A new model reduction scheme for k-power bilinear systems

Al-Baiyat, S.A.; Bettayeb, Maâmar

doi:10.1109/cdc.1993.325196

Cited by 35 publications

(49 citation statements)

References 14 publications

(7 reference statements)

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“…which for certain types of system can be computed efficiently [13]. An interesting discussion of the physical interpretation of such a Gramian can also be found in [10].…”

Section: Cmentioning

confidence: 99%

“…The Gramians in [12] are defined in terms of the kernels of the Volterra series expansion of the state. A benefit of such an approach is that they can be computed as the solution of Generalized Lyapunov equations [13] of the form…”

Section: Cmentioning

confidence: 99%

“…They are defined in terms of the kernels of the Volterra series expansion of the system state. In [13] they are shown to be easily computed as the solution of simple linear matrix equalities; for example, the bilinear observability Gramian must satisfy (3), a Generalized Lyapunov equation. The work presented in [7] then uses the bilinear observability and controllability Gramians to reduce the order of the bilinear system.…”

Section: A Bilinear System Gramiansmentioning

confidence: 99%

“…Efficient interior point algorithms exist for the construction of solutions to such problems [18] with the additional structure of the specific problem being examined in [22] and a polynomial time algorithm being proposed in [23]. In contrast, the solution to a generalized Lyapunov equation required for methods revolving around the bilinear system Gramians is known explicitly [13]. However, it does involve m + 2 Kronecker products between matrices of size n × n and hence for largescale problems storage requirements are large.…”

Section: Choice Of D-gramian and E-gramianmentioning

confidence: 99%

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Model reduction of homogeneous-in-the-state bilinear systems with input constraints

Couchman¹,

Kerrigan²,

Böhm³

2011

Automatica

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Abstract-Homogeneous-in-the-state bilinear systems, appended by an additive disturbance, appear both from the discretization of some partial differential equations and from the bilinearization of certain nonlinear systems. They often have large state vectors that can be cumbersome for simulation and control system design. Our aim is to define a method, invariant to time transformations, for finding a reduced-order model with similar disturbance-output characteristics to those of the plant for all admissible input sequences. The inputs considered satisfy simple upper and lower bound constraints, representing saturating actuators. The approximation is based on a model truncation approach and a condition for the existence of such an approximation is given in terms of the feasibility of a set of linear matrix inequalities. A novelty of our work is in the definition of a new Gramian for this class of system. Explicit error bounds on the scheme are included. The paper concludes with a demonstration of the proposed approach to the model reduction of a solar collector plant.

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“…which for certain types of system can be computed efficiently [13]. An interesting discussion of the physical interpretation of such a Gramian can also be found in [10].…”

Section: Cmentioning

confidence: 99%

Section: Cmentioning

confidence: 99%

Section: A Bilinear System Gramiansmentioning

confidence: 99%

Section: Choice Of D-gramian and E-gramianmentioning

confidence: 99%

See 2 more Smart Citations

Model reduction of homogeneous-in-the-state bilinear systems with input constraints

Couchman¹,

Kerrigan²,

Böhm³

2011

Automatica

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“…Designing control for the bilinear system have been proposed by some authors [6,11,12], and model reduction method to obtain the reduced bilinear system have been published in literatures, such as, the balanced truncation [1,2,5,7,13], moment matching through Krylov subspaces [3,4,8,9,15,16]. The other methods can be found in [13].…”

Section: Introductionmentioning

confidence: 99%

Model Reduction of Bilinear System using Genetic Algorithm

Saragih

2014

IJCA

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Direct methods and ADI‐preconditioned Krylov subspace methods for generalized Lyapunov equations

Damm

2008

Numerical Linear Algebra App

109

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We consider linear matrix equations where the linear mapping is the sum of a standard Lyapunov operator and a positive operator. These equations play a role in the context of stochastic or bilinear control systems. To solve them efficiently one can fall back on known efficient methods developed for standard Lyapunov equations. In this paper, we describe a direct and an iterative method based on this idea. The direct method is applicable if the generalized Lyapunov operator is a low-rank perturbation of a standard Lyapunov operator; it is related to the Sherman-Morrison-Woodbury formula. The iterative method requires a stability assumption; it uses convergent regular splittings, an alternate direction implicit iteration as preconditioner, and Krylov subspace methods.

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A new model reduction scheme for k-power bilinear systems

Cited by 35 publications

References 14 publications

Model reduction of homogeneous-in-the-state bilinear systems with input constraints

Model reduction of homogeneous-in-the-state bilinear systems with input constraints

Model Reduction of Bilinear System using Genetic Algorithm

Direct methods and ADI‐preconditioned Krylov subspace methods for generalized Lyapunov equations

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