Abstract
A new method providing necessary and sufficient conditions to test delay-independent stability for general linear time-invariant systems with constant delays is proposed. The method is utilized for single delay and incommensurate multiple delay systems. The proposed method offers an approach to determine the exact boundaries of unknown parameters such as controller gains or system parameters ensuring delay-independent stability, in addition to exhibiting an efficient test for real parameters. The technique is based on nonexistence of unitary complex zeros of an auxiliary characteristic polynomial obtained via extended Kronecker summation. A special feature of the polynomial, i.e., the self-inversive property, is proved and utilized to check its unitary zeros to determine delay-independent stability by an efficient zero location test. The methodology is executed employing simple algebraic operations and inspection of the number of sign variations in the obtained sequence. For the single delay case, the procedure does not require parameter (or frequency) sweeping, equation solving, and pointwise testing even for the determination of the delayindependent stabilizing regions of unknown parameters. In the case of systems with p multiple delays, (p - 2) agent parameters in the range [0, 2π] and one agent parameter in the range [0, π] are swept to determine delay-independent stability without the requirement of solving equations. A graphical projection approach for multiple delays is proposed in the case in which unknown parameters exist. The complete delay-independent stability analysis of a second order PD-controlled system with single delay is presented. Moreover, the method is applied to find the exact delay-independent stabilizing regions of unknown parameters of systems with two and three delays.
Original language | English |
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Pages (from-to) | 2661-2683 |
Number of pages | 23 |
Journal | SIAM Journal on Control and Optimization |
Volume | 55 |
Issue number | 4 |
DOIs | |
Publication status | Published - 2017 |
Bibliographical note
Publisher Copyright:© 2017 Society for Industrial and Applied Mathematics.
Funding
∗Received by the editors May 31, 2016; accepted for publication (in revised form) June 2, 2017; published electronically August 30, 2017. A preliminary version of this work for single delay systems was presented without details at the IFAC Workshop on Time Delay Systems 2015 held in Ann Arbor, MI. http://www.siam.org/journals/sicon/55-4/M107772.html Funding: This work was supported by the Scientific Research Centre of Istanbul Technical University under project 37993. †Department of Control and Automation Engineering, Istanbul Technical University, Istanbul 34469, Turkey ([email protected], [email protected]).
Funders | Funder number |
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Istanbul Teknik Üniversitesi | 37993 |
Keywords
- Delay-independent stability
- Kronecker sum
- Linear time-delay systems
- Multiple delays
- Self-inversive polynomials