effort, the authors found that the method of [23] is also infeasible for solving the stability bound.
Utilizing the LMI toolbox, the following solutions via (38) can be given by is noticed that the method in [23] and [31] is infeasible for this system. To facilitate simulation, given the initial condition (0) ( 1.5 0.9) T x =− , then the simulations with 21 ( ) (1 ) w k k − =+ and ( ) 5cos w k k = are shown in Figs. 1 and 2, respectively. It can be seen from the simulations that the closed-loop system (45) is asymptotically stable when the disturbance input tends to zero, and ultimately bounded when the disturbance input is bounded, respectively.

Conclusion
A unified sufficient condition for the existence of isolate root and ISS property has been presented by marrying the reduced technique and LMI method. Thus the numerically ill-conditioned problem is avoided. Based on the established results, a proper state feedback control law has been constructed to render the closed-loop systems ISS. Moreover, a workable way for evaluating the stability bound has been obtained via solving GEVP problem, in which a prescribed stability bound is not required. Thus, we has improved and generalized the results and technique in the literature. Finally, two illustrative examples have been given to verify the feasibility of the methods.