Passivity analysis and passification for uncertain signal processing systems

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.

This article addresses the problem of robust stability of piecewise affine (PWA) uncertain systems with unknown time-varying delay in the state. It is assumed that the uncertainty is norm bounded and that upper bounds on the state delay and its rate of change are available. A set of linear matrix inequalities (LMIs) is derived providing sufficient conditions for the stability of the system. These conditions depend on the upper bound of the delay. The main contributions of the article are as follows. First, new delay-dependent LMI conditions are derived for the stability of PWA time-delay systems. Second, the stability conditions are extended to the case of uncertain PWA time delay systems. Numerical examples are presented to show the effectiveness of the approach.

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).

This paper develops a necessary and sufficient condition for the following relationship to exist between two given uncertain systems defined in terms of IQCs: Every input-output behaviour of the first uncertain system is an input-output behaviour of the second uncertain system.

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 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.

This paper presents a new dilated linear matrix inequality (LMI) representation to design a state feedback controller and a dynamic feedforward disturbance compensator for linear parameter-varying (LPV) systems. The improved LMIs are convex and finite-dimensional without any iterative approach. The designs are based on a new proposed equivalent bounded real lemma (BRL) by means of matrix dilation for LPV systems and uncertain linear systems under time-varying parametric uncertainties (TVPUs). This dilated BRL provides lower conservative results than existing methods in terms of robust stability. Accordingly, a dynamic disturbance compensator is designed in addition to a state feedback controller. This paper mainly focuses on the design of compensators against disturbances in addition to the design of state feedback controllers. The dynamic matrices of the compensator change with the time-varying parameters of the LPV or uncertain system during operation, assuming that the disturbances and the parameters are measurable or observable. The compensator can be designed to attenuate the disturbances/noises or to improve reference tracking. Finally, numerical and simulation outcomes are presented to prove both the effectiveness and lower conservativeness of the proposed LMIs.

We consider the sliding-mode output feedback controller (SMOFC) design problem for a class of uncertain multivariable systems. We first design a stabilizing SMOFC for matched uncertain systems. Using linear matrix inequalities (LMIs), we derive a necessary and sufficient condition for the existence of a linear sliding surface depending on outputs and compensator states. Using the solution of the LMI existence condition, we characterize the gain matrices. We give the nonlinear switching feedback gain guaranteeing the reachability condition. Second, we give an LMI-based design method to combine various useful performance criteria which can be used to guarantee a desired robust performance in spite of mismatched uncertainties. The performance criteria include alpha-stability, LQ performance, H 2/H infin performance, and peak-to-peak gain bound. In particular, we show that by including H infin performance constraints, we can easily solve the SMOFC design problem for challenging system models to which the previous methods are not easily applicable. Finally, we give a numerical design example showing that our method can be successfully applied to the problem of designing reduced-order SMOFCs for uncertain time-delay systems or mismatched uncertain systems.

This paper addresses the problem of robust stability of piecewise affine (PWA) uncertain systems with unknown time-varying delay in the state. It is assumed that the uncertainty is norm-bounded and that upper bounds on the state delay and its rate of change are available. A set of linear matrix inequalities (LMI) is derived providing sufficient conditions for the stability of the system. These conditions depend on the upper bound of the delay. The main contributions of the paper are as follows. First, new delay-dependent LMI conditions are derived for the stability of PWA time-delay systems. Second, the stability conditions are extended to the case of uncertain PWA time-delay systems. Numerical examples are presented to show the effectiveness of the approach.

This paper considers robust stabilization problem for plants with uncertainties/nonlinearities satisfying a certain class of integral quadratic constraints (IQCs). The restricted class of IQCs can still characterize many kinds of uncertainties/nonlinearities. We show that this problem reduces to the feasibility of nonconvex matrix inequalities. Owing to their special structure, they can be solved with reasonable computational effort - which is not the case with bilinear matrix inequalities (BMIs) - by combining a nonconvex optimization technique with an interior-point methodology to solve linear matrix inequalities (LMIs). Finally, an illustrative numerical example is given, where the proposed approach is applied to achieve specified gain and phase margins.

A robust stability test for a class of constrained cross-directional controllers is proposed. The cross-directional controller involves a quadratic program embedded within an internal model control structure and achieves optimal steady-state performance when the plant is fully known. Both the non-linearity in the controller and the unmodeled plant dynamics satisfy certain integral quadratic inequalities. This allows us to obtain conditions for robust stability that can be expressed as linear matrix inequalities via the KYP Lemma. Circumstances in which the results can be exploited to decompose the control design into modes are also discussed.

H In this paper we address the problems of robust stability and robust perform2 ance for uncertain discrete time systems with nonlinear parametric uncertainties. We consider two families of systems with parametric uncertainties described by state-space models which offer a fairly general representation of most uncertain systems with one or two parameters (the approach can be extended to more parameters). For these two families we obtain explicit expressions for the Schur stability H radius and for the robust performance radius in the case of uncertainties with a 2 single parameter. Moreover, we provide a line search algorithm for these two problems in the case of two parameters. Both for the robust stability and the robust performance problem, explicit necessary and sufficient conditions are derived.

The problem of guaranteed cost robust Hinf reliable control is investigated for time-delayed uncertain systems against actuator failures. In the considered system, the parameter uncertainty satisfies a generalized matching condition, and all the outputs of the actuator failures are assumed to be zero. Based on Lyapunov stability theory, a sufficient condition of the existence of guaranteed cost robust Hinf reliable controller is derived and furthermore transformed to a linear matrix inequality (LMI). At the same time, the designing approach of state feedback controller is presented and the upper bound of guaranteed cost function is given. The resultant control systems not only could retain asymptotic stability and disturbance attenuation with Hinf-norm bounds but also possess the performance index of guaranteed cost irrespective of any outages within a prespecified subset of actuators. A numerical example shows the validity of the proposed design method.

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.