Computation of the Structured Singular Value via Moment LMI Relaxations

Piga, Dario

doi:10.1109/tac.2015.2438452

Cited by 4 publications

(7 citation statements)

References 27 publications

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“…Consequently we expect to be able to improve the lower bound. The obtained result compares favorably with the approximate value 2.1007 computed in [16]. …”

Section: Numerical Testssupporting

confidence: 75%

A Novel Iterative Method To Approximate Structured Singular Values

Guglielmi¹,

Rehman²,

Kreßner³

2017

SIAM J. Matrix Anal. & Appl.

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Abstract. A novel method for approximating structured singular values (also known as µ-values) is proposed and investigated. These quantities constitute an important tool in the stability analysis of uncertain linear control systems as well as in structured eigenvalue perturbation theory. Our approach consists of an inner-outer iteration. In the outer iteration, a Newton method is used to adjust the perturbation level. The inner iteration solves a gradient system associated with an optimization problem on the manifold induced by the structure. Numerical results and comparison with the well-known Matlab function mussv, implemented in the Matlab Control Toolbox, illustrate the behavior of the method.Key words. Structured singular value, µ-value, spectral value set, block diagonal perturbations, stability radius, differential equation, low-rank matrix manifold. AMS subject classifications. 15A18, 65K051. Introduction. The structured singular value (SSV) [14] is an important and versatile tool in control, as it allows to address a central problem in the analysis and synthesis of control systems: To quantify the stability of a closed-loop linear time-invariant systems subject to structured perturbations. The class of structures addressed by the SSV is very general and allows to cover all types of parametric uncertainties that can be incorporated into the control system via real or complex linear fractional transformations. We refer to [1,3,4,8,9,10,14,17,20] and the references therein for examples and applications of the SSV.The versatility of the SSV comes at the expense of being notoriously hard, in fact NP hard [2], to compute. Algorithms used in practice thus aim at providing upper and lower bounds, often resulting in a coarse estimate of the exact value. An upper bound of the SSV provides sufficient conditions to guarantee robust stability, while a lower bound provides sufficient conditions for instability and often also allows to determine structured perturbations that destabilize the closed loop linear system.The widely used function mussv in the Matlab Control Toolbox computes an upper bound of the SSV using diagonal balancing / LMI techniques [19,5]. The lower bound is computed by a generalization of the power method developed in [18,15]. This algorithm resembles a mixture of the power methods for computing the spectral radius and the largest singular value, which is not surprising, since the SSV can be viewed as a generalization of both. When the algorithm converges, a lower bound of the SSV results and this is always an equilibrium point of the iteration. However, in contrast to the standard power method, there are, in general, several stable equilibrium points and not all of them correspond to the SSV. In turn, one cannot guarantee convergence to the exact value but only to a lower bound. We remark that, despite this drawback, mussv is a very reliable and powerful routine, which reflects the state of the art in the approximation of the SSV.

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“…Consequently we expect to be able to improve the lower bound. The obtained result compares favorably with the approximate value 2.1007 computed in [16]. …”

Section: Numerical Testssupporting

confidence: 75%

A Novel Iterative Method To Approximate Structured Singular Values

Guglielmi¹,

Rehman²,

Kreßner³

2017

SIAM J. Matrix Anal. & Appl.

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“…The worse ratio of µ/μ has been reported to be equal to 0.85 while in most cases the ratio is close to unity [14]. The lower bound μ is therefore generally close to µ [29]. For the aforementioned reasons, the lower bound μ is used as a good estimate of µ in this study.…”

Section: Theoretical Frameworkmentioning

confidence: 84%

“…However, for the sake of simplicity, a temperature coefficient of resistance (α) of 0.004 / o C as for a copper wire is assumed for the aforementioned components in this study. From (29) and Table VIII, it can be seen that the variations in temperature (T) of ±60 o C cause variations in the resistive components of ±24% around their nominal values denoted as Res o .…”

Section: A Uncertain Parametersmentioning

confidence: 99%

“…It has been reported in the literature that the method can be computationally expensive for the analysis of complex systems with a large number of uncertainties [25], [26]. However, a number of algorithms has been developed to reduce the gap between the µ bounds while maintaining reasonable computational time [25], [27], [28], [29]. Tools to optimise state space system models and model reduction methods may be employed to decrease the computational burden [30].…”

Section: Introductionmentioning

confidence: 99%

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Robust Stability Analysis of a DC/DC Buck Converter Under Multiple Parametric Uncertainties

Sumsurooah

Odavić

Bozhko

et al. 2018

IEEE Trans. Power Electron.

