- Research
- Open Access
The stability analysis of a system with two delays
- Sertaç Erman^{1}Email author,
- Hulya Kodal Sevindir^{2} and
- Ali Demir^{2}
- Received: 15 December 2017
- Accepted: 19 June 2018
- Published: 5 July 2018
Abstract
This paper presents new results of stability analysis for a linear system with two delays. We attempt to determine the asymptotic stability regions of the system in a parameter space by using D-partition method. Moreover, some stability and instability conditions in terms of coefficient inequalities have been obtained for the system.
Keywords
- Stability analysis
- Delay differential equations
- Multiple delays
1 Introduction
While modeling by using ordinary differential equations, the delay in the system is always ignored. However, even a small amount of delay may cause large changes in the system solution. Therefore, the use of delay differential equations is more realistic when any encountered problems are modeled.
For a long time, many problems in the fields of engineering [1–4], biology [5–8], chemistry [9], physics [10, 11], economy [12], psychology [13, 14], etc. have been modeled by delay differential equations.
We attempt to determine the stability regions of the system in a parameter space by using D-partition method [24] which is explained in Sect. 2.
2 D-partition method
The method originated from paper [24]. This method consists in obtaining a “partition” of the parameter space in several regions, so that each region is bounded by a hyper surface which corresponds to the case when at least one root lies on the imaginary axis. Furthermore, for all the parameters lying in a given region, the corresponding characteristic equation has the same number of roots with positive real parts [25]. Following theorems and definitions and more details on them can be found in references [26–28] and [29].
The roots \(\lambda= ib\) are called critical roots of the characteristic equation since stability regions of the system are determined by the analysis of the system at critical roots.
It is well known that the solution of system (4)–(5) exists under the assumption \(P_{1}(b)Q_{2}(b) - P_{2}(b)Q_{2}(b) \ne0\).
Definition 1
Theorem 1
The D-curves divide the complex plane up into a finite number of regions [29].
Theorem 2
The characteristic equation \(g(\lambda)\) has a root on the imaginary axis if and only if \((k_{1},k_{2})\) is on the D-curves [29].
Theorem 3
In each region, determined by the D-curves in the \((k_{1},k_{2})\) plane, \(g(\lambda)\) has the same number of roots with positive real parts [29].
For every region \(u_{k}\) of the D-partition, bounded by D-curves, it is possible to assign a number k which is the number of roots with positive real parts of the characteristic equation defined by the points of this region. Among the regions of this decomposition are also found regions \(u_{0}\) (if they exist) on which the characteristic equation does not have any root with positive real part. On these regions, the solutions are asymptotically stable. The determination of these numbers for the individual domains is not an easy task. One technique is analysis of sign of partial derivative along the D-curves. Alternatively, without calculating partial derivatives, the following Stepan’s formulas [30] can also be used to determine the number of roots with positive real parts.
Theorem 4
Since the delay terms have a direct effect on the solution of the characteristic equation, the delay differential equations with the same coefficients but different delay terms \(r_{1},r_{2}, \ldots,r_{n}\) may have different stability regions.
The following definitions are given for the delay differential equations with delay terms \(r_{1},r_{2}, \ldots,r_{n}\).
Definition 2
The system the stability of which depends on the delay terms is called delay-dependent stable system.
Definition 3
The system for which the stability is preserved for every value of delay terms is called delay independent stable.
Methodology
- (i)
Find a parametric equation of D-curves of the system.
- (ii)
Construct the graph of the D-curves. (In this paper, D-curves are obtained by means of MATLAB.)
- (iii)
Select specific points in the regions whose boundaries are the D-curves.
- (iv)
Determine the number of roots with positive real parts for the specific points by using Theorem 4 and generalize them to the relevant regions.
- (v)
Denote the region, on which the number of roots with positive real parts is k for a chosen specific point, by \(u_{k}\).
3 Stability regions and main results
In this section, system (1)–(2) is studied for two different cases in two different spaces. One of these cases is \(r_{1} = r_{2} = r\) for which the characteristic equation becomes much simpler. In other case, \(r_{1} \ne r_{2}\) state, which constitutes the main part of our analysis.
3.1 \((\alpha_{1} - \alpha_{2})\) parameter space
In this subsection, we determine the conditions under which the system is unstable. Moreover, the regions on which the characteristic equation has roots with the same number of positive real parts are shown for fixed delay values.
Lemma 1
If \(0 \le2A \le\frac{1}{r\mu} \), then \(x^{2} + 2Ax\sin (xr) \ge0\) for \(\forall x \in IR\), where r is a positive real number and \(\mu= \sup(\frac{ - \sin x}{x}) \approx0.218\).
Proof
Lemma 2
Let μ be defined as in Lemma 1. If \(r_{1} = r_{2} = r\), \(\beta_{2}\theta_{1} = \theta_{2}\beta_{1}\), \(\vert \theta_{1} \vert < \beta_{1}\), and \(0 \le2\theta_{2} \le\frac{1}{r\mu} \), then \(\alpha_{2}(b) \le0\) for \(\forall b \in IR\).
Proof
(i) If \(\vert \theta_{1} \vert < \beta_{1}\) for \(\forall b \in IR\), then inequality \(\theta_{1}\cos(br) + \beta_{1} > 0\) holds.
