Singular perturbation analysis of system order reduction via system balancing

Gajić, Zoran; Lelic, Muhidin

doi:10.1109/acc.2000.878615

Cited by 12 publications

(7 citation statements)

References 28 publications

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“…Clearly, our approach belongs to the second category as we use thatż 2 → 0 in the associated system (3.5). Note, however, that simply settingż 2 = 0 as is stipulated by residualization methods (see, e.g., [47,29,27]) and solving the resulting algebraic equations for z 2 is different from lettingż 2 → 0; in fact when the time scales of the two subsystems are clearly separated, the point-wise conditionż 2 = 0 of the residualization may not be very meaningful, e.g., when the fast variables oscillate infinitely fast around the stationary mean value zero in which caseż 2 = 0. However assuming a certain degree of "hyperbolicity" of the fast subsystem and suitable decay properties of the controls, it is possible to show (e.g., see [31]) that the fast variables weakly converge to an invariant measure on time scales of order one (e.g., a Gaussian with mean zero in the previous oscillatory scenario).…”

Section: 3mentioning

confidence: 99%

“…We do so by means of a multiscale analysis of the balanced equations of motion which are shown to collapse to a dimension reduced system when some of the Hankel singular values go to zero; see, e.g., [35,36] for a related approach or [22,47,59,29] in which low-rank perturbative approximations of transfer functions of linear systems are sought. To the best of our knowledge our approach is new, and, although it is based on computing an appropriate balanced form of the system equations which certainly becomes infeasible if the system is extremely high-dimensional (n ∼ 10 6 or larger), we see it not merely as an alternative, but rather as an extension to existing projectionbased methods such as Krylov subspace [54,4,15,45,46,12,10] or interpolation (moment-matching) methods [9,23,67], and empirical POD [37,44,13,58]; see also [7,60].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Balanced Averaging of Bilinear Systems with Applications to Stochastic Control

Hartmann¹,

Schäfer-Bung²,

Thöns‐Zueva³

2013

SIAM J. Control Optim.

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Abstract. We study balanced model reduction for stable bilinear systems in the limit of partly vanishing Hankel singular values. We show that the dynamics can be split into a fast and a slow subspace and prove an averaging principle for the slow dynamics. We illustrate our method with an example from stochastic control (density evolution of a dragged Brownian particle) and discuss issues of structure preservation and positivity.

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Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Balanced Averaging of Bilinear Systems with Applications to Stochastic Control

Hartmann¹,

Schäfer-Bung²,

Thöns‐Zueva³

2013

SIAM J. Control Optim.

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“…High order of the calculated controller is sometimes the implementation obstacle. In this case, techniques for model order reduction should be used as in the Henkel degree approximation method [25,26] for reduction order of controller transfer function k mix (s). After several iterations of the reduction method, the controller was reduced to fifth order while it still fulfills the criteria (15)(16)(17) k reduce (s) = 7.3×10 9 s 4 +8.6×10 10 s 3 +2.6×10 11 s 2 +4.1×10 11 s +2.2×10 11 s 5 +4.3×10 4 s 4 +1.8×10 8 s 3 +4.8×10 9 s 2 +5.5×10 10 s +5.79×10 7 .…”

Section: A Comparison Of the Proposed Synthesis Methods With The Methomentioning

confidence: 99%

“…The starting point of the synthesis of a robust controller ensures robust stability of the system according to the changes of the parameters of the model (23)(24)(25). It fulfills the set dynamic control requirements and minimizes the impact if there are external disturbances v 1 (s), v 2 (s) and v 3 (s) (18) on the system control.…”

Section: Systems Of Balance Control (Ball On Beam) (mentioning

confidence: 99%

Robust controller synthesis with consideration of performance criteria

Chowdhury¹,

Sarjaš

Cafuta

et al. 2010

Optim Control Appl Methods

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SUMMARYThis paper presents interactive controller synthesis methods based on an inward approach and offers some advantages over the traditional loop gain shaping methods. The indirect use of weighting functions for the robustness controller tuning is introduced. Besides that, the method allows transparent closed-loop pole placement to achieve desirable transient performance. The obtained controllers are of lower order for comparable performance than those synthesized with H 2 , H ∞ or -synthesis. The method is particularly well suited for robust control problems where frequency domain constraints emerge from the analyses of non-parametric uncertainties and also for control problems where the frequency domain loop shaping is used to achieve time domain performance specifications.

