Improvement of system order reduction via balancing using the method of singular perturbations

Gajić, Zoran; Lelic, Muhidin

doi:10.1016/s0005-1098(01)00139-x

Cited by 24 publications

(5 citation statements)

References 19 publications

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“…An additional term is added to the transfer function of the fast subsystem as seen in the new scheme depicted in Figure 8. It was shown in [4] that such a corrected DC gain approach works well for reduced-order models obtained via balancing [4], [13]. The corrected response of the example considered earlier is shown in Figure 9.…”

Section: Discussion On the Singular Perturbation Approach For Order R...mentioning

confidence: 86%

Remarks on Time-Scale Decomposition Using Singular Perturbations With Applications

Kodra

Zhong

Gajić³

2020

AnnalsARSciMath

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In this paper, we point out important observations on time-scale decomposition of linear singularly perturbed systems. It has been established in the control literature that the asymptotically stable fast modes of a singularly perturbed system decay rapidly in a boundary layer interval when the perturbation parameter is very small hence the slow subsystem can serve as a good approximation of the original model. We observe that while this is the case in the steady state, it is not true during the transient response for a strictly proper system with highly damped and highly oscillatory modes. Instead, the fast subsystem provides a very good approximation of the original model’s response but with a DC gain offset. We propose a correction to rectify the DC gain offset and illustrate the findings using an islanded microgrid electric power system model.

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Section: Discussion On the Singular Perturbation Approach For Order R...mentioning

confidence: 86%

Remarks on Time-Scale Decomposition Using Singular Perturbations With Applications

Kodra

Zhong

Gajić³

2020

AnnalsARSciMath

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“…The previous strategy plus some extra (technical) interconnection conditions guarantee that the origin (x, z) = (0, 0) ∈ R ns × R n f is an asymptotically stable equilibrium point of the closed-loop system ( 6) for ε > 0 but sufficiently small [14,24,25,33]. This feature of normal hyperbolicity has been exploited in many applications, a few examples are [10,34,38,43,51,52,54,55].…”

Section: Composite Control Of Slow-fast Control Systemsmentioning

confidence: 99%

Stabilization of a class of slow–fast control systems at non-hyperbolic points

2019

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In this paper we study the stabilization problem of a general class of slow-fast systems with one fast and arbitrarily many slow states. Moreover, the class of systems under study is slowly actuated, meaning that only the slow states are subject to the action of a controller. Furthermore, we are particularly interested in the case where normal hyperbolicity is lost. We show that by using the Geometric Desingularization method, it is possible to design controllers to locally stabilize non-hyperbolic points of any finite degeneracy. The main novelty of this paper is that, unlike previous research on the topic, we make use of more than one chart of the blow up space to enhance the region of attraction of the operating point. A couple of numerical examples highlight our contribution.

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“…Balancing singularly perturbed linear systems has been considered in several research papers [18–24]. Due to the structure of the singularly perturbed matrices (15) and (16), the controllability and observability Gramians are partitioned as follows [25]

P false(ε false) = (][, \begin{matrix} P_{1} false(ε false) & P_{2} false(ε false) \\ P_{2}^{T} false(ε false) & \frac{1}{ε} P_{3} false(ε false) \end{matrix}), Q false(ε false) = (][, \begin{matrix} Q_{1} false(ε false) & ε Q_{2} false(ε false) \\ ε Q_{2}^{T} false(ε false) & ε Q_{3} false(ε false) \end{matrix})

To simplify notation, in the rest of the paper we have omitted explicit dependency of

P_{j}

j = 1, 2, 3

ε

.…”

Section: Balancing Singularly Perturbed Systemsmentioning

confidence: 99%

“…Its system matrix eigenvalues indicate that this system has three slow and five fast state variables. The eigenvalues are given by

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ \begin{aligned} λ_{FC} = & falsefalse{- 1.4038, - 1.6473, - 2.9151, - 18.2582, \\ - 22.4040, - 46.1768, - 89.4853, - 219.6262 falsefalse} \end{aligned} \end{matrix}

Since the small singular perturbation parameter is not exactly known, it can be evaluated as the ratio of the fastest eigenvalue of the slow cluster with the slowest eigenvalue of the fast cluster, namely

ε = 2.9151 / 18.2582 = 0.157

[24, 30]. The system, input, and output matrices are obtained by permuting rows and columns of the original mathematical model matrices from [29] in order to get the explicit singularly perturbed form defined in (15) and (16).…”

Section: Balancing Singularly Perturbed Systemsmentioning

confidence: 99%

“…The idea was further developed in [18] where it was shown that direct truncation of a balanced system and the slow singular perturbation approximation both yield minimal, stable, and balanced reduced‐order systems. Recent research involving combination of both methods has focused on order reduction in large‐scale systems [19], approximation of bounded balanced and stochastically balanced transfer matrices [20], balancing of non‐linear singularly perturbed systems [21, 22], and the effects that the balancing realisation has on singular perturbations [23, 24].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Finding Hankel singular values for singularly perturbed linear continuous‐time systems

Kodra¹,

Skataric

Gajić³

2017

IET Control Theory & Applications

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Improvement of system order reduction via balancing using the method of singular perturbations

Cited by 24 publications

References 19 publications

Remarks on Time-Scale Decomposition Using Singular Perturbations With Applications

Remarks on Time-Scale Decomposition Using Singular Perturbations With Applications

Stabilization of a class of slow–fast control systems at non-hyperbolic points

Finding Hankel singular values for singularly perturbed linear continuous‐time systems

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