Abstract
The problem of absolute stability of Lur’e systems with sector and slope restricted nonlinearities is revisited. Novel time-domain and frequency-domain criteria are established by using the Lyapunov method and the well-known Kalman-Yakubovich-Popov (KYP) lemma. The criteria strengthen some existing results. Simulations are given to illustrate the efficiency of the results.
1. Introduction
Absolute stability of nonlinear systems has been investigated comprehensively for the past several decades [1–12]. It is well known that the Popov criterion and the circle criterion are two classical results with the forms of frequency-domain inequalities (FDIs), which are turned out to be equivalent to some linear matrix inequalities (LMIs). This not only gives the opportunity to use the powerful LMI toolbox [13] to study absolute stability, but also gives the opportunity to consider the controller design problems. In [14], absolute stability of single-input and single-output Lur’e systems with a sector and slope restricted nonlinearity is brought forward. It is pointed out that the slope restriction on the nonlinearity strengthens the Popov criterion by adding an additional term to the original FDI of the criterion. Much work [15–22] on the slope restricted and multivariable problem has been done by using a Lur’e-Postnikov function or an extended Lur’e-Postnikov function.
In this paper, both time-domain criterion and frequency-domain criterion for absolute stability of Lur’e systems with sector and slope restricted nonlinearities are presented based on the Lyapunov method and the KYP lemma. Some mathematical tools are used through the derivation of the absolute stability criterion. Compared with some existing results, the proposed results are less conservative. This should be owed to the effect of the slope restricted conditions on the nonlinearities. The rest of the paper is organized as follows. In Section 2, the system description and some preliminaries are presented. Time-domain and frequency-domain criteria for absolute stability of the system are given in Section 3. Numerical examples are given in Section 4 and some concluding remarks are given in Section 5.
Throughout this paper, the superscript means transpose of real matrices and conjugate transpose of complex matrices. For a Hermitian matrix , () denotes that is a positive definite (semidefinite) matrix and denotes that is a negative definite matrix. means for any real or complex square matrix .
2. Problem Statement
Consider the following multi-input and multioutput Lur’e system where , , and are real matrices, , is the output, is piecewise continuously differentiable on , and are assumed to satisfy where , , , and . The inequalities (2) and (3) denote sector restriction and slope restriction on , respectively. Let , , , . Then , , , and . Setting , (3) is formulated as follows: The transfer function from to is denoted as .
System (1) is called to be absolutely stable if the equilibrium point is globally asymptotically stable for all nonlinear vector valued functions satisfying (2) and (3). In the following sections, less conservative absolute stability criteria including time-domain criterion and frequency-domain criterion for system (1) are given. Before studying these problems, first we introduce the KYP lemma and Schur complement. These lemmas will be used repeatedly in this paper to get our main results.
Lemma 1 (KYP lemma [23]). Given that , , and symmetric matrix , with for , and the pair is controllable, the following two statements are equivalent.(i), for all .(ii)There exists a matrix such that . The equivalence for strict inequalities holds even if is not controllable.
Lemma 2 (Schur complement [24]). The LMI is equivalent to one of the following statements:(i) and ;(ii) and .
3. Main Results
We choose the following Lur’e-Postnikov function: as the Lyapunov function, where and are necessary to be determined. It should be pointed out that is not necessary to be positive definite and are not necessary to be nonnegative.
Theorem 3. System (1) is absolutely stable for all satisfying (2) and (3) if is Hurwitzian and there exist diagonal matrices , , , and symmetric matrices such that the LMI is feasible: where
Proof. We will demonstrate that the given conditions imply the negative definiteness of and the positive definiteness of .
Taking the derivative of along the trajectory of (1), we have
Conditions (2) and (4) for are equivalent to
For any and , , it follows
where and . Then
The given condition (6) guarantees the negative definiteness of the right hand of (11). Consequently, is negative definite.
Now we are only left to demonstrate that is positive definite. In (5), is only a symmetric matrix but not a positive definite matrix and may be a positive or negative number. Therefore, the proof of the positive definiteness of is a little difficult and complex. Without loss of generality, letting and , then has the following form:
where and . Since (2) implies , is satisfied. Then is positive definite if is positive definite, which is proved in what follows.
