Robust Stability Analysis of Sparsely Interconnected Uncertain Systems

Time delay is prejudicial to power system stability and generally worsens the performance of control system. In this paper, the approaches for stability analysis and stabilization of a power system consisting of time-varying delay are established. The paper starts with modeling of the power system according to time-delay differential algebraic equations. An Integral Quadratic Constraints (IQC) method is constructed to guarantee the stability of proposed feedback controller. The frequency-domain IQC and time-domain dissipation inequality are effectively combined to perform stability analysis and stabilization of power system considering time delays, without complex numerical computation. A 2-area 4-machine system, an IEEE ten-machine thirty nine-bus system and a WECC 179-bus system are employed to support the stability criteria and IQC method. The simulation results indicate that the combined methods effectively damp the oscillation and improve the performance.

The author analyses results obtained by Megretski and Rantzer (ibid., vol.42, p.819-30, 1997) on the stability analysis of nonlinear systems. He establishes the connections and relations between their results and the abstract quadratic criterion developed by Yakubovich (Automation Remote Control, p.5-30, 1967).

The problem of passivity analysis finds important applications in many signal processing systems such as digital quantizers, decision feedback equalizers and digital and analog filters. This paper considers the passivity analysis problem for a large class of systems which involve uncertain parameters, time delays, quantization errors, and unmodeled high order dynamics. By characterizing these and many other types of uncertainty using a general tool called integral quadratic constraints (IQCs), we present a solution to the problem of robust passivity analysis. More specifically, we determine if a given uncertain system is robustly passive. The solution is given in terms of the feasibility of a linear matrix inequality (LMI) which can be solved efficiently.

The goal of this paper is to assess the robustness of an uncertain linear time-varying (LTV) system on a finite time horizon. The uncertain system is modeled as a connection of a known LTV system and a perturbation. The state matrices of the LTV system are assumed to be rational functions of time. This is used to model the uncertain LTV system as an connection of a time invariant system and an augmented perturbation that includes time. The input/output behavior of the perturbation is described by time-domain, integral quadratic constraints (IQCs). Static and dynamic IQCs are developed for the multiplication by time. A sufficient condition to bound the induced L2 gain is formulated using dissipation inequalities and IQCs. The approach is demonstrated with two simple examples.

IQC arguments for analysis of the 3-state Moore-Greitzer compressor system

Studies the description, by means of integral quadratic constraints (IQCs), of a class of static diagonal operators with equal diagonal entries. In particular, the diagonal entries considered are monotonic and slope-restricted nonlinearities. First, we give new IQCs for this type of operators which exploit the equality among nonlinearities. Then we show how these results can be used to derive new stability conditions for systems containing repeated nonlinearities. A simple example shows that the new description can substantially reduce conservativeness, with respect to previous existing techniques, in the analysis of systems with diagonal operators.

The problem of passivity analysis finds important applications in many signal processing systems such as digital quantizers, decision feedback equalizers, and digital and analog filters. Equally important is the problem of passification, where a compensator needs to be designed for a given system to become passive. This paper considers these two problems for a large class of systems that involve uncertain parameters, time delays, quantization errors, and unmodeled high-order dynamics. By characterizing these and many other types of uncertainty using a general tool called integral quadratic constraints (IQCs), we present solutions to the problems of robust passivity analysis and robust passification. More specifically, for the analysis problem, we determine if a given uncertain system is passive for all admissible uncertainty satisfying the IQCs. Similarly, for the problem of robust passification, we are concerned with finding a loop transformation such that a particular part of the uncertain signal processing system becomes passive for all admissible uncertainty. The solutions are given in terms of the feasibility of one or more linear matrix inequalities (LMIs), which can be solved efficiently.

The stability and performance of nonlinear and linear parameter varying (LPV) time delayed systems are analyzed. First, the input/output behavior of the time delay operator is bounded in the frequency domain by integral quadratic constraints (IQCs). A simple geometric interpretation is used to derive new IQCs for both constant and varying delays. Second, the performance of nonlinear and LPV delayed systems is bounded using dissipation inequalities that incorporate IQCs. The nonlinear or LPV part of the system is treated directly in the analysis and not bounded by IQCs. This step makes use of recent results that show, under mild technical conditions, that an IQC has an equivalent representation as a finite-horizon time-domain constraint. A numerical example with a nonlinear delayed system is provided to demonstrate the effectiveness of the method.

