TY - JOUR

T1 - Analysis of Positive Systems Using Copositive Programming

AU - Kato, Teruki

AU - Ebihara, Yoshio

AU - Hagiwara, Tomomichi

N1 - Funding Information:
Manuscript received August 14, 2019; accepted October 3, 2019. Date of publication October 10, 2019; date of current version November 1, 2019. This work was supported in part by JSPS KAKENHI under Grant JP18K04200. Recommended by Senior Editor M. Arcak. (Corresponding author: Teruki Kato.) The authors are with the Department of Electrical Engineering, Graduate School of Engineering, Kyoto University, Kyoto 615-8510, Japan (e-mail: tkato@jaguar.kuee.kyoto-u.ac.jp; ebihara @kuee.kyoto-u.ac.jp; hagiwara@kuee.kyoto-u.ac.jp). Digital Object Identifier 10.1109/LCSYS.2019.2946620
Publisher Copyright:
© 2017 IEEE.

PY - 2020/4

Y1 - 2020/4

N2 - In the field of control, a wide range of analysis and synthesis problems of linear time-invariant (LTI) systems are reduced to semidefinite programming problems (SDPs). On the other hand, in the field of mathematical programming, a class of conic programming problems, so called the copositive programming problem (COP), is actively studied. COP is a convex optimization problem on the copositive cone, and the completely positive cone, the doubly nonnegative cone, and the Minkowski sum of the positive semidefinite cone and the nonnegative cone are also closely related to COP. These four cones naturally appear when we deal with optimization problems described by nonnegative vectors. In this letter, we show that the stability, the H2 and the H∞ performances of LTI positive systems are basically characterized by the feasibility/optimization problems over these four cones. These results can be regarded as the generalization of well-known LMI/SDP-based results on the positive semidefinite cone. We also clarify that in some performances such direct generalization is not possible due to inherent properties of the copositive or the completely positive cone. We thus capture almost entire picture about how far we can generalize the SDP-based results for positive systems to those on the four cones related to COP.

AB - In the field of control, a wide range of analysis and synthesis problems of linear time-invariant (LTI) systems are reduced to semidefinite programming problems (SDPs). On the other hand, in the field of mathematical programming, a class of conic programming problems, so called the copositive programming problem (COP), is actively studied. COP is a convex optimization problem on the copositive cone, and the completely positive cone, the doubly nonnegative cone, and the Minkowski sum of the positive semidefinite cone and the nonnegative cone are also closely related to COP. These four cones naturally appear when we deal with optimization problems described by nonnegative vectors. In this letter, we show that the stability, the H2 and the H∞ performances of LTI positive systems are basically characterized by the feasibility/optimization problems over these four cones. These results can be regarded as the generalization of well-known LMI/SDP-based results on the positive semidefinite cone. We also clarify that in some performances such direct generalization is not possible due to inherent properties of the copositive or the completely positive cone. We thus capture almost entire picture about how far we can generalize the SDP-based results for positive systems to those on the four cones related to COP.

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U2 - 10.1109/LCSYS.2019.2946620

DO - 10.1109/LCSYS.2019.2946620

M3 - Article

AN - SCOPUS:85073163510

VL - 4

SP - 444

EP - 449

JO - IEEE Control Systems Letters

JF - IEEE Control Systems Letters

SN - 2475-1456

IS - 2

M1 - 8864008

ER -