Absolute stability problem of systems with parametric uncertainties

In this chapter, we revisit the classical Lur’e-Postnikov approach to the stability of nonlinear control systems. The problems considered in this chapter include the problem of absolute stability and also its extensions such as the problem of absolute stabilization and the problem of structured dissipativity. As in previous chapters of this book, the approach taken here is to convert the problem under consideration into an equivalent H∞ control problem. Such a conversion is accomplished by making use of the corresponding S-procedure theorem.

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).

In this paper some connections of mathematical system theory with the problem of absolute stability of control systems are pointed out. The absolute stability problem was formulated in the early 1940s starting from some practical control problems. The considerable development of this problem in the last 25 years has been profitable for mathematical system theory. At the same time concepts of system theory provided an adequate context for a deep understanding of this problem. Such system theory concepts as blocks, interconnections, isomorphism between time domain representations, and complex domain characteristics showed themselves very useful in absolute stability and hyperstability studies. Among the main contributions of absolute stability and hyperstability studies to mathematical system theory development one can cite the concept of “block plus integral index” and the system theory interpretation of Bochner's theorem on positive functions.

The electromagnetoelastic actuator on the piezoelectric, piezomagnetic, electrostriction and magnetostriction effects is used in nanoresearch, nanotechnology, nanobiology and adaptive optics. The piezo-actuator is applied in nanotechnology and nanomechanics. The Yakubovich absolute stability criterion of the control system with the condition on the derivative for the hysteresis nonlinearity of the electromagnetoelastic actuator is used. This criterion with the condition on the derivative is development of the Popov absolute stability criterion. The stationary set of the control system for the electromagnetoelastic actuator with the hysteresis deformation is the segment of the straight line. This segment has the points of the intersection of the hysteresis partial loops and the straight line. The absolute stability conditions on the derivative for the control systems with the piezo-actuator at the longitudinal, transverse and shift piezo-effect are received. The condition of the absolute stability on the derivative for the control system for the deformation of the electromagnetoelastic actuator under random impacts in nanoresearch is obtained. For the Lyapunov stable control system this Yakubovich absolute stability criterion has the simplest representation of the result of the investigation of the absolute stability.

This paper deals with the problems of robust stability and stabilization for uncertain continuous descriptor systems. We propose a new necessary and sufficient condition in terms of a strict linear matrix inequality (LMI) for a nominal continuous descriptor system to be admissible (stable, regular, and impulse‐free). Based on this, the state‐feedback admissibility problem is solved and the solution is extended to the case of uncertain descriptor systems. Finally, numerical examples are given to illustrate the results.

In this paper, we consider the problems of robust stability and stabilization of Lurpsilae control systems with time-varying delay. Firstly, by making use of an integral inequality, some delay-dependent robust stability conditions of the system are presented. Then, a sufficient stabilization criterion is established in the terms of linear matrix inequality (LMI), based on the above stability condition and matrix decomposition technique. Finally, numerical examples are given to illustrate the effectiveness of the proposed results.

This paper is concerned with the problem of robust stability and stabilization of singular systems with uncertainties in both the derivative and state matrices. By using a parameter dependent Lyapunov function, we derive the LMI-based sufficient conditions for the stabilization of the singular systems. Secondly, by solving these LMIs, a proportional plus derivative (PD) state feedback controller is designed for the closed-loop systems to be quadratically normal and quadratically stable (QNQS). Finally, the numerical example is given to show the effectiveness of the proposed theorems.

Robust stabilization for systems with parametric uncertainty and time delay

Absolute stability of third-order systems: A numerical algorithm

Absolute stability analysis of bilateral teleoperation systems are typically model-based. Under borderline conditions of absolute stability, depending on the degree of uncertainty in the dynamic model of the teleoperator and existing noise, the system may behave as potentially unstable when the model-based analysis predicts otherwise. In this article, we propose a methodology to experimentally verify the absolute stability of master-slave teleoperation systems. Since absolute stability demands bounds of all possible environments, we achieve this by conducting only three experiments that are often experienced in teleoperation: free slave, mass-carrying slave and locked slave (rigid environment). We will validate and compare our proposed method with the benchmark Llewellyns absolute stability criterion. Furthermore, we will examine the robustness of the proposed method and will provide guidelines for choosing the mass for the mass-carrying load condition.

The stochastic stability and robust stochastic stabilization for time-delay discrete Markovian jump singular systems with parameter uncertainties are discussed. Based on stochastic Lyapunov functional, a delay-dependent linear matrix inequalities (LMIs) condition for the time-delay discrete Markovian jump singular systems to be regular, causal and stochastically stable is given. With this condition, the problem of robust stochastic stabilization is solved. A numerical example to illustrate the effectiveness of the method is given in the paper.

In this contribution we consider the absolute stability problem. We derive necessary and sufficient conditions for the existence of periodic motion using an operator approach. The results yield an efficient algorithm to approximate the maximum sector bounds for absolute stability numerically.

The absolute stabilizability problem is a generalization of the well-known absolute stability problem to controlled dynamical systems. The absolute stability problem was posed by Lur'e, who also suggested a method for solving this problem by constructing a global Lyapunov function of a special form (a quadratic form plus an integral of the nonlinearity). The mathematical techniques for the problem were comprehensively developed in [1, 2]; further they were extended to more general statements. Popov and later Yakubovich used sufficient conditions for absolute stability in the form of "frequency" inequalities, that is, inequalities containing the Fourier transforms of functions of time determined by given linear relations. This form of sufficient conditions proved to be convenient in automatic regulation and control problems. The present paper deals with the problem of absolute stabilizability via output variables in the general statement and gives a method for solving this problem on the basis of frequency inequalities and an auxiliary minimax problem (see Section 2). This method was suggested in [3, 4] for special

A novel mathematical formulation to deal with interfacial stability problems of the Kelvin-Helmholtz type with heat and mass transfer in the presence of oblique electric fields is presented. The perturbed system is composed of two homogeneous, inviscid, incompressible, dielectric, and streaming fluids sep-arated by a horizontal interface, and bounded by two rigid planes. The effect of a phase transition on the instability is considered, and the linear dispersion relations are obtained and discussed. It is found that the electric field has a major effect and can be chosen to stabilize or destabilize the flow. For Ray-leigh-Taylor instability problems of a liquid-vapor system it is found that the effect of mass and heat transfer enhances the stability of the system when the vapor is hotter than the liquid, although the clas-sical stability criterion is still valid. For Kelvin-Helmholtz instability problems, however, the classical stability criterion is found to be substantially modified due to the effects of the electric field, mass and heat transfer. A new stability condition relating the magnitude and orientation of the electric field and the dielectric constants is obtained. Oblique electric fields are found to have stabilizing effects which are reduced by the normal components of the electric fields. The effects of orientation of the electric fields and fluid depths on the stability configuration are also discussed.

This paper shows how value sets and the zero exclusion principle can be used to obtain results on the robust stability and design of controllers against a special infinite dimensional l/sub 1/ norm bounded parametric uncertainty in both numerator and denominator of scalar discrete-time plants. This parametric uncertainty has properties of both parametric and unstructured uncertainty and allows standard H/sup /spl infin// design tools to be used without conservatism.