Denote , , , and . Firstly, the given conditions imply that is Hurwitzian for any diagonal matrix satisfying . Actually, the matrix is Hurwitzian for in virtue of the given conditions. So we will demonstrate that is Hurwitzian for any diagonal matrix satisfying . We assume there exists a diagonal matrix satisfying such that the matrix is not Hurwitzian. On the one hand, a number satisfying can be found such that
holds for certain . Since is Hurwitzian, and are followed. The latter formula indicates that there exists a vector such that
where and . Then we derive
On the another hand, pre- and postmultiplying both sides of (6) by and , we have
where
By the Schur complement, (16) implies
From the KYP lemma, we derive that (18) holds if and only if
where and . Inequality (19) is equivalent to
in terms of the equalities and . Letting in (20) and pre- and postmultiplying both sides of the resulting inequality by and , it follows that
We can observe that (15) and (21) are contradictive, which means that the assumption is not true and is Hurwitzian for any diagonal matrix satisfying . Therefore, the matrix is Hurwitzian. Secondly, the given conditions imply that is positive definite. Actually, pre- and postmultiplying both sides of (16) by and yield
where
Inequality (22) implies . According to , , , , is followed. The matrix is Hurwitzian, which results in the positive definiteness of and . This completes the proof.
It is found in the proof of Theorem 3, more exactly in inequality (16), that if (6) holds, then is Hurwitzian if and only if .
Theorem 4. System (1) is absolutely stable for all satisfying (2) and (3) if there exist diagonal matrices , , , symmetric matrices , such that and the LMI (6) holds.
Remark 5. Theorem 3 is derived directly by using the time-domain method and can be used to study multi-input and multioutput Lur’e systems. Inequality (6) in Theorem 3 is in the form of LMI, which is easier to be solved by means of the LMI toolbox.
The LMI (6) can be transformed into an equivalent FDI. Thus, a frequency-domain criterion for (1) is given as follows.
Theorem 6. System (1) is absolutely stable for all satisfying (2) and (3) if the matrix is Hurwitzian and there exist diagonal matrices , , such that the following frequency-domain inequality holds
Proof. Let , , and . Inequality (6) can be rewritten as where According to the KYP lemma, (25) is equivalent to By simple computations, we have where . Substituting (28) into (27), the equivalence between (6) and (24) is derived.
Remark 7. For the case , the FDI (24) reduces to
which corresponds to the FDI as given in Theorem in [4]. However, the results there only aim at single-input and single-output Lur’e systems.
If the slope restrictions on are removed, another absolute stability criterion is derived by choosing (5) as the Lyapunov function.
Theorem 8. System (1) is absolutely stable for all satisfying (2) if the matrix is Hurwitzian and there exist diagonal matrices , , symmetric matrices , , such that the following LMI is feasible: where , .
Proof. The proof is similar to that of Theorem 3.
Remark 9. Theorem 8 gives absolute stability conditions for sector restricted Lur’e systems. In fact, the slope restricted condition (3) plays an important role in improving the condition of absolute stability. The forthcoming example shows that Theorem 3 is less conservative than Theorem 8.
Similar to Theorem 3, an equivalent frequency-domain criterion to Theorem 8 can be given as follows.
Theorem 10. System (1) is absolutely stable for all satisfying (2) if the matrix is Hurwitzian and there exist diagonal matrices , such that the following FDI holds:
Proof. From the KYP lemma, (30) is equivalent to The equivalence between (30) and (31) is derived from and .
Remark 11. Theorem 10 includes two particular cases. For the case , (31) is reduced to Correspondingly, Theorem 10 is in the form of the circle criterion. For the case , (31) reduces to Theorem 10 has the same form as the Popov criterion.
4. Numerical Example
In this section, a numerical example is presented to illustrate the effectiveness of the proposed results.
Consider Chua’s oscillator [25] with the following dimensionless equations where , , , , , and are numbers. System (35) can be reformulated in the form of (1) with , , , , , and . The nonlinearity satisfies Thus, and .
When , , , and are taken, system (35) is absolutely stable for by applying Theorem 3. However, we derive that system (35) is absolutely stable for and , respectively, by Theorem 8 and the Popov criterion. This shows that Theorem 3 is an improvement with respect to Theorem 8 and the Popov criterion, and the slope restrictions could improve the absolute stability condition. The states of system (35) with at the initial value are given in Figure 1, from which it is illustrated that system (35) is absolutely stable.
5. Conclusion
We have proposed new absolute stability criteria for Lur’e systems with sector and slope restricted nonlinearities from time-domain and frequency-domain points of view. The slope restrictions on nonlinearities improve the absolute stability conditions. We have shown that the criteria are less conservative than some existing results.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research is supported by the National Natural Science Foundation of China (61004050, 61172095) and the Natural Science Foundation of Scientific Research of Hebei Education Department (2009482).