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New stability conditions for class of nonlinear discrete-time systems with time-varying delay

The problem of stability analysis for a class of nonlinear discrete time systems with time varying delay is studied in this work. Such systems are modeled by delayed difference equations. Subsequently, this system is transformed into an arrow form matrix representation. Using M-matrix properties, novel sufﬁcient stability conditions are determined. It is shown how to use our method to design a state feedback controller that stabilizes a discrete time Lure system with time varying delay and sector bounded nonlinearity. The originalities of our ﬁndings are shown in their explicit representation, using system’s parameters, as well as in their easiness to be employed. The obtained results demonstrate also that checking stability of a nonlinear discrete time systems with time varying delay can be reduced to an M-matrix test. Several examples are provided to show the effectiveness of the introduced technique.


I. INTRODUCTION
Stability of delay systems has been examined intensively by the academics from the control community [1]- [13], because several physical systems, like networked control systems, biological systems and chemical systems, are generally associated with time delays, [14]- [19]. Indeed, time delay can vary over time. For example, in real time communication and control systems, the signals are transmitted through the networks and subject to variable time delays because of the network traffic changes/uncertainties. Thus, stability analysis and control of such dynamical systems with time varying delays is essential. To attain stability conditions, two main strategies can be followed due to the time varying nature of the delay. Independent of delay (i.o.d ) results applicable when the size of the delay is arbitrary or if there is no information about the delay. This deficiency leads to conservative criteria, particularly if the delay is relatively small. When information about the size of delay can be included, less conservative delay-dependent ( d.d ) conditions can be provided [20].
The majority of the literature on stability of delay systems gave stability conditions in terms of linear matrix inequalities (LMIs) [21]. This remains true till now in a huge volume of the publications on the topic. And the size of LMIs increases with order/complexity of the systems. Obviously, it is desirable to have a very few number of stability conditions, regardless of order/complexity of a delay system; that is why we try in this study, to determine easy to test stability conditions for nonlinear discrete time systems with time varying delay. New delay dependent stability conditions are obtained by transforming the studied system under an arrow form state space representation [9], [10] and [11], using the Koteleyanski lemma [22] and by applying Lyapunov functional technique and M-matrix properties. The main obtained result is simple, and in fact it consists of verifying a scalar condition, without the need of solving any LMIs. It allows a great freedom by a judicious choice of some scalar parameters. The obtained results can be applied to large class of systems. As an example of these systems, we may mention the famous Lure Postnikov system, see [9] and the references therein. Moreover, we show how to use our method to design a state feedback controller that stabilizes a discrete time Lure system with time varying delay and sector bounded nonlinearity [23]- [28]. Note that this system is one of the most important classes of nonlinear control systems and remains one of the main problems in control theory which is intensively examined due to its various practical applications [29]- [35]. This paper is organized as follows: the utilized notations, the definition of M-matrices as well as some preliminary results are described in Section 2. The main results of this paper are represented in Section 3. Subsequently, the utility of these results if applied to the well known Lure systems is shown in Section 4. Finally, in Section 5 and Section 6, we provide some illustrative examples and a brief conclusion, respectively.

II. PRELIMINARIES
We present, in this part, some preliminary results including some definitions and lemmas used in the proof of the main results. Let us first fix the notation used throughout this paper. The set of real number is denoted by R, N designates the set of non-negative integers, and R n denotes an n−dimensional linear vector space over the reals with the norm ||.||. The notations ||.|| refers to the Euclidean vector norm or the induced matrix norm, as appropriate. Let I n denotes the n × n identity matrix and M T denotes the transpose of matrix M . Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions.
To characterize an M matrix, several equivalent conditions are given in the below definitions: . . , n, and for any real vector η > 0, the algebraic equation: M y = η, has a solution y = w > 0.
The below two lemmas play an important role in the proof of the main results. Consider the following arrow form matrix Λ, which will be used in the next section where λ i,n , λ n,i > 0, i = 1, . . . , n − 1 ; λ i,i < 1, i = 1, . . . , n . Lemma 2. Any matrix having the form presented in (1) is (-M)-matrix when the following conditions are satisfied: Proof. In case the matrix Λ is (-M)-matrix, −Λ is an M-matrix. Based on Kotelyanski lemma and Remark 1, successive principal minors of −Λ with positive signs yields to λ i,i < 1, ∀i = 1, . . . , n − 1. It comes the first condition of lemma. For i = n, it comes, which completes the proof.

III. MAIN RESULTS
The class of nonlinear delay systems studied in this manuscript are governed by the following difference equation: .., 1 and ∀i = 0, . . . , n − 1, where y is the system output, and h(k) : N → N denotes a time varying delay. In practice, the time delay may be unknown and can vary over time in a certain interval. It is thus assumed that h(k) has an upper limit h m so that h(k) ≤ h m , h m ∈ N.
The system (3) becomes where for any i = 1, ..., n − 1, We treat in the rest of this part the two cases of constant delay and variable delay.

A. Constant delay case
Before stating the main result, let us define the following matrix: where Theorem 1. The time delay system (3) with constant delay, h(k) = h, is asymptotically stable if there exist distinct real numbers, |α i | < 1, i = 1, . . . , n − 1, such that the following inequality holds true, Proof: Choosing a radially unbounded, positive definite Lyapunov function candidate such that where p(X(k)) = ( p 1 (X(k)) p 2 (X(k)) ... p n (X(k)) ) T , with Because ρ > 0, so that V (k) > 0. We obtain the difference V (k + 1) − V (k) under the solution of (8) as follows: We notice that and the substitution of (22) in (20) gives it comes from (18), (22) and (20), As nonlinear elements of M T 1 (S 1 ), in the last column, are isolated, we obtain constant eigenvector v(.) relative to the eigenvalue Then from (23), we obtain: It is noted, from Lemmas 1 and 2, that when Thus, the proof is finished.

