Beam analogy for optimal control of linear dynamic systems

Szyszkowski, W.; Grewal, I. S.

doi:10.1007/s004660050496

Cited by 7 publications

(7 citation statements)

References 12 publications

(11 reference statements)

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“…(1)), and by eliminating the costates. The optimality equation in terms of DOFs of the problem takes the form (see [12], for details of the derivation):…”

Section: Optimal Vibration Controlmentioning

confidence: 99%

“…After some formal manipulations (see [12,13]) the optimality condition (4), in terms of DOFs can be transformed into the optimality equation in terms of the modal variables in the form:…”

Section: Optimal Vibration Controlmentioning

confidence: 99%

“…The beam's results can be converted into the modal variables and the modal controls by applying (24) and (25). The beam analogy was successfully implemented in [12,13] to problems related to open loop control, in which the optimal actuator forces and the trajectories were obtained in function of time. Its application is extended here to calculate optimal gains for closed loop control applications.…”

Section: Beam Analogymentioning

confidence: 99%

“…It is an extension of the beam analogy presented in [12] for discrete and in [13] for continuous systems, where it was used for open-loop optimal active vibration attenuations. For such problems the optimal trajectories, actuator forces etc.…”

Section: Introductionmentioning

confidence: 99%

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Optimal gains for active vibration control by the beam analogy

Szyszkowski

Baweja

2004

Computational Mechanics

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A numerical approach to determine optimal gains for active optimal vibration control of continuous structures by the finite element method is presented. The approach uses an analogy between the optimality equations for control in the time domain and the governing equations for a set of static beams in the spatial domain. The finite element model of control is generated and analyzed in a fictitious spatial domain. The results are then transferred to the time domain to describe the optimal dynamic response of the system. The analogy (referred to as the beam analogy) allows for an efficient application of the finite element method to solve two-point-boundaryvalue problems for the finite time control cases. Here it is applied to the infinite time control cases. An algorithm for a direct calculation of optimal gains for closed loop control of time-invariant problems without using Riccati's equation is discussed in detail. IntroductionOptimal gains for active vibration attenuation are typically determined by minimizing a quadratic performance index. The corresponding Riccati equations for the problem are in the form of coupled non-linear matrix equations, difficult to solve for larger matrices [1,2]. Although several iterative algorithms have been proposed to get optimal gains from these equations, 'the solutions of the Riccati equation are the most timeconsuming part of any optimal control problem' ([1], p. 109). Besides being very intensive numerically these algorithms 'do not universally guarantee stable and accurate computation' ([2], p. 248).Potter's method was used in [3] to obtain the optimal gains for control of piezoelectric plates. In such a method the gains are determined by first solving the complex eigenvalue problem of an asymmetric matrix (referred to as Hamiltonian's matrix) formed of the matrices defining the states and the performance index (weighting matrices).A different approach, in which the gains are iteratively improved, was proposed in [4]. However, the corresponding algorithm is essentially intuitive without a proof of convergence. The approach was somewhat refined in [5].Optimal gains can also be calculated by using the general optimization techniques. In such an approach first the controls and then the states are expressed in terms of gains and substituted into the performance index. Next, treating the gains as the optimization variables, the performance is gradually improved until minimum is reached [6]. A genetic algorithm was used for that purpose in [7], while gradient-based algorithms were applied in [8-10]. The above approach is numerically very intensive. In order to facilitate its convergence, various auxiliary optimality conditions were derived and used as extra constraints while minimizing the performance. The constrained optimization process was converted into unconstrained one by using either the Lagrange multipliers [8,10], or a penalty method [9].The concept of independent modal-space control methodology was used in [11] in which the gains for each independent mode are calculat...

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“…(1)), and by eliminating the costates. The optimality equation in terms of DOFs of the problem takes the form (see [12], for details of the derivation):…”

Section: Optimal Vibration Controlmentioning

confidence: 99%

“…After some formal manipulations (see [12,13]) the optimality condition (4), in terms of DOFs can be transformed into the optimality equation in terms of the modal variables in the form:…”

Section: Optimal Vibration Controlmentioning

confidence: 99%

Section: Beam Analogymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimal gains for active vibration control by the beam analogy

Szyszkowski

Baweja

2004

Computational Mechanics

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“…The resulting de-coupled equations contain only even order derivatives and are similar to the governing equations of a set of the static beams on an elastic foundation under axial forces. This similarity constitutes the beam analogy References [9,10], in which the dynamic problem of optimal vibration control of an arbitrary structure in the time domain can be converted into the corresponding static problem of beam bending in the spatial domain. The analogy was successfully implemented to optimize a single degreeof-freedom (DOF) system in Reference [9] and to multi-DOF discrete systems in Reference [10].…”

Section: Introductionmentioning

confidence: 99%

Optimal vibration control of continuous structures by FEM: Part I—the optimality equations

Szyszkowski

Grewal

2002

Commun. Numer. Meth. Engng.

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SUMMARYThe governing equations of the problem of optimal vibration control of continuous linear structures are derived in the form of a set of fourth-order ordinary di erential equations in the time domain. The equations decouple in the modal space and become suitable for handling by the FEM technique with the time domain subdivided into 'ÿnite time' elements of class C 1 . It is demonstrated that the standard beam element with cubic Hermitian interpolation functions, routinely used in a static analysis of beams, can conveniently be substituted for the required 'ÿnite time' element.

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The FEM optimization of active vibration control in structures by solving the corresponding two-point-boundary-value problem

Szyszkowski

Baweja

2007

Struct Multidisc Optim

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Beam analogy for optimal control of linear dynamic systems

Cited by 7 publications

References 12 publications

Optimal gains for active vibration control by the beam analogy

Optimal gains for active vibration control by the beam analogy

Optimal vibration control of continuous structures by FEM: Part I—the optimality equations

The FEM optimization of active vibration control in structures by solving the corresponding two-point-boundary-value problem

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