A new upper bound for the skewed structured singular value

The structured singular value (s.s.v)μ enables the study of robust stability and performance of a controller in the presence of real parametric uncertainties and complex uncertainties corresponding to neglected dynamics. In spite of the NP-hard characteristic of the problem, it is now possible to compute an interval for the s.s.v. μ using polynomial-time algorithms. The skewed s.s.v. ν was introduced by Fan and Tits in the context of robust performance analysis. The primary aim of this paper is to propose a new mixed ν upper bound, which is applicable to problems with a special, but practically important, structure. We then illustrate through a realistic missile example that certain problems naturally require the ν tool rather than the μ tool. Copyright © 1999 John Wiley & Sons, Ltd.     

Polynomial methods for the structured singular value with real parameters

We consider the structured singular value problem with real parametric uncertainty only. Using techniques from algebraic geometry, we propose two algorithms that in principle can yield the precise value of the structured singular value at a fixed frequency. Their ability to do so depends upon their ability to find all common roots to a system of polynomial equations. The first algorithm is applicable to problems with two real parameters each of multiplicity two. The second algorithm is applicable to problems with n distinct real parameters. These algorithms have proved useful in applications to aerospace control law analysis.     

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.     

A study of the gap between the structured singular value and itsconvex upper bound for low-rank matrices

The size of the smallest structured destabilizing perturbation for a linear time-invariant system can be calculated via the structured singular value (μ). The function μ can be bounded above by the solution of a convex optimization problem, and in general there is a gap between μ and the convex bound. This paper gives an alternative characterization of μ which is used to study this gap for low-rank matrices. The low-rank characterization provides an easily computed bound which can potentially be significantly better than the standard convex bound. This is used to find new examples with larger gaps than previously known     

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.     

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.     

An upper bound for the gain of stabilizing proportional controllers

A stable linear system controlled by a proportional controller is closed-loop stable provided the controller has sufficiently small gain. If the system has an unstable zero then any proportional controller with sufficiently large gain is destabilizing.In this note we give an upper bound for the gain of stabilizing proportional controllers of stable systems that have one or more unstable zeros.     

Robust control for nonlinear systems with structured 2-gain bounded uncertainty

This paper focuses on the problem of robust control design for uncertain nonlinear systems with ₂-gain bounded dynamic uncertainty and periodically time-varying memoryless uncertainty. The robust performance problem is solved via nonlinear ^∞ control with scaling factors. It is shown that the scaling factors can be functions of state variables in contrast with linear robust control.     

Reduction of the small gain condition for large‐scale interconnections

The small gain condition is sufficient for input‐to‐state stability (ISS) of interconnected systems. However, verification of the small gain condition requires large amount of computations in the case of a large size of the system. To facilitate this procedure, we aggregate the subsystems and the gains between the subsystems that belong to certain interconnection patterns (motifs) by using three heuristic rules. These rules are based on three motifs: sequentially connected nodes, nodes connected in parallel, and almost disconnected subgraphs. Aggregation of these motifs keeps the structure of the mutual influences between the subsystems in the network. Furthermore, fulfillment of the reduced small gain condition implies ISS of the large network. Thus, such reduction allows to decrease the number of computations needed to verify the small gain condition. Finally, an ISS‐Lyapunov function for the large network can be constructed using the reduced small gain condition. Applications of these rules is illustrated on an example. Copyright © 2013 John Wiley & Sons, Ltd.     

An almost necessary and sufficient condition for robust stability of closed-loop systems with disturbance observer

The disturbance observer (DOB)-based controller has been widely employed in industrial applications due to its powerful ability to reject disturbances and compensate plant uncertainties. In spite of various successful applications, no necessary and sufficient condition for robust stability of the closed loop systems with the DOB has been reported in the literature. In this paper, we present an almost necessary and sufficient condition for robust stability when the Q-filter has a sufficiently small time constant. The proposed condition indicates that robust stabilization can be achieved against arbitrarily large (but bounded) uncertain parameters, provided that an outer-loop controller stabilizes the nominal system, and uncertain plant is of minimum phase.     

Stability analysis of nonlinear processes using empirical state affine models and LMIs

A novel methodology is proposed for the analysis of robust stability of a nonlinear process under PI (Proportional-Integral) control. The analysis is based on state-affine empirical models regressed from input-output data. The state model is represented by a set of polynomial matrices nonlinear with respect to the manipulated variables. This model in combination with a linear PI controller results in a closed loop model that can be shown to lie in a polytope of matrices. This allows for the formulation of a Lyapunov stability test in terms of a simple set of LMIs (linear matrix inequalities). This set of inequalities can be also expanded to account for input saturation. The stability analysis produces regions of stability, in terms of the PI controller parameters, that are significantly larger than the regions previously calculated by a singular value test. The issues of saturation and modeling error are also incorporated into the analysis. The technique permits also to test the stability of the closed loop system with a gain scheduling PI controller. The conservativeness of the analysis is assessed by comparison to closed loop simulations of a highly nonlinear CSTR (continuous stirred tank reactor) under PI control.     

Estimating the conservatism of Popov''s criterion for real parametric uncertainties

We consider the problems of robust stability and performance analysis of linear systems subject to parametric uncertainties. The Popov criterion provides lower bounds of stability margins and upper bounds on robust performance. In this paper we propose a method for obtaining upper bounds on stability margin or lower bounds on robust H₂ performance based on the outcome of the lower bound computation for the stability problem or the upper bound computation for the H₂ performance problem. We make several numerical tests.     

Explicit construction of quadratic lyapunov functions for the small gain,positivity, circle,and popov theorems and their application to robust stability. part II: Discrete-time theory

In a companion paper (‘Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle, and Popov theorems and their application to robust stability. Part I: Continuous-time theory’), Lyapunov functions were constructed in a unified framework to prove sufficiency in the small gain, positivity, circle, and Popov theorems. In this Part II, analogous results are developed for the discrete-time case. As in the continuous-time case, each result is based upon a suitable Riccati-like matrix equation that is used to explicitly construct a Lyapunov function that guarantees asymptotic stability of the feedback interconnection of a linear time-invariant system and a memoryless nonlinearity. Multivariable versions of the discrete-time circle and Popov criteria are obtained as extensions of known results. Each result is specialized to the linear uncertainty case and connections with robust stability for state-space systems is explored.     

An upper bound for the distance between a zero and a critical point of a solution of a second order linear differential equation

This paper presents an upper bound for the distance between a zero and a critical point of a solution of the second order linear differential equation (p(x)y^′)^′+q(x)y(x)=0, with p(x),q(x)>0. It also compares it with previous results.