Assessing the stability of linear time-invariant continuous interval dynamic systems

Kolev, Lubomir V.; Petrakieva, Simona

doi:10.1109/tac.2005.843857

Cited by 40 publications

(79 citation statements)

References 6 publications

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“…The first four refer to the case of the interval matrix description (5). As has been predicted analytically in Section 3.4, the examples provided show that the present method outperforms a similar method of polynomial numerical complexity [6] regarding both its numerical efficiency (reduced volume of computations) and its domain of applicability (capacity to solve problems of increased uncertainty intervals). It should, however, be borne in mind that the present implementation of the new method is based on full-matrix operations, limiting its application to the problems of moderate size.…”

supporting

confidence: 65%

“…Analogous results are obtained for t = 0.13 (Table VII). As is seen from the above three tables, the present method requires about half the time needed by the previous method from [6].…”

Section: Examplementioning

confidence: 76%

“…We now compare the numerical efficiency of the new method M1 with that of another method of polynomial complexity suggested in [6]. As mentioned in Section 1, the method of [6] …”

Section: Comparison With Another Methodsmentioning

confidence: 99%

“…These methods referred to as exact methods [5,6] are based on the use of the so-called outer solutions [7,24,25] …”

Section: Remarkmentioning

confidence: 99%

“…Since Procedure 1 is applicable, the eigenvalue considered is guaranteed to be simple and, hence, structurally stable [6] so Let x = (x 1 , . .…”

Section: Real Eigenvaluesmentioning

confidence: 99%

See 4 more Smart Citations

Eigenvalue range determination for interval and parametric matrices

Kolev

2010

Circuit Theory & Apps

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SUMMARYStudy of the dynamic behaviour of linear limped parameter circuits or systems under parametric uncertainties is a well-established research area. The present paper addresses the problem of determining the exact (within rounding errors) ranges for the real and imaginary parts of an eigenvalue of real matrices whose elements are either independent intervals or linear (affine) functions of independent interval parameters. A unified method for solving the above problem is suggested in the paper. It is iterative and reduces to forming and solving, at each iteration, two real (incomplete) quadratic systems to find outer interval bounds on the set of the right and left eigenvectors associated with the eigenvalue considered. The method is applicable if: (i) the solutions of both quadratic systems involved are positive and (ii) the outer interval bounds on the eigenvectors satisfy certain constant sign conditions. It is shown that its numerical complexity is polynomial in the size n of the matrix studied. Numerical examples illustrating the applicability of the new method are provided.

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supporting

confidence: 65%

“…Analogous results are obtained for t = 0.13 (Table VII). As is seen from the above three tables, the present method requires about half the time needed by the previous method from [6].…”

Section: Examplementioning

confidence: 76%

“…We now compare the numerical efficiency of the new method M1 with that of another method of polynomial complexity suggested in [6]. As mentioned in Section 1, the method of [6] …”

Section: Comparison With Another Methodsmentioning

confidence: 99%

“…These methods referred to as exact methods [5,6] are based on the use of the so-called outer solutions [7,24,25] …”

Section: Remarkmentioning

confidence: 99%

“…Since Procedure 1 is applicable, the eigenvalue considered is guaranteed to be simple and, hence, structurally stable [6] so Let x = (x 1 , . .…”

Section: Real Eigenvaluesmentioning

confidence: 99%

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Eigenvalue range determination for interval and parametric matrices

Kolev

2010

Circuit Theory & Apps

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Robust and guaranteed output‐feedback force control of piezoelectric actuator under temperature variation and input constraints

Hammouche

Lutz

Rakotondrabe

2020

Asian Journal of Control

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This paper addresses the control of manipulation force in a piezoelectric tube actuator (piezotube) subjected to temperature variation and input constrains. To handle this problem a robust output-feedback design is proposed using an interval state-space model, which permits consideration of the parameter uncertainties caused by temperature variation. The design method is robust in the sense that the eigenvalues of the interval system are designed to be clustered inside desired regions. For that, an algorithm based on Set Inversion Via Interval Analysis (SIVIA) combined with interval eigenvalues computation is proposed. This recursive SIVIA-based algorithm allows to approximate with subpaving the set solutions of the feedback gain [K] that satisfy the inclusion of the eigenvalues of the closed-loop system in the desired region, while at the same time ensuring the control inputs amplitude is bounded by specified saturation. The effectiveness of the control strategy is illustrated by experiments on a real piezotube of which the environmental temperature is varied. KEYWORDS input constraint, interval models, piezoelectric tube actuator, robust output-feedback, set inversion via interval analysis 1 INTRODUCTION Piezoelectric actuators, such as a piezoelectric tube and piezoelectric multimorph cantilever, are among the most used actuator in micro/nano-scales applications, particularly in micro/nano manipulation, Scanning Probe Microscopy (SPM), and Atomic Force Microscopy (AFM) due to their high speed (large bandwidth up to 1kHz), high precision (sub-nanometric), high resolution, and multi-degrees of freedom [1-6]. Unfortunately, they are characterized by nonlinearities (hysteresis, time varying parameters, creep, etc). They are also sensitive to the environment and especially to ambient temperature variation [7]. Actually, there are several sources that may cause this thermal variation during experimentation: the lamps used to illuminate the tasks at the microscale and related cameras, the heating of the surrounding devices (voltage amplifiers …), and all other natural sources. This temperature variation considerably impacts the approximated model of the actuator and induces the change in its dynamics and its steady-state behavior. Furthermore, in micro/nano manipulation, the manipulated object is usually so fragile that if the desired performance (overshoot and rapidity) is not sufficiently respected under

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Determining the stability margin in linear interval parameter electric circuits via a DAE model

Kolev

2011

Circuit Theory & Apps

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SUMMARYThis paper addresses the problem of determining the stability margin M s in linear time-invariant electric circuits whose parameters R, L, C, M and controlled source coefficients are allowed to take on fixed but unknown values within corresponding preset intervals. A method for solving the above robust stability problem is suggested. Unlike previous results resorting to the state-space formulation, the new method is based on an appropriate description of the circuit considered using a specific model in semi-state form. The set of differential algebraic equations (DAEs) thus obtained contain the circuit parameters in a linear manner. The latter property permits M s to be determined in an efficient way using the right endpoint of the range related to the real part of the first (closest to the imaginary axis) eigenvalue of a corresponding interval generalized eigenvalue problem. The new method is shown to have polynomial numerical complexity with respect to the size of the generalized eigenvalue problem. Two numerical examples illustrating the new DAE approach are provided.

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Assessing the stability of linear time-invariant continuous interval dynamic systems

Cited by 40 publications

References 6 publications

Eigenvalue range determination for interval and parametric matrices

Eigenvalue range determination for interval and parametric matrices

Robust and guaranteed output‐feedback force control of piezoelectric actuator under temperature variation and input constraints

Determining the stability margin in linear interval parameter electric circuits via a DAE model

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