Reliable state-space upper bounds for the peak structured singular value

A state-space method for computing upper bounds for the peak of the structured singular value over frequency for both real and complex uncertainties is presented. These bounds are based on the positivity and Popov criteria for one-sided, sector-bounded and for norm-bounded, block-structured linear uncertainty. These criteria are restated and used to derive upper bounds for the peak structured singular value by equating the feasibility of a linear matrix inequality which involves a plant state-space realization to the strict positive realness of a transfer function. Numerical examples are given to illustrate these upper bounds. © 1998 John Wiley & Sons, Ltd.     

State‐space µ analysis for an experimental drive‐by‐wire vehicle

This paper considers the application of the skewed structured singular value to the robust stability of systems subject to strictly real parametric uncertainty. Three state‐space formulations that counteract the discontinuous nature of this problem are detailed. It is shown that the calculation of the supremum of the structured singular value over a frequency range using these formulations transforms into a single skewed structured singular value calculation. Similar to the structured singular value, the exact calculation of the skewed structured singular value is an NP‐hard problem. In this work, two efficient algorithms that determine upper and lower bounds on the skewed structured singular value are presented. These algorithms are critically assessed using a series of robustness analysis tests on a safety‐critical experimental drive‐by‐wire vehicle. Copyright © 2008 John Wiley & Sons, Ltd.     

Robust stability theory using both singular value and numerical range data

An approach to robust stability analysis for block structured multiplicative plant perturbations is presented where off-diagonal block uncertainty is described by singular value bounds and diagonal block uncertainty by (mixed) singular value bounds and inclusion conditions on the numerical range.     

Robustness analysis of uncertain linear singular systems withoutput feedback control

The robustness analysis for a linear singular system with uncertain parameters and static output feedback control is considered. The problem is transformed into a robust nonsingularity problem. Based on the linear fractional transformation (LFT) approach, the robustness bounds to preserve regularity, impulse immunity, and stability are found in terms of the structured singular value μ with respect to parametric uncertainties. The LFT approach provides a unified framework for robustness analysis of both uncertain linear continuous/discrete-time singular systems     

Analysis and design for ill-conditioned plants

A study is made of a special case of the robust performance problem given by Freudenberg (1989). When the weightings used to describe the uncertainty and performance specifications vary only with frequency, then it is possible to strengthen the results in the above-mentioned work by deriving both upper and lower bounds upon the structured singular value. Both sets of bounds are stated in terms of the coupling coefficients introduced by Freudenberg (1989), and essentially yield necessary and sufficient conditions for the structured singular value to be small. This information is used to suggest a strategy for compensator design to achieve robust performance despite plant ill-conditioning. Applying this strategy to an example, it can be seen how the design trade-offs quantified by the Bode gain-phase relation manifest themselves in the robust performance problem. Finally, the design is compared with one obtained using the'µ-synthesis' approach.     

A new method for singular value loop shaping in design of multiple-channel controllers

In this note, simple symmetric interval bounds on the singular values of a matrix based on its Gershgorin disks are proposed. This allows the Gershgorin theorem to be used not only to provide information about the location of the eigenvalues of a matrix but also its singular values. This is utilized for the proposition of a new design technique for singular value loop shaping based on the diagonal dominance methodology for design of linear multivariable plants. In return, this allows multiple-channel simply structured controllers to be designed with a view to robustness and to meet constraints and specifications on the behavior of its singular values. A design example is given demonstrating the effectiveness of this approach.     

Some new properties of the structured singular value

J.C. Doyle et al. (1982) have shown that a necessary and sufficient condition for robust stability or robust performance in the H^∞-frame work may be formulated as a bound on the structured singular value (μ) of a specific matrix M which includes information on the system model, the controller, the model uncertainty, and the performance specifications. Often it is desirable to express the robust stability and performance conditions as norm bounds on transfer matrices (T) which are of direct interest to the engineer, e.g. sensitivity or complementary sensitivity. The present paper shows how to derive bounds on σ(T) from bounds on μ(M)     

Improved upper bounds for the mixed structured singular value

In this paper, we take a new look at the mixed structured singular value problem, a problem of finding important applications in robust stability analysis. Several new upper bounds are proposed using a very simple approach which we call the multiplier approach. These new bounds are convex and computable by using linear matrix inequality (LMI) techniques. We show, most importantly, that these upper bounds are actually lower bounds of a well-known upper bound which involves the so-called D-scaling (for complex perturbations) and G-scaling (for real perturbations)     

A non‐smooth lower bound on ν

A non‐smooth optimization technique to directly compute a lower bound on the skew structured singular value ν is developed. As corroborated by several real‐world challenging applications, the proposed technique can provide tighter lower bounds when compared with currently available techniques. Moreover, in many cases, the determined lower bound equals the true value of ν. Thanks to the efficiency of the non‐smooth technique, the algorithm can be applied to problems involving even a significant number of uncertain parameters. Another appealing feature of the proposed non‐smooth approach is that the dimension of repeated scalar uncertainties in the overall structured uncertainty matrix has little impact on the computational time. The technique can be used to compute a lower bound on the structured singular value μ as well. Copyright © 2012 John Wiley & Sons, Ltd.     

Characterization and efficient computation of the structured singular value

The concept of structured singular value was recently introduced by Doyle [1] as a tool for the analysis and synthesis of feedback systems with structured uncertainties. It is a key to the design of control systems under joint robustness and performance specifications and it very nicely complements theH^{infty}approach to control system design. in this paper, it is shown that the structured singular value can be obtained as the solution of several smooth optimization problems. Properties of these optimization problems are exhibited, leading to a fast algorithm that always yields the structured singular value for block structures of size no larger than 3, and often does for block structures of larger size.     

