Absolute Stability Criteria for Large-Scale Lurie Direct Control Systems with Time-Varying Coefficients

The absolute stability of large-scale Lurie direct control systems with time-varying coefficients is investigated. Based on the decomposition method for large-scale systems and technique of the nonsingular M-matrix, a suitable scalar Lyapunov function as a weighted sum is constructed. By estimating its total time derivative, some absolute stability criteria and practical corollaries are derived. Furthermore, the results are extended to multiple nonlinearities. The salient feature of this paper is that the criteria which we propose allow for the situation that the norms of time-varying coefficients are unbounded. The main idea of the methodology is that even if the coefficients are norm-unbounded, by restricting their relative magnitude, the problem of negative definiteness for the derivative can also be changed into the problem of stability for a constant matrix. Finally, some numerical examples are included to illustrate the effectiveness of the proposed criteria.


Introduction
The absolute stability problem has an important position in the analysis and design of control systems. In fact, as a typical class of nonlinear system, the problem of the absolute stability of the Lurie control system has been studied for almost 70 years [1][2][3] and has proved a fruitful area of research [4][5][6][7][8]. Recently, the problem of Lurie control systems has been extended. Among the studies, [9,10] studied the absolute stability of large-scale Lurie systems, [11][12][13] considered the robust stability of uncertain Lurie systems, and [14,15] discussed Lurie systems with time delays. In addition, there are many monographs on these topics [16,17].
Nevertheless, most of the papers on Lurie control systems are confined to the norm-bounded coefficients. The absolute stability of Lurie control systems with time-varying and norm-unbounded coefficients has received little attention. In [18], a research method was introduced for the stability of large-scale systems with time-varying coefficients. The core result was that if the order of infinities for the interconnected elements is far less than the order of isolated subsystems, then the large-scale system is still asymptotically stable.
Reference [19] promoted the results and considered the problem of robust exponential control for a class of largescale systems with uncertainties and unbounded coefficients. On the other hand, the approach is also an effective way to investigate the Lurie control systems with time-varying coefficients. In particular, in the case of the Lurie indirect control systems, it is relatively easier to study them using this method. In [20,21], this approach was applied to the Lurie indirect control systems with norm-unbounded coefficients, and some absolute stability criteria for this kind of system were obtained. Reference [22] subsequently extended the criteria to systems with multiple nonlinearities and largescale Lurie indirect systems. However, because of the linear relationship between ( is not an independent variable) and the other state variables in Lurie direct control systems, this became the main roadblock in judging the total time derivative of the Lyapunov function. Reference [23] overcame this difficulty and derived the sufficient condition of absolute stability for the Lurie direct control systems with normunbounded coefficients.
The problem proposed in this paper is more general than that described above: it focuses on large-scale Lurie 2 The Scientific World Journal direct control systems with time-varying coefficients and systems with multiple nonlinearities. The study of stability for large-scale systems is not a new one; it began in the 1960s [24], and the research method is basically the decomposition method for large-scale systems [25,26]. From the viewpoint of cybernetics, this is known as decentralized control [27,28]. The basic idea is decomposing the large-scale system into a certain number of lower-order isolated subsystems and constructing the Lyapunov function of the large-scale system through the isolated subsystems. Our objective in the following analysis is to extend this approach and study largescale Lurie direct control systems with norm-unbounded coefficients. For brevity, we will not involve the concept of the isolated subsystem but will directly give the Lyapunov function by employing the isolated subsystem and estimate the upper bound of its total derivative. Notation. Throughout this paper, ‖ ‖ denotes the Euclidean norm ‖ ‖ = √ , where = ( 1 2 ⋅ ⋅ ⋅ ) is a column vector and denotes the transposition. For vectors = ( 1 2 ⋅ ⋅ ⋅ ) , = ( 1 2 ⋅ ⋅ ⋅ ) , ≤ ( < ) means that ≤ ( < ) ( = 1, 2, . . . , ). ‖ ‖ represents the Euclidean norm of a matrix which is induced by the Euclidean vector norm ‖ ‖; that is, ‖ ‖ = max ‖ ‖=1 ‖ ‖.
( ) denotes an arbitrary eigenvalue of matrix , and > 0 ( < 0) means that is positive definite (negative definite). The symbol stands for the time interval ( , +∞), where ∈ or = −∞. lim → +∞ ( ) represents the upper limit of the function ( ); that is, lim → +∞ ( ) = lim → +∞ sup ≥ ( ( )). When the limit of function exists, the upper limit must exist and be equal to its limit, so if we change the upper limit involved in this paper into the limit, the conclusions still hold. The reason why we employ an upper limit to describe the theorems and corollaries is that the upper limit makes the stability conditions less conservative.

Absolute Stability of Large-Scale Lurie Systems with Single Nonlinearity
Consider the following large-scale Lurie direct control system with time-varying coefficients and single nonlinearity: System (1) is said to be absolutely stable if its zero solution is globally asymptotically stable for any nonlinearity ( ) ∈ [0, +∞) [16,17]. Basic assumptions about system (1) are proposed in the following.
(A2) We assume that where ( ) > 0 is a known function.