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Abstract-Stability studies are a crucial part of the design of power electronic systems, especially for safety critical applications. Standard methods can guarantee stability under nominal conditions but do not take into account the multiple uncertainties that are inherent in the physical system or in the system model. These uncertainties, if unaccounted for, may lead to highly optimistic or even erroneous stability margins. The structured singular value-based µ method justifiably takes into account all possible uncertainties in the system. However, the application of the µ method to power electronic systems with multiple uncertainties is not widely discussed in the literature. This work presents practical approaches to applying the µ method in the robust stability analysis of such uncertain systems. Further, it reveals the significant impact of various types of parametric uncertainties on the reliability of stability assessments of power electronic systems. This is achieved by examining the robust stability margin of the dc/dc buck converter system, when it is subject to variations in system load, line resistance, operating temperature and uncertainties in the system model. The µ predictions are supported by time domain simulation and experimental results.Index Terms-Robust stability analysis, dc/dc buck power electronic converter, Linear fractional transformation, Structured singular value, µ analysis.

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“…The following theorem (based on a proper extension of the results recently proposed by one of the authors in [41] to compute the structured singular value of a matrix) provides necessary and sufficient conditions to check determinist (robust) D-stability of the matrix A(ρ) against the uncertainty set ∆.…”

Section: Checking Determinist D-stabilitymentioning

confidence: 99%

A Unified Framework for Deterministic and Probabilistic $\mathscr {D}$-Stability Analysis of Uncertain Polynomial Matrices

Piga¹,

Benavoli²

2017

IEEE Trans. Automat. Contr.

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ForewordThis report is an extended version of the paper A unified framework for deterministic and probabilistic D-stability analysis of uncertain polynomial matrices submitted by the authors to the IEEE Transactions on Automatic Control. AbstractIn control theory, we are often interested in robust D-stability analysis, which aims at verifying if all the eigenvalues of an uncertain matrix lie in a given region D of the complex plane. Although many algorithms have been developed to provide conditions for an uncertain matrix to be robustly D-stable, the problem of computing the probability of an uncertain matrix to be D-stable is still unexplored. The goal of this paper is to fill this gap by generalizing algorithms for robust D-stability analysis in two directions. First, the only constraint on the stability region D that we impose is that its complement is a semialgebraic set described by polynomial constraints. This comprises main important cases in robust control theory. Second, the D-stability analysis problem is formulated in a probabilistic framework, by assuming that only few probabilistic information is available on the uncertain parameters, such as support and some moments. We will show how to efficiently compute the minimum probability that the matrix is D-stable by using convex relaxations based on the theory of moments. We will also show that standard robust D-stability is a particular case of the more general probabilistic D-stability problem. Application to robustness and probabilistic analysis of dynamical systems is discussed.In control theory, we are often interested in robust D-stability analysis, which aims at verifying if all the eigenvalues of an uncertain matrix lie in a given region D of the complex plane. Although many algorithms have been developed to provide conditions for an uncertain matrix to be robustly D-stable, the problem of computing the probability of an uncertain matrix to be D-stable is still unexplored. The goal of this paper is to fill this gap by generalizing algorithms for robust D-stability analysis in two directions. First, the only constraint on the stability region D that we impose is that its complement is a semialgebraic set described by polynomial constraints. This comprises main important cases in robust control theory. Second, the D-stability analysis problem is formulated in a probabilistic framework, by assuming that only few probabilistic information 1 arXiv:1604.02031v3 [math.OC] 17 Jun 2018 is available on the uncertain parameters, such as support and some moments. We will show how to efficiently compute the minimum probability that the matrix is D-stable by using convex relaxations based on the theory of moments. We will also show that standard robust D-stability is a particular case of the more general probabilistic D-stability problem. Application to robustness and probabilistic analysis of dynamical systems is discussed.To this end, we develop a unified framework for deterministic (robust) and probabilistic D-stability analysis. A semi-infinite linear progr...

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Computation of the Structured Singular Value via Moment LMI Relaxations

Cited by 4 publications

References 27 publications

A Novel Iterative Method To Approximate Structured Singular Values

A Novel Iterative Method To Approximate Structured Singular Values

Robust Stability Analysis of a DC/DC Buck Converter Under Multiple Parametric Uncertainties

A Unified Framework for Deterministic and Probabilistic $\mathscr {D}$-Stability Analysis of Uncertain Polynomial Matrices

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