Double Imaginary Root Degeneracies in Time-Delayed Systems and CTCR Treatment

2017 ◽  
Author(s):  
Ryan R. Jenkins ◽  
Nejat Olgac
Keyword(s):  
Linear Time ◽  
Arbitrary Parameter ◽  
Imaginary Root ◽  
Delayed Systems ◽  
Stability Behavior ◽  
Linear Time Invariant ◽  
Characteristic Roots ◽  
Point Of Interest ◽  
The Stability ◽  
Time Invariant

The dynamics we treat here is a very special and degenerate class of linear time-invariant time-delayed systems (LTI-TDS) with commensurate delays, which exhibit a double imaginary root for a particular value of the delay. The stability behavior of the system within the immediate proximity of this parametric setting which creates the degenerate dynamics is investigated. Several recent investigations also handled this class of systems from the perspective of calculus of variations. We approach the same problem from a different angle, using a recent paradigm called Cluster Treatment of Characteristic Roots (CTCR). We convert one of the parameters in the system into a variable and perturb it around the degenerate point of interest, while simultaneously varying the delay. Clearly, only a particular selection of this arbitrary parameter and the delay enforce the degeneracy. All other adjacent points would be free of the mentioned degeneracy, and therefore can be handled with the CTCR paradigm. Analysis then reveals that the parametrically limiting stability behavior of the dynamics can be extracted by simply using CTCR. The results are shown to be very much aligned with the other investigations on the problem. Simplicity and numerical speed of CTCR may be considered as practical advantages in analyzing such systems. This approach also exhibits the capabilities of CTCR in handling these degenerate cases contrary to the convictions in earlier reports. An example case study is provided to demonstrate these features.

Kernel and Offspring Concepts for the Stability Robustness of Multiple Time Delayed Systems (MTDS)

2006 ◽  
Vol 129 (3) ◽  
pp. 245-251 ◽  
Author(s):  
Rifat Sipahi ◽  
Nejat Olgac
Keyword(s):  
Time Delays ◽  
Linear Time ◽  
Characteristic Root ◽  
Invariance Property ◽  
Multiple Time ◽  
Delayed Systems ◽  
Linear Time Invariant ◽  
Characteristic Roots ◽  
Stability Robustness ◽  
The Stability

A novel treatment for the stability of linear time invariant (LTI) systems with rationally independent multiple time delays is presented in this paper. The independence of delays makes the problem much more challenging compared to systems with commensurate time delays (where the delays have rational relations). We uncover some wonderful features for such systems. For instance, all the imaginary characteristic roots of these systems can be found exhaustively along a set of surfaces in the domain of the delays. They are called the “kernel” surfaces (curves for two-delay cases), and it is proven that the number of the kernel surfaces is manageably small and bounded. All possible time delay combinations, which yield an imaginary characteristic root, lie either on this kernel or its infinitely many “offspring” surfaces. Another hidden feature is that the root tendencies along these surfaces exhibit an invariance property. From these outstanding characteristics an efficient, exact, and exhaustive methodology results for the stability assessment. As an added uniqueness of this method, the systems under consideration do not have to be stable for zero delays. Several example case studies are presented, which are prohibitively difficult, if not impossible to solve using any other peer methodology known to the authors.

Equivalency of Stability Transitions Between the SDS (Spectral Delay Space) and DS (Delay Space)

2013 ◽  
Author(s):  
Qingbin Gao ◽  
Umut Zalluhoglu ◽  
Nejat Olgac
Keyword(s):  
Linear Time ◽  
Broad Class ◽  
Delay Systems ◽  
Multiple Delays ◽  
Delayed Systems ◽  
Linear Time Invariant ◽  
Time In Literature ◽  
The Stability ◽  
Time Invariant ◽  
Delay Space

It has been shown that the stability of LTI time-delayed systems with respect to the delays can be analyzed in two equivalent domains: (i) delay space (DS) and (ii) spectral delay space (SDS). Considering a broad class of linear time-invariant time delay systems with multiple delays, the equivalency of the stability transitions along the transition boundaries is studied in both spaces. For this we follow two corresponding radial lines in DS and SDS, and prove for the first time in literature that they are equivalent. This property enables us to extract local stability transition features within the SDS without going back to the DS. The main advantage of remaining in SDS is that, one can avoid a non-linear transition from kernel hypercurves to offspring hypercurves in DS. Instead the potential stability switching curves in SDS are generated simply by stacking a finite dimensional cube called the building block (BB) along the axes. A case study is presented within the report to visualize this property.