Remark 2.
In [23,29], we know that < 0 is the necessary condition for absolute stability in the case of Lurie direct control systems with constant coefficients.
(A4) We assume that where ( , = 1, 2, . . . , ; ̸ = ) are constant. As we know, the norm-unbounded and time-varying coefficients in system (1) are the main roadblocks in estimating the total time derivative of the Lyapunov function. But, by using the ( ), ( ) ( = 1, 2, . . . , ) in conditions A1 and A2 and placing them in the denominator, the relative magnitude of the norm-unbounded coefficients can be restricted in conditions A3 and A4. That is, the "infinite" nature can be expressed by the "finite" form, which makes the study of the absolute stability of large-scale systems (1) feasible. Therefore, we have the following results.
The Scientific World Journal 3 Theorem 3. Under A1, A2, A3, and A4, the system described by (1) is absolutely stable if the matrix is stable.
Proof. Choose a candidate Lyapunov function for system (1) as where 1 , . . . , , +1 are positive numbers that will be determined later. From condition A1, we get that ∑ =1 is a positive definite quadratic form, and, by the properties of ( ), we know that in (8) is radially unbounded, is positive definite, and has an infinitesimal upper bound.
Remark 3. This is very different from the Lurie indirect control system. For the Lurie indirect control system, is an independent component of the state vector. Because of this, in order to guarantee that in (8) is positive definite in +1 and is radially unbounded, we need to also assume ∫ ±∞ 0 ( ) = +∞ in [20][21][22].
It should be noted that A3 and A4 can be weakened by establishing upper limits. Therefore, the following corollaries are obtained.

Corollary 4.
Under A1, A2, A3 , and A4 , the system described by (1) is absolutely stable if the matrix is stable.
Namely, is a matrix induced by whose every off-diagonal entry is increased by .
According to the properties of the matrix, if is stable, then there exists a sufficiently small scalar > 0 such that is also stable. We choose an > 0 that can allow for the stability of .
From the assumptions here and the definition of upper limit, as to the above , there exists a number (≥ ), which satisfies that if > , then Thus, according to Theorem 3, system (1) is absolutely stable. If = 0 ( , = 1, 2, . . . , ; ̸ = ), then the following corollary is proposed.
Proof. According to A4 , the eigenpolynomial of the matrix can be obtained as where = −1 is an eigenvalue with multiplicity ( − 1) and the other two eigenvalues satisfy It is easy to see that all roots of the above equation have a real part if and only if ∑ =1 < 1. So is stable if ∑ =1 < 1. This implies that system (1) is absolutely stable by Corollary 4.
Consider a more specific case; we have the following corollary. holds.
Proof. Choosing the undetermined coefficients of the Lyapunov function in Theorem 3 as 1 = ⋅ ⋅ ⋅ = = +1 = 1 and combining them with Corollary 4, we just need to prove that is negative definite. Note that each eigenvalue of + is a real number (since + is a real symmetric matrix) and If inequalities (30) hold, then each eigenvalue of + satisfies < 0, and this implies that + is negative definite. The proof is completed.
Moreover, the conditions of Corollary 7 can be weakened as follows.

(38)
If inequalities (34) hold, then Re( ) < 0, and this implies that is stable. Similarly, if inequalities (35) hold, we can prove that is stable; that is, is stable. With the conditions of Corollary 4 being satisfied, we conclude that system (1) is absolutely stable.

Absolute Stability of Large-Scale Lurie Systems with Multiple Nonlinearities
Consider the following large-scale Lurie direct control systems with time-varying coefficients and multiple nonlinearities:̇= System (39) is said to be absolutely stable if its zero solution is globally asymptotically stable for any nonlinearity ( ) ∈ [0, +∞) [16,17].
At the same time, we can get the following corollaries. The proof for corollaries is similar to that in large-scale Lurie systems with single nonlinearity and thus is omitted.

Numerical Examples
In this section, two simple numerical examples are introduced to demonstrate the effectiveness of our criteria. Example 1. Consider system (1) with Note that the norms of coefficient matrices for the above system are unbounded; we choose Therefore, 1 ( ) = 2 , 1 = 2; 2 ( ) = 10 5 , 2 = 10; This means that assumptions A1 and A2 are satisfied. From lim → +∞ 2 1 1 ( ) we know that = 0, = 0 ( = 1, 2) in assumption A3 . Since lim → +∞ 2 1 12 ( ) we know that assumption A4 is satisfied and It is easy to verify that is stable. In summary, according to Corollary 4 (or Corollary 8), the system is absolutely stable.
Let ( ) = 3 and 1 = ( 1 2 ) , 2 = 3 ; then system (1) can be written aṡ Simulation results are shown in Figure 1. Although the coefficients of the large-scale Lurie direct control system with time-varying coefficients are norm-unbounded, we can clearly see from Figure 1 that the convergence rate of the system is very fast. This illustrates the availability of our results.
Similarly, we choose Then, 1 ( ) = 2 , 2 ( ) = 10 5 , So, assumption A7 is satisfied. From we know that assumptions A8 and A9 are satisfied and It is easy to see that is stable. Hence, according to Corollary 11, the system is absolutely stable. Let 1 ( 1 ) = 3 1 , 2 ( 2 ) = 5 2 , and 1 = ( 1 2 ) , 2 = 3 ; then system (39) can be written aṡ as shown in Figure 2. For the large-scale Lurie direct control system with time-varying coefficients and multiple nonlinearities, although the coefficients are norm-unbounded, we can see from Figure 2 that the large-scale system is absolutely stable.

Conclusions
The absolute stability of large-scale Lurie direct control systems with time-varying coefficients and systems with multiple nonlinearities is studied in this paper. By restricting the relative magnitude of the time-varying coefficients and employing the decomposition theory of large-scale systems, some absolute stability criteria were obtained. The criteria, introduced in this paper, can be used not only in largescale Lurie direct control systems with norm-unbounded coefficients but also in systems with norm-bounded coefficients. Two numerical examples are introduced to illustrate the availability of our results.

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