Robust performance with respect to block diagonal input uncertainty

The authors consider the robust performance problem of maintaining a small output sensitivity function despite the presence of block diagonal uncertainty at the plant input. For the case of two blocks of uncertainty, they study this problem by analyzing bounds on the structured singular value. It is shown that the potential difficulty of attaining robust performance is primarily determined by the difficulty of maintaining a small set of interaction parameters. The authors derive a robustness indicator which is useful in detecting this potential difficulty. The indicator is a function of both the plant condition number and its input directionality properties     

Real-μ bounds based on fixed shapes in the Nyquist plane: Parabolas, hyperbolas, cissoids, nephroids, and octomorphs

In this paper we introduce new bounds for the real structured singular value. The approach is based on absolute stability criteria with plant-dependent multipliers that exclude the Nyquist plot from fixed plane curve shapes containing the critical point − + _jO. Unlike half-plane and circle-based bounds the critical feature of the fixed curve bounds is their ability to differentiate between the real and imaginary components of the uncertainty. Since the plant-dependent multipliers have the same functional form at all frequencies, the resulting graphical interpretation of the absolute stability criteria are frequency independent in contrast to the frequency-dependent off-axis circles that arise in standard real-μ bounds.     

Real-μ bounds based on fixed shapes in the Nyquist plane: Parabolas, hyperbolas, cissoids, nephroids, and octomorphs

In this paper we introduce new bounds for the real structured singular value. The approach is based on absolute stability criteria with plant-dependent multipliers that exclude the Nyquist plot from fixed plane curve shapes containing the critical point − + _jO. Unlike half-plane and circle-based bounds the critical feature of the fixed curve bounds is their ability to differentiate between the real and imaginary components of the uncertainty. Since the plant-dependent multipliers have the same functional form at all frequencies, the resulting graphical interpretation of the absolute stability criteria are frequency independent in contrast to the frequency-dependent off-axis circles that arise in standard real-μ bounds.     

An implicit small gain condition and an upper bound for the real structured singular value

In this paper we develop an upper bound for the real structured singular value that has the form of an implicit small gain theorem. The implicit small gain condition involves a shifted plant whose dynamics depend upon the uncertainty set bound and, unlike prior bounds, does not require frequency-dependent scales or multipliers. Numerical results show that the implicit small gain bound compares favorably with real-μ bounds that involve frequency-dependent scales and multipliers.     

On H/sub /spl infin// model reduction using LMIs

In this note, we deal with the problem of approximating a given nth-order linear time-invariant system G by an rth-order system G/sub r/ where r相似文献   

Structured singular values with nondiagonal structures. II.Computation

In part I we introduced a robust stability measure-termed generalized structured singular value. In part II we address computational issues pertaining to this notion. Our main contribution is a computational method which would render the computation of the generalized structured singular value both more efficient and potentially more accurate. As in the computation of the usual structured singular value, the key in our method is to compute an upper norm bound scaled via a similarity transformation. It is shown that this bound is as tight as those obtained elsewhere and that it can be computed considerably more efficiently. Furthermore, it is shown that this bound is actually equal to the generalized structured singular value when the uncertainty has four blocks     

Robustness under bounded uncertainty with phase information

The authors consider uncertain linear systems where the uncertainties, in addition to being bounded, also satisfy constraints on their phase. In this context, the authors define the “phase-sensitive structured singular value” (PS-SSV) of a matrix and show that sufficient (and sometimes necessary) conditions for stability of such uncertain linear systems can be rewritten as conditions involving PS-SSV. They then derive upper bounds for PS-SSV, computable via convex optimization. They extend these results to the case where the uncertainties are structured (diagonal or block-diagonal, for instance)     

A detailed comparative analysis of all practical algorithms to compute lower bounds on the structured singular value

μ analysis is one of the most efficient techniques to evaluate the stability margins and the performance levels of linear time-invariant systems in the presence of structured time-invariant uncertainties. The exact computation of the structured singular value μ is known to be NP hard in the general case, but several methods have been developed in the last 30 years to compute accurate and reliable bounds. In this paper, all existing μ lower bound algorithms are reviewed and the most relevant ones are evaluated on a wide set of real-world benchmarks, corresponding to various fields of application, system dimensions and structures of the uncertainties. The results are thoroughly analyzed and simple improvements to the existing algorithms are proposed to approach the exact value of μ with a reasonable computation cost. Conclusions show that non-conservative values can be obtained in almost all cases. A brief extension to skew-μ analysis confirms the good results obtained in the classical μ case.     

Minimum phase robustness for uncertain state-space systems

The minimum phase robustness of an uncertain state-space system with affine parametric uncertainties in the state-space matrices is studied. A tolerable margin in terms of the structured singular value is given for uncertain parameters to guarantee the minimum phase property of the system. Based on the linear fractional transformation methodology, the matrix sizes involved in the computation of structured singular value are reduced significantly to improve computational burden. The approach can be applied to the proper or strictly proper linear uncertain systems.     

Structured singular values with nondiagonal structures. I.Characterizations

The purpose of this two-part series is to provide a robustness analysis framework for a class of problems with highly structured modeling uncertainties. This framework is more general than that of the usual block, diagonally structured uncertainties, and it corresponds to a structure consisting of block-by-block matrix perturbations. We study the structured singular value with respect to this structure, and we establish a number of novel results for this notion. This paper contains a study on the properties of the structured singular value. We give an alternative characterization of this notion as the solution of a smooth optimization problem. Furthermore, we show that under a certain circumstance the structured singular value reduces to a vector-induced matrix norm