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)     

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.     

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.     

Bounds on generalized structured singular values via the Perron root of matrix majorants

The purpose of this paper is to present several bounds upon the structured singular value. We first adopt a generalized notion of the structured singular value which is useful for problems where uncertainties are assumed to be bounded in an l_p-induced matrix norm. Two different type of bounds, in terms of Perron root and interaction parameters respectively, are given for the new structured singular value and their relations are discussed. These bounds are useful in that they are easy to compute and may be further analyzed to provide insights useful in design.     

Bounds for uncertain matrix root-clustering in a union of subregions

The research for robustness bounds for systems whose behaviour is described by a linear state-space model is addressed. The paper lays stress on the location of the eigenvalues of the state matrix when this matrix is subject either to an unstructured additive uncertainty or to a structured additive uncertainty. In the first case, upper bounds on the spectral norm of the uncertainty matrix are determined whereas in the second case, upper bounds on the maximal real perturbation in the state matrix are derived. In both cases, the fact that these bounds are not exceeded ensures that the eigenvalues of the uncertain state matrix lie in a specified region 𝒟 of the complex plane in which those of the nominal state matrix already lie. These bounds are obtained through a linear matrix inequalities approach. This approach allows to specify 𝒟, not only as a simple convex region, symmetric with respect to the real axis, but also as a non-convex (but symmetric with respect to the real axis) region defined itself as a union of convex subregions, each of them being not necessarily symmetric with respect to the real axis. Copyright © 1999 John Wiley & Sons, Ltd.     

Stability robustness bounds for linear state-space models with structured uncertainty

In this note, we consider the robust stability analysis problem in linear state-space models. We consider systems with structured uncertainty. Some lower bounds on allowable perturbations which maintain the stability of a nominally stable system are derived. These bounds are shown to be less conservative than the existing ones.     

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     

Parameter bounds for root clustering of uncertain matrices

This paper deals with robust root-clustering in a region of order one or two of the complex plane. We consider continuous or discrete linear state-space models with unstructured and structured perturbations. Some bounds on allowable perturbations that maintain the eigenvalues of a system matrix in a desired region are derived. These bounds are shown to be less conservative than existing ones.     

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)     

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.     

Application of real/mixed μ computational techniques to an H∞ missile autopilot

A multivariable missile autopilot is synthesized using an H_∞ approach. A tradeoff is achieved between performance, actuators solicitation and uncertainties in the actuators and bending modes dynamics. Robust stability and performance of the control law are then studied in the face of large real parametric aerodynamic uncertainties: computational techniques for real and mixed μ analysis (namely De Gaston and Safonov's, Dailey's, Jones’, Young and Doyle's, Fan, Tits and Doyle's and Safonov and Lee's methods) are briefly reviewed before being used to compute either the exact value, or an interval of the structured singular value (SSV). For small amounts of parameters, the upper and lower bounds provided by these methods are compared to the exact value, computed by De Gaston and Safonov's method. For larger amounts of parameters, NP hardness of the problem prohibits the use of algorithms which compute the exact value: these algorithms are indeed necessarily exponential-time. As an alternative in this case, the use of polynomial-time methods for computing upper and lower bounds leads in our examples to accurate approximates of the real and mixed structured singular values.     

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.     

Time response bounds for linear parametric uncertain systems

Parametric uncertainties in linear time-invariant state-space models introduce perturbations in the eigenvalues and eigenvectors (eigenstructure), which modify the properties of a nominal system. Besides stability by the perturbed eigenvalues, controllability, observability, transient and steady-state values, etc., are some basic properties of a linear system significantly influenced by the perturbed eigenvectors. In this paper, time response analysis of parametric uncertain systems is considered. First the bounds on eigenvectors and parametric uncertainties are established for the perturbed eigenvalues to reside in a set of distinct circular regions. Secondly, analysis of such parametric uncertain systems is discussed by presenting bounds on the perturbed state variables in time coordinate. These bounds are then investigated for norm-bounded and structured uncertainty classifications. Input-output analysis of uncertain systems is illustrated using the initial conditions and input functions.     

Interpolation for nonlinear BVP in circular membrane with known upper and lower solutions

A successive nonextrapolatory linear interpolation is described to solve a singular two-point boundary value problem arising in circular membrane theory. The problem is associated with a second-order nonlinear ordinary differential equation for which the upper and lower bounds of the solution is analytically established/known. The importance and the scope of these bounds in solving the problem is stressed. Also depicted graphically are the lower and upper solutions as well as the true and iterated solutions. In addition, discussed are the reasons why linear interpolation, and not nonlinear interpolation or bisection which are possible procedures, has been employed.     

Perturbation bounds for root-clustering of linear discrete-time systems

Sufficient conditions are obtained for the eigenvalues of a discrete-time system to remain in a specified subregion in the unit circle with the centre at the origin in the complex plane in the presence of perturbations. The perturbations are allowed to be non-linear and time varying. Bounds are determined for linear state-space models using the Lyapunov theory. The bounds are derived for both highly and weakly structured perturbations. The results of the structured perturbations are then extended to interval matrices.     

Stability robustness bounds for discrete-time linear regulators

Stability robustness analysis and design for linear multivariable discrete-time systems with bounded uncertainties are discussed. Robust stability of the full-state feedback linear quadratic (LQ) regulator in the presence of perturbations (modelling errors) of the system matrices is investigated. These results are based on a recently developed bound on elemental (structured) time-varying perturbations of an asymptotically stable linear time-invariant discrete-time system. Lyapunov theory and singular value decomposition techniques are employed in deriving these bounds. Extensions of these results to linear stochastic systems with the Kalman filter as the stale estimator (LQG regulators) and to reduced-order dynamic compensator feedback are described. A state feedback control design method is presented for LQ regulators, using a quantitative measure called the Stability Robustness Index. Simple examples illustrate these new results.     

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.     

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.