(iii) If \(0 \le\theta_{2} \le\frac{1}{r\mu} \), then it follows from Lemma 1 that \(b^{2} + 2\theta_{2}b\sin(br) \ge0\) holds for \(\forall b \in IR\).
Lemma 3
If \(r_{1} = r_{2} = r\), \(\beta_{2}\theta_{1} = \theta_{2}\beta_{1}\), \(\vert \theta_{1} \vert < \beta_{1}\), and \(2 \vert \theta_{2} \vert \le \vert \beta_{2} \vert \), then \(\alpha_{2}(b) \le 0\) for \(\forall b \in IR\).
Proof
It follows from the proof of Lemma 2 that the denominator of equality (15) is positive. Now let us investigate the following cases for the numerator of equality (15). Since the numerator of equality (15) is an even function of b, it is sufficient to show for \(\forall b \ge0\).
Theorem 5
Proof
It follows from Lemma 2 or Lemma 3 that D-curves of system (1)–(2) are located outside of the region which is determined by inequality system (16) in \((\alpha_{1} - \alpha_{2})\) parameter space. Suppose that there were two points within the region (16) with different numbers of roots with positive real parts. Then along any arc within that region connecting the points there must be a point where some of the roots of the characteristic equation lie on the imaginary axis. This point must lie on the D-curves, giving a contradiction. As a result, the number of roots of the characteristic equation with positive real parts does not change in the region (16). In other words, if the system is unstable for specific values of the parameters which satisfy the conditions of the theorem, the instability of the system has been shown. Using Stepan’s formula (6) with \(\beta_{1} = \beta_{2} = 1.2\), \(\theta_{1} = \theta_{2} = 1\), \(r_{1} = r_{2} = 1\), \(\alpha_{1} = 0.1\), \(\alpha_{2} = 1\) values, it is obtained that the characteristic equation of the system has one root with a positive real part. □
Theorem 6
Proof
The proof follows the lines of the proof of Theorem 5. □
It follows from Theorems 5 and 6 that since the region on which \(\alpha_{2} > 0\) has no curves but straight line (9), the stability of the system does not change. Moreover, the straight line (9) splits this region into two subregions on which the characteristic equation of the system has either one or two roots with positive real parts. As a result, the system is unstable under the assumption of Theorems 5 and 6.
3.2 \((\beta_{1} - \beta_{2})\) parameter space
Proposition 1
There exist real numbers m and M such that \(m < \beta_{2}(b) < M\) for \(\forall b \in IR\).
Proof
Theorem 7
If \(\frac{\alpha_{2}\theta_{1} - \alpha_{1}\theta _{2} + \alpha_{2}\beta_{1}}{\alpha_{1}} < \beta_{2} < m\) is satisfied, then system (1)–(2) is stable.
Proof
From equation (9) and Proposition 1, the region which is determined by \(\frac{\alpha_{2}\theta_{1} - \alpha_{1}\theta_{2} + \alpha_{2}\beta_{1}}{\alpha_{1}} < \beta_{2} < m\) does not include D-curves in (\(\beta_{1} - \beta_{2}\)) parameter space.
In order to prove that on each region the characteristic equation of the system has the same number of roots with positive real parts, let us suppose that in a region there were two points for which the numbers of roots with positive real parts are different. Then along any arc within that region connecting these points there must be a point where some of the roots of the characteristic equation lie on the imaginary axis. This point must lie on the D-curves, giving a contradiction. As a result, the number of roots of the characteristic equation with positive real part does not change in the region. Thus, it is sufficient to show that it is stable for specific values which satisfy the condition of the theorem. Using Stepan’s formula (6) with \(\alpha_{1} = 0.2\), \(\alpha_{2} = 0.2\), \(\beta_{1} = - 5\), \(\beta_{2} = - 2\), \(\theta_{1} = 1.25\), \(\theta_{2} = 1\), \(r_{1} = 1\), \(r_{2} = 2\), it is shown that the characteristic equation of the system has no root with apositive real part, which implies the stability of the system (1)–(2). □
Theorem 8
Proof
The proof is similar to the proof of Theorem 7. Figure 2 can be used to find the specific values that satisfy the conditions. □
Proposition 2
Let \(\mu= \sup(\frac{ - \sin x}{x})\). If \(0 \le \theta_{2} \le\frac{1}{r_{2}\mu} \) and \(\alpha_{2} > 0\), then there exists a real number N such that \(\beta_{1}(b) < N\) for \(\forall b \in IR\).
Proof
4 Conclusion
It shown that the delay plays an important role on the stability of the system for both cases. The stability region gets either expanded or contracted in one direction as the delay increases. Having examined the stability locally, we found that a certain range of delays gain stability. However, this is not a common result for all values of coefficients of the system. In Fig. 2, increasing delay \(r_{1}\) does not change stability considerably.
In Theorems 5–9, new conditions in terms of coefficients are obtained for stability and instability of the system. These conditions are derived by exploiting D-partition method.
In the future work, we will develop the conditions of theorems as figures imply.
Declarations
Acknowledgements
The authors are grateful to the reviewer for valuable comments that improved the manuscript.
Availability of data and materials
Not applicable.
Funding
Not applicable.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read, checked, and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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