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In this paper, we propose a new dual class of stability condition for MIMO single-delay systems which is based on the implicit existence of a Lyapunov-Krasovskii functional but does not explicitly construct such a functional. This new type of stability condition allows the controller synthesis problem to be formulated as a convex optimization problem with little or no conservatism using a variable transformation. Furthermore, we show how to invert this variable transformation in order to obtain the stabilizing controller. The stability and controller synthesis conditions are then enforced using the SOS framework exploiting recent advances in this field. Numerical testing verifies there is little to no conservatism in either the “dual” stability test or the controller synthesis condition.
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A Dual to Lyapanov's Second Method for Linear Systems with Multiple Delays and Implementation using SOS
We present a dual form of Lyapunov-Krasovskii functional which allows the problem of controller synthesis for multi-delay systems to be formulated and solved in a convex manner. First, we give a generalized version of the dual stability condition formulated in terms of Lyapunov operators which are positive, self-adjoint and preserve the structure of the state-space. Second, we provide a class of such operators and express the stability conditions as positivity and negativity of quadratic Lyapunov-Krasovskii functional forms. Next, we adapt the SOS methodology to express positivity and negativity of these forms as LMIs, describing a new set of polynomial manipulation tools designed for this purpose. We apply the resulting LMIs to a battery of numerical examples and demonstrate that the stability conditions are not significantly conservative. Finally, we formulate a test for controller synthesis for systems with multiple delays, apply the test to a numerical example, and simulate the resulting closed-loop system.
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- Award ID(s):
- 1739990
- PAR ID:
- 10073374
- Date Published:
- Journal Name:
- IEEE Transactions on Automatic Control
- ISSN:
- 0018-9286
- Page Range / eLocation ID:
- 1 to 1
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/TAC.2018.2832470
