Controllability of bilinear systems

Piechottka, Uwe; Frank, P.M.

doi:10.1016/0005-1098(92)90160-h

Cited by 8 publications

(8 citation statements)

References 10 publications

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“…• Our second example illustrates the tightness of the Gramianbased lower bound (16) for the input energy functional.…”

Section: Minimum Input Energy For Reachabilitymentioning

confidence: 99%

“…. , K − 1 and noting V (x(0)) = 0, we get (16). The sufficient condition (15) is a magnitude constraint at every actuator.…”

Section: Minimum Input Energy For Reachabilitymentioning

confidence: 99%

“…Bilinear systems [12], [13], [14] are one of the simplest classes of nonlinear systems but can be used to represent a wide range of physical, chemical, economical, and biological systems that cannot be effectively modeled using linear systems. While reachability/controllability of bilinear systems as a binary property has been widely investigated, see e.g., [15], [16], [17], [18], [13] and references therein, few results are available for quantitative metrics. A notion of reachability Gramian exists for bilinear systems, but its relation with the input energy functional is not fully understood.…”

Section: Introductionmentioning

confidence: 99%

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Gramian-Based Reachability Metrics for Bilinear Networks

Zhao

Cortés

2017

IEEE Trans. Control Netw. Syst.

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Abstract-This paper studies Gramian-based reachability metrics for bilinear control systems. In the context of complex networks, bilinear systems capture scenarios where an actuator not only can affect the state of a node but also interconnections among nodes. Under the assumption that the input's infinity norm is bounded by some function of the network dynamic matrices, we derive a Gramian-based lower bound on the minimum input energy required to steer the state from the origin to any reachable target state. This result motivates our study of various objects associated to the reachability Gramian to quantify the ease of controllability of the bilinear network: the minimum eigenvalue (worst-case minimum input energy to reach a state), the trace (average minimum input energy to reach a state), and its determinant (volume of the ellipsoid containing the reachable states using control inputs with no more than unit energy). We establish an increasing returns property of the reachability Gramian as a function of the actuators, which in turn allows us to derive a general lower bound on the reachability metrics in terms of the aggregate contribution of the individual actuators. We conclude by examining the effect on the worst-case minimum input energy of the addition of bilinear inputs to difficult-to-control linear symmetric networks. We show that the bilinear networks resulting from the addition of either inputs at a finite number of interconnections or at all self loops with weight vanishing with the network scale remain difficult-to-control. Various examples illustrate our results.

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“…• Our second example illustrates the tightness of the Gramianbased lower bound (16) for the input energy functional.…”

Section: Minimum Input Energy For Reachabilitymentioning

confidence: 99%

“…. , K − 1 and noting V (x(0)) = 0, we get (16). The sufficient condition (15) is a magnitude constraint at every actuator.…”

Section: Minimum Input Energy For Reachabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Gramian-Based Reachability Metrics for Bilinear Networks

Zhao

Cortés

2017

IEEE Trans. Control Netw. Syst.

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“…The main interest of bilinear systems lies in the fact that many important processes, not only in engineering, but also in biology, socio-economics, and ecology, can be modeled by bilinear systems [2][3][4][5][6][7][8][9][10][11]. The fundamental property of bilinear systems, say, controllability, has been extensively studied for both continuous-time and discrete-time cases [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Since a discrete-time approximation to a bilinear system is essential for computer simulation or digital control purpose [28][29][30], a natural question is that whether the discretization changes the system controllability.…”

Section: Introductionmentioning

confidence: 99%

“…Secondly, we show that, in comparison with the controllability criterion of this paper, the counterexample presented by [1] is a special case of the proposed necessary and sufficient conditions. Finally, by noting that system (1) is uncontrollable in any finite dimension [4,14,22], it is shown that the discretization counterpart of (1), say system (2), is also uncontrollable if its dimension is greater than two.…”

Section: Introductionmentioning

confidence: 99%

On controllability of discrete-time bilinear systems

Tie

Cai

Lin

2011

Journal of the Franklin Institute

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Infinite‐time partially observed nonzero‐sum bilinear affine‐quadratic stochastic differential game and its application to competitive advertising

Tao

Zhang

Yang

2023

Asian Journal of Control

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Due to the conjunction of huge expenditure and latent inefficiencies, competitive advertising has always been occupying a front‐and‐center place in marketing, among which a bilinear system is one of the most important. Because much real business is two giants playing among other emerging challengers and the players can only inspect the market through a related noisy process by the random sampling survey, the sales‐advertising dynamics is a partially observed nonzero‐sum stochastic differential game. However, by the intrinsic difficulties of mathematics, there is no extant research to augment the analysis on the infinite‐time horizon, which is of particular significance to implementing the online sales campaign and evaluating the consequence of endless competition. Motivated by this, controllability and stability analysis are explored first for the iterative approximation to bilinearity. Then, the simulation provides a rapid prototyping and automatic approach for the most efficient strategies. Moreover, four implications have been discovered by comparison with eight examples as the application of four encounters on finite‐ and infinite‐time horizons. Several new insights have been revealed for those struggling to develop successful advertising strategies.

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Controllability of bilinear systems

Cited by 8 publications

References 10 publications

Gramian-Based Reachability Metrics for Bilinear Networks

Gramian-Based Reachability Metrics for Bilinear Networks

On controllability of discrete-time bilinear systems

Infinite‐time partially observed nonzero‐sum bilinear affine‐quadratic stochastic differential game and its application to competitive advertising

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