We consider the problemSeeSeeintegral quadratic constraint SeeSeelower hemicontinuous SeeSeefinely lower hemicontinuous SeeSeeweakly lower hemicontinuous SeeSeelower hemicontinuous SeeSeefinely lower hemicontinuous SeeSeeweakly lower hemicontinuous SeeSeeinput/output system of determining the input/output stability of the feedback interconnection of two systems. Dissipativity and graph separation techniques are two related and popular approaches to this problem, and they include well-known passivity and small-gain methods. The use of block diagram transformations with dynamic multipliers can greatly reduce the conservativeness of such approaches, but for the stability of the transformed system to imply that of the original one, these multipliers should admit appropriate factorizations. An alternative approach which circumvents the need to factorize multipliers was provided by Megretski and Rantzer in their seminal 1997 paper on integral quadratic constraints. Their approach is based on homotopy: one constructs a continuous transformation of a trivially stable system into the target system of interest, and by satisfying certain conditions along the homotopy path one guarantees that the target system is also stable. This method assumes that the feedback interconnection is well-posed along the homotopy path, namely, that the feedback equations have solutions for all possible exogenous inputs and that the mapping from these inputs to the solutions is causal. In this chapter we will explore the role of well-posedness in this homotopy method. In so doing we demonstrate that what suffices for the homotopy analysis is a property significantly weaker than well-posedness, one which involves a certain lower hemicontinuity of the feedback interconnection along with a certain controllability of its domain. Moreover, we show that these methods can be applied to general signal spaces, including extended Sobolev spaces, spaces of smooth functions, and spaces of distributions.

An IQC-based stability criterion for systems with slowly varying parameters

SummaryThis paper provides a comprehensive treatment of full‐block multipliers within the integral quadratic constraints framework for stability analysis of feedback systems containing repeated, slope‐restricted scalar nonlinearities. We develop a novel stability result that offers more flexibility in its application because it allows for the inclusion of general Popov and Yakubovich criteria in combination with the well‐established Circle and Zames‐Falb stability tests within integral quadratic constraint theory. A particular focus lies on the formulation of stability criteria in terms of full‐block multipliers, some of which are new, and thus typically involve less conservatism than current methods. Furthermore, a new asymptotically exact parametrization of full‐block Zames‐Falb multipliers is given that allows to exploit the complete potential of this stability test. Copyright © 2017 John Wiley & Sons, Ltd.

A feedback interconnection of a neutrally stable, linear time-invariant system and a nonlinearity with 0/spl les/x/spl phi/(x)/spl les/kx/sup 2/ is called critical because the worst case linearization is at best neutrally stable. This characteristic makes the stability analysis of such systems particularly hard. It is shown that an integrator and a sector bounded nonlinearity can be encapsulated in a bounded operator that satisfies several useful integral quadratic constraints, which gives powerful tools for stability analysis of a general class of critically stable systems.

Enhanced robust stability analysis of large hydraulic control systems via a bifurcation-based procedure

We have derived the largest class of bounded linear time-invariant operators that preserve positivity of repeated monotone nonlinearities. It reduces conservativeness of integral quadratic constraint (IQC) stability analysis of systems having repeated monotone nonlinearities to the minimum possible. Utility of this characterization stems from the fact that such nonlinearities are typically encountered in stability analysis of anti windup schemes among other applications. We have also derived the time varying counterpart of this class of operators.

In this paper, the problem of robust performance analysis of interconnected uncertain systems with hierarchical structure is investigated. The computational load associated with such problems does not allow a direct application of robustness analysis usual tools. To overcome this difficulty, we exploit the hierarchical structure of the problem and propose an algorithm to perform robustness analysis using integral quadratic constraint “propagation” along the hierarchical structure. This algorithm allows us to establish a tradeoff between computation time required to perform the analysis and the conservatism of the obtained results. Furthermore, it is possible to perform parallel computation using the proposed algorithm.