An averaging result for switched DAEs with multiple modes

Proc Appl Math and Mech

et al. 2013

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Switched differential-algebraic equations (switched DAEs) E σ(t)ẋ (t) = A σ(t) x(t) are suitable for modeling many practical systems, e.g. electrical circuits. When the switching is periodic and of high frequency, the question arises whether the solutions of switched DAEs can be approximated by an average non-switching system. It is well known that for a quite general class of switched ordinary differential equations (ODEs) this is the case. For switched DAEs, due the presence of the so-called consistency projectors, it is possible that the limit of trajectories for faster and faster switching does not exist. Under certain assumptions on the consistency projectors a result concerning the averaging for switched DAEs is presented.

Section: Avaraging Results For Switched Daesmentioning

confidence: 83%

“…We are now ready to state our main result concerning averaging for switched DAEs whose proof can be found in [10].…”

Section: Avaraging Results For Switched Daesmentioning

confidence: 99%

Averaging for switched DAEs

Iannelli

Pedicini

Proc Appl Math and Mech

et al. 2013

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“…4] a straight-forward generalization of the averaging result to more than two modes is not possible, but with a different proof technique (e.g. the one used in [4] or [8]) it may be possible to also show convergence of the impulsive part for more than two modes. In the special case of commuting A diff matrices it follows that all consistency projectors commute with each other as well as with all A diff -matrices [6,Lem.…”

Section: Remark 3 (Commuting Consistency Projectors)mentioning

confidence: 99%

“…[2] or [3]) that convergence towards an average model is always guaranteed. However, due to the presence of jumps in the solutions of switched DAEs (1) convergence towards an average system does not always takes place and additional assumptions have to be made [5], [4], [9], [7], [8]. In addition to jumps, the solutions of (1) can also contain Dirac impulse (the observation that inconsistent initial values can induce Dirac impulses was already made in [17], for a recent discussion on the effect of inconsistent initial values see [16,Sec.…”

Section: Introductionmentioning

confidence: 99%

Distributional averaging of switched DAEs with two modes

2015 54th IEEE Conference on Decision and Control (CDC)

2015

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The averaging technique is a powerful tool for the analysis and control of switched systems. Recently, classical averaging results were generalized to the class of switched differential algebraic equations (switched DAEs). These results did not consider the possible Dirac impulses in the solutions of switched DAEs and it was believed that the presence of Dirac impulses does not prevent convergence towards an average model and can therefore be neglected. It turns out that the first claim (convergence) is indeed true, but nevertheless the Dirac impulses cannot be neglected, they play an important role for the resulting limit. This note first shows with a simple example how the presence of Dirac impulses effects the convergence towards an averaged model and then a formal proof of convergence in the distributional sense for switched DAEs with two modes is given.

“…An alternative approach is to directly consider the averaged model for the switched DAE (without introducing the approximation parameter ε > 0), i.e. utilizing our averaging results [2][3][4][7][8][9] for the problem of stabilization via fast switching. Assume the switching signal σ : [0, ∞) → {1, 2, .…”

Section: A Direct Averaging Approachmentioning

confidence: 99%

Stabilization of switched DAEs via fast switching

Proc Appl Math and Mech

2016

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Switched differential algebraic equations (switched DAEs) can model dynamical systems with state constraints together with sudden structural changes (switches). These switches may lead to induced jumps and can destabilize the system even in the case that each mode is stable. However, the opposite effect is also possible; in particular, the question of finding a stabilizing switching signal is of interest. Two approaches are presented how to stabilize a switched DAE via fast switching.