Stability Analysis of LTI Systems With Three Independent Delays—A Computationally Efficient Procedure

2009 ◽  
Vol 131 (5) ◽  
Author(s):  
Rifat Sipahi ◽  
Hassan Fazelinia ◽  
Nejat Olgac
Keyword(s):  
Linear Time ◽  
Numerical Procedure ◽  
Computationally Efficient ◽  
Linear Time Invariant ◽  
Characteristic Roots ◽  
Stability Robustness ◽  
Linear Time Invariant Systems ◽  
The Stability ◽  
Operation Results ◽  
Time Invariant

A practical numerical procedure is introduced for determining the stability robustness map of a general class of higher order linear time invariant systems with three independent delays, against uncertainties in the delays. The procedure is based on an efficient and exhaustive frequency-sweeping technique within a single loop. This operation results in determination of the complete description of the kernel and the offspring hypersurfaces, which constitute exhaustively the potential stability switching loci in the space of the delays. The new numerical procedure corresponds to the first step in the overarching framework, called the cluster treatment of characteristic roots. The results of this treatment can also be represented in another domain (called the spectral delay space) within a finite dimensional cube called the building block, which is much simpler to view and analyze. The paper also offers several case studies to demonstrate the practicality of the new numerical methodology.

Kernel and Offspring Concepts for the Stability Robustness of Multiple Time Delayed Systems (MTDS)

2005 ◽  
Author(s):  
Rifat Sipahi ◽  
Nejat Olgac
Keyword(s):  
Time Delays ◽  
Linear Time ◽  
Characteristic Root ◽  
Test Point ◽  
Invariance Property ◽  
Multiple Time ◽  
Delayed Systems ◽  
Linear Time Invariant ◽  
Characteristic Roots ◽  
The Stability

A novel treatment for the stability of linear time invariant (LTI) systems with rationally independent multiple time delays is presented. The independence of delays makes the problem much more challenging compared to the systems with commensurate time delays (where the delays have rational relations). It is shown that the imaginary characteristic roots can all be found along a set of curves in the domain of the delays. They are called the “kernel curves”, and it is proven that their number is small and bounded. All possible time delay combinations, which yield an imaginary characteristic root, lie on a curve so called the offspring of the kernel curves within the domain of the delays. We also claim that the root tendencies show a very interesting invariance property as a test point crosses these curves. An efficient, exact and exhaustive methodology results from these outstanding characteristics. It is unique to the new methodology that, the systems under consideration do not have to possess stable behavior for zero delays. Several example case studies are presented, which are prohibitively difficult, if not impossible to solve using any other peer methodology.

The Cluster Treatment of Characteristic Roots and the Neutral Type Time-Delayed Systems

2004 ◽  
Author(s):  
Nejat Olgac ◽  
Rifat Sipahi
Keyword(s):  
Time Delay ◽  
Linear Time ◽  
Neutral Type ◽  
Direct Consequence ◽  
Necessary Condition ◽  
Delayed Systems ◽  
Linear Time Invariant ◽  
Characteristic Roots ◽  
Stability Intervals ◽  
The Stability

A new methodology is presented for assessing the stability posture of a general class of linear time-invariant – neutral time-delayed systems (LTI-NTDS). It is based on a “Cluster Treatment of Characteristic Roots CTCR” paradigm. The technique offers a number of unique features: It returns exact bounds of time delay for stability, furthermore it yields the number of unstable characteristic roots of the system in an explicit and non-sequentially evaluated function of time delay. As a direct consequence of the latter feature, the new methodology creates entirely, all existing stability intervals of delay, τ. It is shown that the CTCR inherently enforces an intriguing necessary condition for τ-stabilizability, which is the main contribution of this paper. This, so called “small delay” effect, was recognized earlier for NTDS, only through some cumbersome mathematics. In addition to the above listed characteristics, the CTCR is also unique in handling systems with unstable starting posture for τ = 0, which may be τ-stabilized for higher values of delay. Example cases are provided.

Degenerate Cases in Using the Direct Method

2003 ◽  
Author(s):  
Rifat Sipahi ◽  
Nejat Olgac
Keyword(s):  
Stability Analysis ◽  
Time Delay ◽  
Linear Time ◽  
Direct Method ◽  
Degenerate System ◽  
Delayed Systems ◽  
Linear Time Invariant ◽  
The Stability ◽  
Time Invariant ◽  
Degenerate Cases

A practical stability analysis, the Direct Method, for linear time invariant, time delayed systems (LTI-TDS) is revisited in this work considering the degenerate system dynamics. The principal strengths and enabling novelties of the method are reviewed along with its structured steps involved for assessing the stability. Uncommon in the literature, the Direct Method can handle large dimensional systems (e.g. larger than 2) very comfortably, it returns an explicit formula for the exact stability posture of the system for a given time delay, as such it reveals the possible detached stability pockets throughout the time delay axis. Both retarded and neutral classes of LTI-TDS are considered in this work. The main contribution here is to demonstrate the ability of the Direct Method in tackling degenerate cases. Along with the analytical arguments, example case studies are provided for a group of degeneracies. It is shown that the new method is capable of resolving them without any difficulty.

