Robust Sub-optimality of Linear-Saturated Control via Quadratic Zero-Sum Differential Games

Abstract

In this paper, we determine the approximation ratio of a linear-saturated control policy of a typical robust-stabilization problem. We consider a system, whose state integrates the discrepancy between the unknown but bounded disturbance and control. The control aims at keeping the state within a target set, whereas the disturbance aims at pushing the state outside of the target set by opposing the control action. The literature often solves this kind of problems via a linear-saturated control policy. We show how this policy is an approximation for the optimal control policy by reframing the problem in the context of quadratic zero-sum differential games. We prove that the considered approximation ratio is asymptotically bounded by 2, and it is upper bounded by 2 in the case of 1-dimensional system. In this last case, we also discuss how the approximation ratio may apparently change, when the system’s demand is subject to uncertainty. In conclusion, we compare the approximation ratio of the linear-saturated policy with the one of a family of control policies which generalize the bang–bang one.

Original language	English
Pages (from-to)	1109–1125
Number of pages	17
Journal	Journal of optimization theory and applications
Volume	184
Issue number	3
DOIs	https://doi.org/10.1007/s10957-019-01611-x
Publication status	Published - 9-Dec-2020

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Robust Sub-optimality of Linear-Saturated Control viaQuadratic Zero-Sum Differential GamesFinal publisher's version, 418 KBLicence: Taverne

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title = "Robust Sub-optimality of Linear-Saturated Control via Quadratic Zero-Sum Differential Games",

abstract = "In this paper, we determine the approximation ratio of a linear-saturated control policy of a typical robust-stabilization problem. We consider a system, whose state integrates the discrepancy between the unknown but bounded disturbance and control. The control aims at keeping the state within a target set, whereas the disturbance aims at pushing the state outside of the target set by opposing the control action. The literature often solves this kind of problems via a linear-saturated control policy. We show how this policy is an approximation for the optimal control policy by reframing the problem in the context of quadratic zero-sum differential games. We prove that the considered approximation ratio is asymptotically bounded by 2, and it is upper bounded by 2 in the case of 1-dimensional system. In this last case, we also discuss how the approximation ratio may apparently change, when the system{\textquoteright}s demand is subject to uncertainty. In conclusion, we compare the approximation ratio of the linear-saturated policy with the one of a family of control policies which generalize the bang–bang one.",

author = "Dario Bauso and Rosario Maggistro and Raffaele Pesenti",

year = "2020",

month = dec,

day = "9",

doi = "10.1007/s10957-019-01611-x",

language = "English",

volume = "184",

pages = "1109–1125",

journal = "Journal of optimization theory and applications",

issn = "0022-3239",

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TY - JOUR

T1 - Robust Sub-optimality of Linear-Saturated Control via Quadratic Zero-Sum Differential Games

AU - Bauso, Dario

AU - Maggistro, Rosario

AU - Pesenti, Raffaele

PY - 2020/12/9

Y1 - 2020/12/9

N2 - In this paper, we determine the approximation ratio of a linear-saturated control policy of a typical robust-stabilization problem. We consider a system, whose state integrates the discrepancy between the unknown but bounded disturbance and control. The control aims at keeping the state within a target set, whereas the disturbance aims at pushing the state outside of the target set by opposing the control action. The literature often solves this kind of problems via a linear-saturated control policy. We show how this policy is an approximation for the optimal control policy by reframing the problem in the context of quadratic zero-sum differential games. We prove that the considered approximation ratio is asymptotically bounded by 2, and it is upper bounded by 2 in the case of 1-dimensional system. In this last case, we also discuss how the approximation ratio may apparently change, when the system’s demand is subject to uncertainty. In conclusion, we compare the approximation ratio of the linear-saturated policy with the one of a family of control policies which generalize the bang–bang one.

AB - In this paper, we determine the approximation ratio of a linear-saturated control policy of a typical robust-stabilization problem. We consider a system, whose state integrates the discrepancy between the unknown but bounded disturbance and control. The control aims at keeping the state within a target set, whereas the disturbance aims at pushing the state outside of the target set by opposing the control action. The literature often solves this kind of problems via a linear-saturated control policy. We show how this policy is an approximation for the optimal control policy by reframing the problem in the context of quadratic zero-sum differential games. We prove that the considered approximation ratio is asymptotically bounded by 2, and it is upper bounded by 2 in the case of 1-dimensional system. In this last case, we also discuss how the approximation ratio may apparently change, when the system’s demand is subject to uncertainty. In conclusion, we compare the approximation ratio of the linear-saturated policy with the one of a family of control policies which generalize the bang–bang one.

U2 - 10.1007/s10957-019-01611-x

DO - 10.1007/s10957-019-01611-x

M3 - Article

SN - 0022-3239

VL - 184

SP - 1109

EP - 1125

JO - Journal of optimization theory and applications

JF - Journal of optimization theory and applications

IS - 3

ER -

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