Passivity Theorem Research Articles

Dissipative controller designs for second-order dynamic systems

The passivity theorem may be used to design robust controllers for structures with positive transfer functions. This paper extends this result to more general configurations using dissipative system theory. A stability theorem for robust control of structures with displacement sensors is given and applied to several common classes of controllers.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stability conditions for systems with parametric uncertainties and nonlinear feedback

The purpose of this paper is to use the passivity theorem to derive new stability conditions for a class of systems that consist of a linear subsystem with parametric uncertainty and nonlinear feedback. To address the inherent conservativism in applying the passivity theorem, causal multipliers are introduced into the analysis and it is shown that a Popov-like stability condition results. The stability results are then extended for noncausal multipliers using the notion of positivity. >

Transient bounds for adaptive control systems

Adaptive control systems are essentially nonlinear and mechanisms to analyze their stability and transient response typically derive from more general nonlinear theories such as small gain arguments or passivity. In this note, we consider how these theories may be applied to adaptive control to quantify transient response. This involves the explicit description of the constants appearing in the passivity theorem and/or the small gain theorem which characterize both system gains and initial condition effects. The result is a fundamental connection between transient response bounds and uniform plant controllability, which connects initial state conditions with the input-output analysis. Applying these general theorems to adaptive control, we are able to interpret the uniform controllability condition as a persistency of excitation requirement and thereby to provide local bounds on transient response. The implication of these sufficient results is that without both bounds upon initial conditions and guarantees of excitation, potentially extreme transient excursions of system variables are possible even though global convergence and asymptotic performance are guaranteed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

New relationships between Lyapunov functions and the passivity theorem

AbstractIn this paper we study new relationships between a class of Lyapunov functions and the passivity theorem. It is proved that under some (sufficient) conditions a Lyapunov‐stable system can be analysed as the feedback connection of two (strictly) passive subsystems. It is also shown that very recent adaptive schemes for linear plants of any relative degree can in a certain sense be unified through a passivity point of view.

Passive Control of a Stiff Flexible Link

It is demonstrated in the article that, for an appropriately chosen output variable, the transfer function for a sufficiently rigid single flexible link is passive. As such, by the passivity theorem, using any strictly passive controller with finite gain will result in an L2 -stable system. One approach to designing such a controller is outlined, and the experimental results are presented. The implications of this work are examined, and future research issues are discussed.

ROBUST STABILITY ANALYSIS OF GENERIC MODEL CONTROL

In this paper, the robust stability of Generic Model Control (GMC) is analyzed under the condition that the explicit control law is available. This anslysis is performed by finding a strict Lyapunov function for the nominal process and applying a perturbation theorem. Based on the passivity theorem, a procedure to synthesize a robust stable GMC controller is proposed for a given set of processes. The significance of this approach is discussed as well as its disadvantages.

Asymptotically stable set point control laws for flexible robots

A general family of asymptotically stabilizing control laws is introduced for a class of nonlinear Hamiltonian systems. The inherent passivity property of this class of systems and the Passivity Theorem are used to show the closed loop input/output stability which is then related to the internal state space stability through an observability condition. Applications of these results include fully actuated robots, flexible-joint robots, and robots with link flexibility.

Nonlinear repetitive control with application to trajectory control of manipulators

AbstractIn this article, a new control scheme named repetitive control is proposed for a class of nonlinear systems described by x(t) = Ax(t) + Bu(t) + n(x(t)) and y(t) = Cx(t), in which the controlled variables follow periodic reference commands. The stability condition is derived by applying the passivity theorem. We show how to apply the repetitive control scheme to the trajectory control of a manipulator. A simple repetitive control scheme is developed for the trajectory control of a manipulator by using nonlinear compensation and feedbacks of position and velocity signals. Experimental results for a three link manipulator verify that the proposed repetitive control reduces the tracking error to a very low level.

Robustness of discrete-time direct adaptive controllers

The problem of preserving stability of discrete-time adaptive controllers in spite of reduced-order modeling and output disturbances is addressed in this paper. Conditions for global stability (convergence of the tracking error with bounded signals) are derived for a discrete-time pole-zero placement adaptive controller where the parameter estimator is modified in terms of normalized signals. Following an input-output perpective, the overall system is decomposed into two subsystems reflecting the parameter estimation and modeling errors, respectively, and its stability is studied using the sector stability and passivity theorems. First the analysis is carried for the class of disturbances and reference inputs that are either decaying or can be exactly hulled by a linear controller of the chosen structure. In this L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> -framework, it is shown that the only substantive assumption to assure stability is the existence of a linear controller such that the closed-loop transfer function verifies certain conicity conditions. The convergence speed and alertness properties of various parameter adaptation algorithms regarding this condition are discussed. The results are further extended to a broader class of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_{\infty}</tex> disturbances and reference inputs.

Off-axis multivariable circle stability criterion

In 1968, the scalar circle stability criterion was extended by Cho and Narendra to the off-axis case by the use of multipliers Q, Q−1 which have an RL or RC realise structure, so that the product of the linear portion of the feedback system and the multiplier was positive-real. Fabl, Freedman and Zames produced a multivariable on-axis circle criterion for systems whose linear part is normal for all frequencies. As yet the off-axis circle criterion has not yet been established for the multivariable case, although Cook (1976) has shown that a criterion of this type does provide the conditions for the absence of limit cycles. Utilising the loop transformation theorem and the passivity theorem, an off-axis multivariable circle stability criterion is established via the method of multipliers for nonlinear feedback systems with a normal linear operator. The criterion is shown to have a simple graphical interpretation based on the Nyquist plots of the eigenvalues of the system.

&amp;lt;tex&amp;gt;L_2&amp;lt;/tex&amp;gt;-stability of interconnected systems using a reformulation of the passivity theorem

In this paper, we give a reformulation of the well-known passivity theorem for single-loop systems, which removes the asymmetry between the conditions on the forward and feedback subsystems. This reformulation leads to stability as well as instability criteria for interconnected systems, and these are shown to contain known results as special cases.

The small-gain theorem, the passivity theorem and their equivalence

Two small-gain theorems and a passivity theorem yielding stability results for general feedback systems are shown to be equivalent.

On passivity of a class of systems with time-varying non-linearities†

In this paper a generalized definition of passivity is introduced. Sufficient conditions of passivity for a certain class of time-varying non-linearities associating with various RL, RC and RLC multipliers are presented. It is pointed out that these passivity results can be used in conjunction with a Popov-type frequency domain condition to obtain many very general stability criteria of non-linear time-varying feedback systems in a systematic and unified manner by means of the passivity theorem.

Stability criterion for a feedback system with backlash--An extension of frequency-domain condition by multiplier method

A stability criterion for a feedback system with backlash is obtained by applying the passivity theorem of Zames in the derivative domain. The result is a generalization of the well-known criterion of the Popov type.

L2_stability of non-linear time-varying systems †

This paper presents two stability criteria for a very broad class of non-linear time-varying as well as time-invariant non-linear feedback systems. The stability criteria are obtained by the use of the passivity theorem and the concept of multipliers in a more systematic, straightforward and unified manner. The results obtained generalize many previously published ones in several aspects.

Input-output properties of multiple-input, multiple-output discrete systems: Part II

Part II considers input-output properties of nonlinear time-varying, discrete systems. Slightly generalized forms of the Small Gain and the Passivity theorems are derived. Some results of Part I and these theorems are used to derive stability criteria. The memoryless nonlinearities and the multipliers are not required to be noninteracting.