Degenerate Cases in Using the Direct Method

2003 ◽  
Vol 125 (2) ◽  
pp. 194-201 ◽  
Author(s):  
Rifat Sipahi ◽  
Nejat Olgac
Keyword(s):  
Stability Analysis ◽  
Time Delay ◽  
Linear Time ◽  
Direct Method ◽  
Degenerate System ◽  
Delayed Systems ◽  
Linear Time Invariant ◽  
The Stability ◽  
Time Invariant ◽  
Degenerate Cases

A recent stability analysis, the Direct Method, for linear time invariant, time delayed systems (LTI-TDS) is revisited in this work considering the degenerate system dynamics. The principal strengths and enabling novelties of the method are reviewed along with its structured steps involved for assessing the stability. Uncommon in the literature, the Direct Method can handle large dimensional systems (e.g., larger than two) very comfortably. It returns an explicit formula for the exact stability posture of the system for a given time delay, as such it reveals the possible detached stability pockets throughout the time delay axis. Both retarded and neutral classes of LTI-TDS are considered in this work. The main contribution here is to demonstrate the ability of the Direct Method in tackling degenerate cases. Along with the analytical arguments, example case studies are provided for a group of degeneracies. It is shown that the new method is capable of resolving them without any difficulty.

Stability Analysis of Multiple Time Delayed Systems Using the Direct Method

2003 ◽  
Author(s):  
Rifat Sipahi ◽  
Nejat Olgac
Keyword(s):  
Stability Analysis ◽  
Time Delays ◽  
Linear Time ◽  
Direct Method ◽  
Multiple Time ◽  
Delayed Systems ◽  
Practical Applications ◽  
Linear Time Invariant ◽  
Multiple Time Delays ◽  
The Stability

A novel treatment for the stability of a class of linear time invariant (LTI) systems with rationally independent multiple time delays using the Direct Method (DM) is studied. Since they appear in many practical applications in the systems and control community, this class of dynamics has attracted considerable interest. The stability analysis is very complex because of the infinite dimensional nature (even for single delay) of the dynamics and furthermore the multiplicity of these delays. The stability problem is much more challenging compared to the TDS with commensurate time delays (where time delays have rational relations). It is shown in an earlier publication of the authors that the DM brings a unique, exact and structured methodology for the stability analysis of commensurate time delayed cases. The transition from the commensurate time delays to multiple delay case motivates our study. It is shown that the DM reveals all possible stability regions in the space of multiple time delays. The systems that are considered do not have to possess stable behavior for zero delays. We present a numerical example on a system, which is considered “prohibitively difficult” in the literature, just to exhibit the strengths of the new procedure.

scholarly journals Torsion Discriminance for Stability of Linear Time-Invariant Systems

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 386
Author(s):  
Yuxin Wang ◽  
Huafei Sun ◽  
Yueqi Cao ◽  
Shiqiang Zhang
Keyword(s):  
Lebesgue Measure ◽  
Linear Time ◽  
Zero Solution ◽  
Positive Lebesgue Measure ◽  
Asymptotically Stable ◽  
Linear Time Invariant ◽  
Measurable Set ◽  
Linear Time Invariant Systems ◽  
The Stability ◽  
Time Invariant

This paper extends the former approaches to describe the stability of n-dimensional linear time-invariant systems via the torsion τ ( t ) of the state trajectory. For a system r ˙ ( t ) = A r ( t ) where A is invertible, we show that (1) if there exists a measurable set E 1 with positive Lebesgue measure, such that r ( 0 ) ∈ E 1 implies that lim t → + ∞ τ ( t ) ≠ 0 or lim t → + ∞ τ ( t ) does not exist, then the zero solution of the system is stable; (2) if there exists a measurable set E 2 with positive Lebesgue measure, such that r ( 0 ) ∈ E 2 implies that lim t → + ∞ τ ( t ) = + ∞ , then the zero solution of the system is asymptotically stable. Furthermore, we establish a relationship between the ith curvature ( i = 1 , 2 , ⋯ ) of the trajectory and the stability of the zero solution when A is similar to a real diagonal matrix.

Admissible Range of Proportional Gain in Stabilizing PID Region

2014 ◽  
Vol 530-531 ◽  
pp. 1068-1077 ◽  
Author(s):  
Yu Dong
Keyword(s):  
Linear Time ◽  
Stability Boundary ◽  
Closed Loop System ◽  
Double Pole ◽  
Linear Time Invariant ◽  
Admissible Range ◽  
Boundary Locus ◽  
The Stability ◽  
Time Invariant ◽  
Integral Gain

This paper considers the problem of stabilizing linear time-invariant plants by a PID controller. If the proportional gain reaches the extreme value, the closed-loop system contains a double pole on the imaginary axis. Using this property, the admissible range of the proportional gain is derived, also the corresponding integral gain and derivative gain are obtained. If the proportional gain is fixed, the stability region in the plane with respect to the integral gain and the derivative gain is determined by plotting the stability boundary locus. The effectiveness of the method presented is illustrated by several examples.

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