On the absolute stability of nonlinear feedback systems

Fundamentals of the theory of non-linear pluse control systems

In this paper, we study the problems of robustness and disturbance rejection in a general setting, whereby the signal space in which the inputs and outputs reside is not necessarily L 2 . It is well known that, if the signal space is taken as L 2 , then both optimal robustness design and optimal disturbance rejection can be formulated as H ∞ -norm minimization problems. Three distinct 'Aizerman' type of conjectures regarding the stability of nonlinear feedback systems are formulated, each of which happens to be true in the special case when the underlying signal space is L 2 . It is shown that, in a general setting, only one of the three Aizerman-type conjectures is true, namely: If a feedback system is stable for all linear, possibly time-varying feedback elements belonging to a specified sector, then the feedback system remains stable for all nonlinear, possibly time-varying feedback elements belonging to the same sector. It is shown that the remaining two conjectures are equivalent to each other, and necessary and sufficient conditions for each of the two conjectures to hold are derived. Next, it is shown that, in general, the problems of optimal disturbance rejection and optimal robustness design are quite distinct. In the special case where the signal space is L 2 and the corresponding Banach algebra of causal stable LTI systems is H ∞ , both problems coincide. But in general, the problem of optimal disturbance rejection is that of minimizing the norm of the weighted sensitivity matrix, whereas the problem of optimal robustness design is that of minimizing something like the spectral radius of the weighted complementary sensitivity matrix.

Current industry practice on determination of necessary weight for lateral stability of offshore pipelines makes many important assumptions on the boundary conditions and initial configurations of a subsea pipeline. The pipe is assumed to be on a flat seabed, the distance between local constraints of the pipeline is assumed to be so large that the boundary conditions at the ends have no influence on the maximum lateral displacement, and the contact between seabed and pipeline is assumed to be uniform. Pipelines are in reality placed on rough seabed, and the contact between pipeline and seabed may be very non-uniform. Therefore it is necessary to investigate the effects of free spans and their influence on lateral stability. Pipelines in free spans experience much lower hydrodynamic loads, including lift, drag and inertia, but at the same time a reduced overall resistance. The local influence of the free span is governed by span gap, span length, soil properties, pipe geometry, effective axial force and pipe submerged weight. Local studies of lateral stability on span shoulders have been performed using a simplified approach according to the absolute stability criterion and penetration models according to DNV-RP-F109, and the influence with respect to design according to the absolute static stability criterion is shown. It is demonstrated in this paper that local displacements on free span shoulders may occur even if the pipe satisfies the absolute stability criterion. However, the overall effects of free spans [2] are fairly minor on the global pipeline behaviour, even for pipes with a high ratio of spans to contact points. Therefore, the displacements demonstrated in this paper are likely to be local to smaller areas on span shoulders.

The global stability of positive continuous-time standard and fractional order nonlinear feedback systems is investigated. New sufficient conditions for the global stability of these classes of positive nonlinear systems are established. The effectiveness of these new stability conditions is demonstrated on simple examples of positive nonlinear systems.

Linear feedback stabilization of nonlinear systems is studied for systems whose linearization possesses either a simple zero or an imaginary pair of eigenvalues that is uncontrollable. The results complement previous work on the synthesis of nonlinear stabilization control laws. Calculations related to the stability of bifurcation branches of related families of systems are employed in the analysis and synthesis of stabilization by linear feedback. One condition for stabilizability by linear feedback is that the open-loop linearized stable subsystem should have an adequate margin of stability.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Consideration was given to the behavior of the third-order systems in phase space. Regularities of motion of the phase trajectories were established, and a criterion for absolute nonoscillation was obtained. For the absolutely nonoscillatory systems, the Hurwitz conditions serve as the absolute stability criterion. For the oscillatory systems, an additional Bulgakov condition was introduced to eliminate the possibility of parametric resonance. This condition which is verified on the invariant set defined using the Poincare transform was shown to be a criterion for absolute stability of the oscillatory systems. The results obtained were used to solve the problem of absolute stability of a third-order control system with nonstationary sectorial nonlinearity.

In this paper we develop a nonsmooth output feedback framework for local and/or global stabilization of a class of inherently nonlinear systems, which may not be achieved by any smooth feedback controller. A systematic design algorithm is presented for the construction of stabilizing, dynamic output compensators that are nonsmooth but Holder continuous. A new ingredient of the proposed output feedback control scheme is the introduction of a recursive observer design procedure, making it possible to construct a reduced-order observer step-by-step, in a naturally augmented fashion. Such nonsmooth design leads to a number of new results on output feedback stabilization of nonlinear systems with unstabilizable/undetectable linearization. One of them is the global stabilizability of a chain of odd power integrators by Holder continuous output feedback. The other one is the local stabilization using nonsmooth output feedback for a wide class of nonlinear systems studied in Qian and Lin (2001), where only global stabilizability by nonsmooth state feedback was proved to be possible.

