Computing the Stochastic ${H} ^\infty $ -Norm by a Newton Iteration

Damm, Tobias; Benner, Peter; Hauth, Jan

doi:10.1109/lcsys.2017.2707409

Cited by 6 publications

(5 citation statements)

References 21 publications

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“…Nevertheless, when it comes to the 𝐻 ∞ problem, previous works assume 𝓁 2 finite-energy exogenous disturbances 8,9,10,11,12,13,14,15 , regardless of deterministic or stochastic approaches. They all presuppose a disturbance dependent-noise or a state-multiplicative disturbance that vanishes when the system reaches equilibrium 10,16,13,17,18 , or deal with the finite horizon case 19,20 . Compared with other literature on system models, a noticeable feature of the CSVIU method is that infinite-energy 𝓁 2 disturbance signals are accounted for in an infinite horizon approach.…”

Section: 𝑥(𝑘 + 1) = 𝐴𝑥(𝑘) + 𝜎𝜔(𝑘)mentioning

confidence: 99%

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The H∞ optimal Control Problem of CSVIU Systems

Val

Campos

2023

Preprint

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The paper devises a H ∞ -norm theory for the CSVIU (control and state variations increase uncertainty) class of stochastic systems. This system model appeals to stochastic control problems to express the state evolution of a possibly nonlinear dynamic system restraint to poor modeling. Contrary to other H ∞ stochastic formulations that mimic deterministic models dealing with finite energy disturbances, the focus is on the H ∞ control with infinity energy disturbance signals. Thus, the approach portrays the persistent perturbations due to the environment more naturally. In this regard, it requires a refined connection between a suitable notion of stability and the systems’ energy or power finiteness. It delves into the control solution employing the relations between H ∞ optimization and differential games, connecting the worst-case stability analysis of CSVIU systems with a perturbed Lyapunov type of equation. The norm characterization relies on the optimal cost induced by the Min-Max control strategy. The rise of a pure saddle point is linked to the solvability of a modified Riccati-type equation in a form known as a generalized game-type Riccati equation, which yields the solution of the CSVIU dynamic game. The emerging optimal disturbance compensator produces inaction regions in the sense that, for sufficiently minor deviations from the model, the optimal action is constant or null in the face of the uncertainty involved. A numerical example illustrates the synthesis.

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Section: 𝑥(𝑘 + 1) = 𝐴𝑥(𝑘) + 𝜎𝜔(𝑘)mentioning

confidence: 99%

“…Note that the difference in ( 14) depends explicitly on 𝑥 𝑘 and 𝜔 𝑘 only, and we denote Δ𝑉 (𝑥 𝑘 , 𝜔 𝑘 ) ∶= 𝑉 (𝑘 + 1, 𝑥 𝑘+1 ) − 𝑉 (𝑘, 𝑥 𝑘 ) for short. By adding and subtracting the terms (16) where…”

Section: Notationmentioning

confidence: 99%

The H∞ optimal Control Problem of CSVIU Systems

Val

Campos

2023

Preprint

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“…Nevertheless, when it comes to the

{H}_{\infty }

problem, previous works assume

{\ell}_2

finite‐energy exogenous disturbances, 8‐15 regardless of deterministic or stochastic approaches. They all presuppose a disturbance dependent‐noise or a state‐multiplicative disturbance that vanishes when the system reaches equilibrium, 10,13,16‐18 or deal with the finite horizon case 19,20 . Compared with other literature on system models, a noticeable feature of the CSVIU method is that infinite‐energy

{\ell}_2

disturbance signals are accounted for in an infinite horizon approach.…”

Section: Introductionmentioning

confidence: 99%

The H∞‐optimal control problem of CSVIU systems

Campos,

do Val

2023

Intl J Robust & Nonlinear

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The article devises an ‐norm theory for the CSVIU (control and state variations increase uncertainty) class of stochastic systems. This system model appeals to stochastic control problems to express the state evolution of a possibly nonlinear dynamic system restraint to poor modeling. It introduces the control with infinity energy disturbance signals, contrary to usual stochastic formulations, which mimic deterministic models dealing with finite energy disturbances. The reason is to portray the persistent perturbations due to the environment more naturally. In this regard, it develops a refined connection between a notion of stability and the system's power finiteness. It delves into the control solution employing the relations between optimization and differential games, connecting the worst‐case stability analysis of CSVIU systems with a perturbed Lyapunov type of equation. The norm characterization relies on the optimal cost induced by the min–max control strategy. The solvability of a generalized game‐type Riccati equation couples to the rise of a pure saddle point, yielding the global solution of the CSVIU dynamic game. In the way to the optimal stabilizing compensator, the article finds it based on linear feedback stabilizing gain from solving the Riccati equation together with a spectral radius test of an ensuing matrix. The optimal disturbance compensator produces inaction regions in the sense that, for sufficiently minor deviations from the model, the optimal action is constant or null in the face of the uncertainty involved. Finally, an explicit solution approximation developed by a Monte Carlo method appears, and a numerical example illustrates the synthesis.

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“…On the other hand, note that most existing literature on ℋ ∞ control of linear, continuoustime stochastic systems usually considers systems with only multiplicative noise, cf. for example (HINRICHSEN D. PRITCHARD, 1998;UGRINOVSKII, 1998;CHEN, 2006;DRAGAN et al, 2006;SHAKED, 2006;SHAKED, 2008;ZHANG et al, 2014;SHENG et al, 2015;DAMM et al, 2017) . Hinrichsen and Pritchard point out in (HINRICHSEN D. PRITCHARD, 1998) that the use of additive white noise might pose additional, restrictive conditions on the the design of ℋ ∞ controllers.…”

Section: Long Run Average Cost and Robust Controlmentioning

confidence: 99%

Norms and stability of stochastic systems for which control and state variation increase uncertainties

Silva¹

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First and foremost I would like to thank my advisor, Prof. João Bosco, for the patience, enthusiasm and guidance during both the masterŠs program and the previous experiences as an undergraduate research assistant. He has not only unwaveringly supported my research and guided me into the right direction when needed, but also gave me freedom to pursue various research projects. His support and my time as an undergraduate research student make up one of the main reasons why I decided to pursue an academic career in the Ąrst place.I would also like to thank the masterŠs thesis committee members, Prof. Geromel and Prof. Costa, for the careful reading of this rather lengthy work. Another special thanks to Prof. Geromel for his assistance with the graduate school application process.A sincere thanks to Prof. Tamer Başar and the Coordinated Science Lab of the University of Illinois at Urbana Champaign for hosting me as a visiting student and accommodating me in his research group; and to Prof. Serdar Yüksel and the Department of Mathematics and Statistics of QueenŠs University at Kingston, Canada, for the shortterm visit and insightful discussions amid the Canadian winter.A special thanks to my family and friends, in particular my parents, José Nilton (in Memoriam) and Valdelice, and my siblings, Karoliny and Matheus, for the constant support through all these years. I would also like to thank my colleagues at the Department of Systems and Energy, for the support and fruitful discussions; as well as the School of Electrical and Computer Engineering and the University of Campinas, where I spent a good many years as an undergraduate and masterŠs student.

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Computing the Stochastic ${H} ^\infty $ -Norm by a Newton Iteration

Cited by 6 publications

References 21 publications

The H∞ optimal Control Problem of CSVIU Systems

The H∞ optimal Control Problem of CSVIU Systems

The H∞‐optimal control problem of CSVIU systems

Norms and stability of stochastic systems for which control and state variation increase uncertainties

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