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“…In other words, we derive approximations of the original dynamical system for any given initial condition, whereas, e.g., standard balanced truncation yields only approximation for zero initial condition x(0) = 0. To the best of our knowledge, a systematic multiscale analysis of the equations of motion in the limit of vanishing Hankel singular values is new; see, e.g., [12,13,14] for approaches in which low-rank perturbative approximations of the transfer function are sought.…”

mentioning

confidence: 99%

Balanced Truncation of Linear Second-Order Systems: A Hamiltonian Approach

Hartmann¹,

Vulcanov²,

Schütte³

2010

Multiscale Model. Simul.

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We present a formal procedure for structure-preserving model reduction of linear second-order and Hamiltonian control problems that appear in a variety of physical contexts, e.g., vibromechanical systems or electrical circuit design. Typical balanced truncation methods that project onto the subspace of the largest Hankel singular values fail to preserve the problem's physical structure and may suffer from lack of stability. In this paper, we adopt the framework of generalized Hamiltonian systems that covers the class of relevant problems and that allows for a generalization of balanced truncation to second-order problems. It turns out that the Hamiltonian structure, stability, and passivity are preserved if the truncation is done by imposing a holonomic constraint on the system rather than standard Galerkin projection. Abstract. We present a formal procedure for structure-preserving model reduction of linear second-order and Hamiltonian control problems that appear in a variety of physical contexts, e.g., vibromechanical systems or electrical circuit design. Typical balanced truncation methods that project onto the subspace of the largest Hankel singular values fail to preserve the problem's physical structure and may suffer from lack of stability. In this paper, we adopt the framework of generalized Hamiltonian systems that covers the class of relevant problems and that allows for a generalization of balanced truncation to second-order problems. It turns out that the Hamiltonian structure, stability, and passivity are preserved if the truncation is done by imposing a holonomic constraint on the system rather than standard Galerkin projection.Key words. structure-preserving model reduction, generalized Hamiltonian systems, balanced truncation, strong confinement, invariant manifolds, Hankel norm approximation AMS subject classifications. 70Q05, 70J50, 93B11, 93C05, 93C70. DOI. 10.1137/0807327171. Introduction. Model reduction is a major issue for control, optimization, and simulation of large-scale systems. In particular, for linear time-invariant systems, balanced truncation is a well-established tool for deriving reduced (i.e., low-dimensional) models that have an input-output behavior similar to the original model [1, 2]; see also [3] and the references therein. The general idea of balanced truncation is to restrict the system onto the subspace of easily controllable and observable states which can be determined by the computing Hankel singular values associated with the system. Moreover the method is known to preserve certain properties of the original system such as stability or passivity and gives an error bound that is easily computable. One drawback of balanced truncation is that there is no straightforward generalization to second-order systems; see, e.g., [4] or the recent articles [5,6,7,8] for a discussion of various possible strategies. Second-order equations occur in modeling and control of many physical systems, e.g., electrical circuits, structural mechanics, or vibroacoustic models (see...

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Singular perturbation analysis of system order reduction via system balancing

Cited by 12 publications

References 28 publications

Balanced Averaging of Bilinear Systems with Applications to Stochastic Control

Balanced Averaging of Bilinear Systems with Applications to Stochastic Control

Robust controller synthesis with consideration of performance criteria

Balanced Truncation of Linear Second-Order Systems: A Hamiltonian Approach

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