IV. APPLICATION TO DELAYED LURE SYSTEMS
Consider the Lure type discrete time system presented in Figure 1. The model consists of a static nonlinearity in cascade with a dynamic linear time delay system. The structure of this system where only the variable ε n is nonlinearly modulated , allows us to investigate the Lure type discrete-time system by the following nonlinear regression equation: Setting the following variable: Fig. 1. Block representation of the studied system with the following notation: . Applying Theorem 1 leads to sufficient condition of the same form (12), but depending on ε k−h , . . . , ε k−h+n−i , . . . , ε k−h+n−1 . The obtained results are often difficult to implement, furthermore its interpretations with respect to the linear and nonlinear characteristics of the studied processes are generally limited. These considerations are due to the fact that the matrix description, from which the study is conducted, comes with a base change. The choice of a prior representation of Frobenius type allows to set similar to the previous stability conditions in which the coefficients depend only ε k−h . By introducing the following variable changes, and by choosing the state vector x k = x n k . . . x 2 k ε k T the corresponding expression in terms of state space representation (41) becomes: andG This system is particular case of (3 ) where D(z, .) =D(z) = z n + n−1 i=0ā n−1−i z i , and N (z, .) = g * (.)N (z) = . , where g(.) is a function satisfying the sector bound condition, Z is the Z transform, B 0 (s) = 1−e −Ts s s is a zero order holder, T s the sampling time and h = τ Ts the time delay.

A. Sufficient stability conditions: autonomous case
Let us first consider the autonomous case (r = 0). The obtained system is a special case of (3) wheref i (.) =ā n−1−i , g i (.) = g * (.)b n−1−i ∀ i = 1, ..., n − 1, γ n (.) = γ n = −ā 0 − n−1 i=1 α i and δ n (.) = −g * (.)b n . The following change of coordinates is employed: where The transformation results in the following system where and In which case we obtain A sufficient stability condition for this system is given in the following theorem. Theorem 3. The Lure type discrete-time system presented in figure. 1 is ( i.o.d ) asymptotically stable, if there exist distinct real numbers, |α i | < 1, i = 1, . . . , n − 1, such that the following inequality holds true Remark 3. L. Hou et al. in [34] established ultimate boundedness results for PWM feedback systems which can be considered a particular case of Theorem 1 when g * is considered as sign() function. They show that the solutions are ultimately bounded only when system is Hurwitz stable. Our result stated in Theorem 3. is obviously more general because it remains true when the system contains one unstable root and with delay. IfN (z) has all its roots z i , i = 1, . . . , n − 1 such that |z i | < 1, and −D(z i )β i > 0, then condition of Theorem 3 simplifies considerably. The following corollary gives this simplified condition. Corollary 2. The Lure type discrete-time system presented in Fig. 1 is ( i.o.d ) asymptotically stable, if there exist distinct real numbers, |α i | < 1, i = 1, . . . , n − 1, such that the following inequality holds true γ n > 0 ,−D(z i )β i > 0 and Proof: It is sufficient to take α i = z i in the condition of Theorem 3, in this caseN (α i ) =N (z i ) = 0 . Another important condition can be obtained whenD(z) admits n − 1 distinct roots with module inside the unit circle and the n th root can be outside the unit circle. This condition is given by the following corollary.
Proof: It is sufficient to take α i equal to the roots ofD that are inside the unit circle. In this case terms in condition of Theorem 3 becomesD(α i ) = 0 and γ n = −ā n−1 − n−1 i=1 z i = −z n .

Remark 4.
The last condition can also be simplified. In fact, if the roots ofD verifyN (z i )β i > 0, i = 1, . . . , n − 1 and b n−1 > 0, we obtain a new simple condition given by the following corollary.

B. Feedback stabilization
In this case, take r(k) = −Kx(k) with K = (k 0 , k 1 , ...., k n−1 ), then the obtained system has the same form as (3), with The stabilizing values of K can be obtained by making the following changes: γ n = −a n−1 − Then a sufficient stability condition for this system is given in the following theorem. |δ n (., k n−1 )| + Remark 5. The above result of Theorem 4 gives an explicit way how to calculate the stabilizing values of the feedback gain vector K.

Example. 3
Consider the study of a DC motor controlled by pulse width modulation from a tachometer. The control pulses are rectangular, with a constant amplitude equal to M and the sign of the error signal is defined at the sampling instants. Let T s be the sampling time, R k = θ|ε k | be the duration of the impulse in unsaturated regime and τ i , i = 1, 2 be the time constants of the DC motor. The output of the modulator is a sequence of pulses of height M and the width of the control pulses is related to the error function at the sampling instants by a relationship of the form: or simply under the following relation: where N (s) D(s) = λ 1 s + 1 (1 + τ 1 s)(1 + τ 2 s) .

VI. CONCLUSION
In this work, we have presented the new stability conditions for delayed nonlinear discrete time systems. The conditions are explicit, scalar and easy to check. Indeed, the application of the proposed method to delayed Lure Postnikov system shows simplicity and effectiveness. Moreover, our approach is self-contained, and systematic, and it does not go through the Linear Matrix Inequalities (LMI). Our theorems can deal with time delays, non-linearity and discrete time systems, and thus have more general applicability than those in the related literature.