Partially stable systems involve dynamical systems whose motion lie in a subspace of the state space resulting in system stability with respect to part of the system's states. In this paper, we develop partial stability theorems for nonlinear continuous-time and discrete-time dissipative feedback systems. Specifically, by invoking additional structural constraints on the forward loop and feedback loop system storage functions, we develop feedback interconnection partial stability results for dissipative nonlinear dynamical systems. Our results provide extensions of the positivity and small gain theorems for guaranteeing partial stability of feedback interconnected systems.

The object of this paper is to outline a stability theory for input-output problems using functional methods. More particularly, the aim is to derive open loop conditions for the boundedness and continuity of feedback systems, without, at the beginning, placing restrictions on linearity or time invariance. It will be recalled that, in the special case of a linear time invariant feedback system, stability can be assessed using Nyquist's criterion; roughly speaking, stability depends on the mounts by which signals are amplified and delayed in flowing around the loop. An attempt is made here to show that similar considerations govern the behavior of feedback systems in general-that stability of nonlinear time-varying feedback systems can often be assessed from certain gross features of input-output behavior, which are related to amplification and delay. This paper is divided into two parts: Part I contains general theorems, free of restrictions on linearity or time invariance; Part II, which will appear in a later issue, contains applications to a loop with one nonlinear element. There are three main results in Part I, which follow the introduction of concepts of gain, conicity, positivity, and strong positivity: THEOREM 1: If the open loop gain is less than one, then the closed loop is bounded. THEOREM 2: If the open loop can be factored into two, suitably proportioned, conic relations, then the closed loop is bounded. THEOREM 3: If the open loop can be factored into two positive relations, one of which is strongly positive and has finite gain, then the closed loop is bounded. Results analogous to Theorems I-3, but with boundedness replaced by continuity, are also obtained.

Stability of single-stage single-bit ∑Δ modulator

In this paper finite-time stabilization of nonlinear affine systems, is considered. First we provide a formula for a finite-time stabilizing state feedback control law, for the last class of system. Then we extend the result to the case of a norm-bounded uncertain system and a polytopic uncertain system stabilization are given. Some numerical examples are given to illustrate the main result.

The absolute stability of large-scale Lurie direct control systems with time-varying coefficients is investigated. Based on the decomposition method for large-scale systems and technique of the nonsingular M-matrix, a suitable scalar Lyapunov function as a weighted sum is constructed. By estimating its total time derivative, some absolute stability criteria and practical corollaries are derived. Furthermore, the results are extended to multiple nonlinearities. The salient feature of this paper is that the criteria which we propose allow for the situation that the norms of time-varying coefficients are unbounded. The main idea of the methodology is that even if the coefficients are norm-unbounded, by restricting their relative magnitude, the problem of negative definiteness for the derivative can also be changed into the problem of stability for a constant matrix. Finally, some numerical examples are included to illustrate the effectiveness of the proposed criteria.

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.

In this paper, we present an improved delay-dependent absolute stability criterion for Lur’e systems with time delays. The guarantee of absolute stability is provided by Lyapunov-Krasovskii theorem with the Lyapunov functional containing the integral of sector-bounded nonlinearities. The Lyapunov functional terms involving delay are partitioned to be associated with each equidistant fragment on the length of time delay. Employing the Jensen inequality and S-procedure, the sufficient condition is derived from time derivative of the Lyapunov functional. Then, the absolute stability criterion expressed in terms of linear matrix inequalities (LMIs) can be efficiently solved using available LMI solvers. The bisection method is used to determine the maximum allowable time delays to ensure the stability of Lur’e systems in the presence of uncertain time-invariant delays. In addition, the stability criterion is extended to Lur’e systems subject to norm-bounded uncertainties by using the matrix eliminating lemma. Numerical results from two benchmark problems show that the proposed criteria give significant improvement on the maximum allowable time delays.

In this paper, the stability of a fluid conduit connected to a mass and spring system or other control elements is studied. If an element is stable in the combination with any length of pipeline, we call the element "absolutely stable." A sufficient condition of the absolute stability is given for a spool valve a poppet valve. The stability problem of a pipeline having an element at one end or elements at both ends and an element having upstream and downstream pipelines is also studied. The absolute stability criterion is generally studied, using vector locus. The viscosity of a fluid flowing in a pipeline has little effect on the absolute stability.