Realization relationships for uncertain systems with integral quadratic constraints

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.

This paper considers a problem of absolute stabilization with a specified level of disturbance attenuation for a class of uncertain systems in which the uncertainty is structured and satisfies a certain integral quadratic constraint. The paper shows that an uncertain system is absolutely stabilizable with a specified level of disturbance attenuation if and only if there exists a solution to a corresponding H/sup /spl infin// control problem. The paper also shows that if an uncertain system can be absolutely stabilized with a specified level of disturbance attenuation via nonlinear output feedback control then it can be absolutely stabilized via linear output feedback control. A key technical result used in this paper is an extension of the S-procedure result of Megretsky and Treil (1993).

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 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.

Equivalent realizations for uncertain systems with an IQC uncertainty description

A robust synthesis algorithm is developed for a class of uncertain, linear parameter varying (LPV) systems. The uncertain system is described as an interconnection of a nominal LPV system and a block structured uncertainty. The nominal part is a “gridded” LPV system with state matrices that are arbitrary functions of the parameter. The input/output behavior of the uncertainty is described by integral quadratic constraints (IQCs). The robust synthesis problem leads to a non-convex optimization. The proposed algorithm is a coordinate-wise descent similar to the well-known DK iteration for μ synthesis. It alternates between an LPV synthesis step and an IQC analysis step. Both steps can be efficiently solved as semidefinite programs. The derivation of the synthesis algorithm is less obvious for LPV systems as compared to its LTI counterpart due to the lack of a valid frequency response interpretation. The main contribution is the construction of the iterative synthesis algorithm using time domain dissipation inequalities and a scaled system analogous to that appearing in μ synthesis. It is shown that the proposed algorithm ensures that the robust performance is non-increasing at each iteration step. The effectiveness of the proposed method is demonstrated on a simple numerical example.

The paper presents a new approach to the robust H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sup> control of an uncertain system via an output feedback controller which is both stable and positive real. The uncertain systems under consideration contain structured uncertainty described by integral quadratic constraints. The controller is designed to achieve absolute stabilization with a specified level of disturbance attenuation. The main result involves solving a state feedback version of the problem by solving an algebraic Riccati equation dependent on a set of scaling parameters. Then two further algebraic Riccati equations are solved which depend on a further set of scaling parameters. The required controller is constructed from the Riccati solutions.

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.

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.

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

Stability margin of systems with mixed uncertainties under the IQC descriptions

This paper considers uncertain systems in terms of the corresponding system graph. The paper develops a necessary and sufficient condition for the graph of a given uncertain system to be contained in the graph of another uncertain system. This result also enables one to consider the equivalence between two uncertain systems. The uncertain systems under consideration are linear time-varying uncertain systems in which the uncertainty is described by a time domain integral quadratic constraint.

This paper is concerned with the robust prediction problem for a class of uncertain systems. In this problem, the noise and uncertainty are modelled deterministically via an integral quadratic constraint. The robust prediction problem involves constructing the set of all possible states at the current time t consistent with given output measurements up to a time /spl tau/<t and the integral quadratic constraint. This set is found to be an ellipsoid which is constructed by solving a Riccati differential equation.

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 paper presents an output feedback optimal guaranteed cost control result for a new class of uncertain linear systems. The cost function considered is a quadratic cost function defined over a finite time interval. The new uncertainty class introduced in the paper involves structured uncertainties which are required to satisfy a certain averaged integral quadratic constraint. The introduction of this new structured uncertainty description enables us to solve a previously unsolved problem of optimal guaranteed cost control via output feedback for an uncertain system with structured uncertainties. The solution is obtained by solving a pair of parametrized Riccati differential equations of the game type. >