TY - JOUR
T1 - A Dual to Lyapunov's Second Method for Linear Systems with Multiple Delays and Implementation Using SOS
AU - Peet, Matthew
N1 - Funding Information:
Manuscript received July 3, 2017; revised January 8, 2018; accepted April 5, 2018. Date of publication May 2, 2018; date of current version February 26, 2019. This work was supported by the National Science Foundation under Grant 1301660, Grant 1538374, and Grant 1739990. Recommended by Associate Editor W. Michiels.
PY - 2019
Y1 - 2019
N2 - We present a dual form of Lyapunov-Krasovskii functional which allows the problem of controller synthesis for multidelay 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 Sum of Squares (SOS) methodology to express positivity and negativity of these forms as Linear Matrix Inequalities (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.
AB - We present a dual form of Lyapunov-Krasovskii functional which allows the problem of controller synthesis for multidelay 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 Sum of Squares (SOS) methodology to express positivity and negativity of these forms as Linear Matrix Inequalities (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.
KW - Controller synthesis
KW - LMIs
KW - delay systems
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U2 - 10.1109/TAC.2018.2832470
DO - 10.1109/TAC.2018.2832470
M3 - Article
AN - SCOPUS:85046364974
VL - 64
SP - 944
EP - 959
JO - IEEE Transactions on Automatic Control
JF - IEEE Transactions on Automatic Control
SN - 0018-9286
IS - 3
M1 - 8353379
ER -