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.

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 Adaptive Fuzzy Tracking Control for a Class of Uncertain Nonlinear Time-Delayed Systems with Saturation Constrains

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Yu-Jun Zhang ◽  
Li-Bing Wu ◽  
Hong-Yang Zhao ◽  
Xiao-Dong Hu ◽  
Wen-Yu Zhang ◽  
...  
Keyword(s):  
Tracking Control ◽  
Mean Value ◽  
Van Der Pol Oscillator ◽  
Mean Value Theorem ◽  
Approximation Technique ◽  
Multiple Time ◽  
Tracking Errors ◽  
Delayed Systems ◽  
Input Nonlinearity ◽  
Adaptive Fuzzy

In this paper, the problem of adaptive fuzzy tracking control is considered for a class of uncertain nonaffine nonlinear systems with external disturbances, multiple time delays, and nonsymmetric saturation constrains. First, the mean value theorem is employed to deal with the nonaffine term with input nonlinearity. Then, a new adaptive fuzzy tracking controller with parameter updating laws is designed by using fuzzy approximation technique. Moreover, it is shown that all the closed-loop signals are bounded and the tracking errors can asymptotically converge to zero via the Lyapunov stability analysis. Finally, the simulation example for van der Pol oscillator system is worked out to verify the effectiveness of the proposed adaptive fuzzy design approach.

scholarly journals Complex Scaling Approach for Stability Analysis and Stabilization of Multiple Time-Delayed Systems

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Jun Zhou ◽  
Ketian Gao ◽  
Xinbiao Lu
Keyword(s):  
Stability Analysis ◽  
Linear Time ◽  
Open Loop ◽  
State Feedback Control ◽  
Multiple Time ◽  
Control Input ◽  
Complex Scaling ◽  
Delayed Systems ◽  
Static State ◽  
Linear Time Invariant

Based on complex scaling, the paper copes with stability analysis and stabilization of linear time-invariant continuous-time systems with multiple time delays in state, control input, and measured output, under state and/or output feedback. More specifically, the paper establishes stability criteria for exponential/asymptotical stability with Hurwitz complex scaling being applied to the related characteristic polynomials. The stability conditions are necessary and sufficient, delay-dependent, and independent of feedback structures and open-loop poles distribution. The criteria can be implemented graphically with locus plotting or numerically without it; moreover, no prior frequency sweeping is involved, and the contour and locus encirclement orientations can be self-defined. Exploiting the complex scaling approach and embracing its technical merits, it is considered to design static state feedback control for robustly stabilizing time-delayed systems. A small-gain interpretation for the suggested stabilization is also elaborated. Examples are included to illustrate the main results.

Optimal control of multiple-time delayed systems based on the control parameterization method

2012 ◽  
Vol 53 ◽  
pp. 66
Author(s):  
Weihua Gui ◽  
Xiaoyu Shen ◽  
Ning Chen ◽  
Chunhua Yang ◽  
L. Y. Wang
Keyword(s):  
Optimal Control ◽  
Multiple Time ◽  
Delayed Systems ◽  
Parameterization Method ◽  
Control Parameterization

scholarly journals Robust fault detection of singular LPV systems with multiple time–varying delays

2016 ◽  
Vol 26 (1) ◽  
pp. 45-61 ◽  
Author(s):  
Amir Hossein Hassanabadi ◽  
Masoud Shafiee ◽  
Vicenç Puig
Keyword(s):  
Fault Detection ◽  
Reference Model ◽  
Residual Error ◽  
Unknown Input ◽  
Multiple Time ◽  
Delayed Systems ◽  
Bounded Real Lemma ◽  
Fault Sensitivity ◽  
Robust Fault Detection ◽  
Minimum Sensitivity

Abstract In this paper, the robust fault detection problem for LPV singular delayed systems in the presence of disturbances and actuator faults is considered. For both disturbance decoupling and actuator fault detection, an unknown input observer (UIO) is proposed. The aim is to compute a residual signal which has minimum sensitivity to disturbances while having maximum sensitivity to faults. Robustness to unknown inputs is formulated in the sense of the ℋ∞-norm by means of the bounded real lemma (BRL) for LPV delayed systems. In order to formulate fault sensitivity conditions, a reference model which characterizes the ideal residual behavior in a faulty situation is considered. The residual error with respect to this reference model is computed. Then, the maximization of the residual fault effect is converted to minimization of its effect on the residual error and is addressed by using the BRL. The compromise between the unknown input effect and the fault effect on the residual is translated into a multi-objective optimization problem with some LMI constraints. In order to show the efficiency and applicability of the proposed method, a part of the Barcelona sewer system is considered.

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