Circle Criterion Research Articles

HYBRID FUZZY CONTROL OF A CLASS OF NONLINEAR SYSTEMS

In this paper, a hybrid fuzzy controller based on a fuzzy proportional-integral plus conventional derivative control structure is proposed to control a class of single-input single-output nonlinear systems. The proposed controller is capable of generating both linear and nonlinear outputs so that the closed-loop system is equivalent to a feedback connection of a linear system and a nonlinear element in the error state space. It is shown that absolute stability of the closed-loop system can be guaranteed if the nonlinear output of the fuzzy controller is bounded by virtue of the circle criterion. The proposed controller is used to control an inverted pendulum and the performance is shown to be superior.

Robust Stability Analysis for Nonlinear Sampled-Data Control Systems in a Frequency Domain

The robust stability condition for sampled-data control systems with a sector nonlinearity was presented in our previous paper. Although it is applicable only to the sampled-data control system of a certain class, a usual discrete-time control system can belong to this type of class. The criterion contains the robust stability condition in regard to the nonlinear allowable band (sector) for continuous-time systems and the linear allowable gain band (interval set) for discrete-time systems. This paper describes the relationship between the robust stability condition and the Popov criterion for continuous-time system. In addition, this paper shows a representation of an off-axis circle criterion on a complex plane, and proposes an evaluation method of the stability from the relative position with the vector locus of the open-loop frequency response characteristics.

An analysis and design method for linear systems subject to actuator saturation and disturbance

We present a method for estimating the domain of attraction of the origin for a system under a saturated linear feedback. A simple condition is derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractively invariant. This condition is shown to be less conservative than the existing conditions which are based on the circle criterion or the vertex analysis. Moreover, the condition can be expressed as linear matrix inequalities (LMIs) in terms of all the varying parameters and hence can easily be used for controller synthesis. This condition is then extended to determine the invariant sets for systems with persistent disturbances. LMI based methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stability requirements. The effectiveness of the developed methods is illustrated with examples.

Nonlinear observers: a circle criterion design and robustness analysis

Globally convergent observers are designed for a class of systems with monotonic nonlinearities. The approach is to represent the observer error system as the feedback interconnection of a linear system and a time-varying multivariable sector nonlinearity. Using LMI software, observer gain matrices are computed to satisfy the circle criterion and, hence, to drive the observer error to zero. In output-feedback design, the observer is combined with control laws that ensure input-to-state stability with respect to the observer error. Robustness to unmodeled dynamics is achieved with a small-gain assignment design, as illustrated on a jet engine compressor example.

Delay-independent circle criterion and popov criterion

In the present paper are studied delay-independent versions of circle criterion and Popov criterion, for some nonlinear delay systenls with sector-bounded nonlinearities. Necessary and sufficient conditions are obtained in terms of solvability of linear matrix inequalities. This constitutes an analogue to the famous results, valid for nonlinear delay-free systems

Lyapunov–Krasovskii functionals and frequency domain: delay‐independent absolute stability criteria for delay systems

The present paper is devoted to the study of absolute stability of delay systems with nonlinearities subject to sector conditions. We construct quadratic candidate Lyapunov–Krasovskii functional, whose decreasingness along trajectories is expressed in terms of linear matrix inequalities. We then show that the feasibility of the latter implies some frequency-domain conditions, which may be seen as delay-independent versions of the circle criterion and the Popov criterion. Copyright © 2001 John Wiley & Sons, Ltd.

Lyapunov–Krasovskii functionals and frequency domain: delay‐independent absolute stability criteria for delay systems

AbstractThe present paper is devoted to the study of absolute stability of delay systems with nonlinearities subject to sector conditions. We construct quadratic candidate Lyapunov–Krasovskii functional, whose decreasingness along trajectories is expressed in terms of linear matrix inequalities. We then show that the feasibility of the latter implies some frequency‐domain conditions, which may be seen as delay‐independent versions of the circle criterion and the Popov criterion. Copyright © 2001 John Wiley & Sons, Ltd.

Friction analysis based on integral quadratic constraints

AbstractPassivity arguments have long been used in the analysis of systems with friction. This paper proves a new set of integral quadratic constraints, which reduces the conservatism of such methods. The result is used to derive a quantitative bound on the amount of integrator leakage that is needed to remove stiction‐induced oscillations in a servo system.Finally, an incremental version of the circle criterion is used to study the existence of subharmonics due to friction in a system subject to externally forced oscillations. Copyright © 2001 John Wiley & Sons, Ltd.

Feasibility conditions for circle criterion designs

We derive necessary and sufficient conditions for the existence of a control law that renders the closed-loop system strictly positive real with respect to a disturbance input. The result is used to establish feasibility of recent nonlinear controller and observer designs based on the circle criterion.

A note on absolute stability of uncertain systems

The absolute stability of nonlinear systems is an important robustness issue which has been studied intensively since its first formulation by Lur'e. Recently, extensions of the solutions to different versions of this problem have been developed for cases with either structured or unstructured uncertainties in the linear part. In this note, the circle criterion and the Popov criterion are recast in an even more general setting; one which accommodates both types of perturbations in the linear plant. Numerical examples are provided to illustrate the application of these frequency domain tools.

Synchronization in arrays of coupled nonlinear systems: passivity, circle criterion, and observer design

It has been shown that synchronization between two nonlinear systems can be studied as a control-theory problem. We show that this relationship can he extended to synchronization in arbitrary coupled arrays of nonlinear systems. In particular, we use several well-known stability conditions to obtain synchronization criteria in arbitrarily coupled arrays: the passivity criterion, the circle criterion and a result on observer design of Lipschitz nonlinear systems. We also study how these synchronization criteria depend on the topology of the coupled networks. In particular, we show that synchronization is improved by using nonlocal connections or introducing random connections.

Conservatism of the circle criterion-solution of a problem posed by A. Megretski

In the collection of open problems in mathematical systems and control theory Blondel et al. (1999), Megretski poses a problem related to the degree of conservativism of the well-known circle criterion follows. We solve this problem.

The spectral coefficients of the response of nonlinear systems to asymptotically almost periodic inputs

It is known that time-invariant systems having approximately finite memory and satisfying some often-satisfied continuity constraints map asymptotically almost periodic inputs into asymptotically almost periodic outputs with the module of the output a subset of the module of the input. Systems described by Volterra integral equations of the second kind that meet the circle criterion and satisfy some additional constraints fall into this class. In this paper we present an analytical basis for numerically evaluating the spectral coefficients of the output of such systems when the input is asymptotically almost periodic.

Controlling the nonlinear dynamics of a beam system

Control based on linear error feedback is applied to reduce vibration amplitudes in a piecewise linear beam system. Hereto small amplitude 1-periodic solutions are stabilized wherever they coexist with two or more long-term solutions. In theory, no control effort is required to maintain the 1-periodic response once it has been stabilized. For the beam system, 1-periodic solutions are stabilized by feedback at one location along the beam. Feedback is represented by servo-stiffness or servo-damping which results from increasing two corresponding control parameters. At appropriate levels of these parameters local, or global, asymptotic stability (of the zero-equilibrium) of the error dynamics, i.e. stability of the underlying 1-periodic solutions, can be guaranteed. Local asymptotic stability can be guaranteed for a large range of actuator locations and excitation frequencies and is indicated by bifurcations. Global asymptotic stability can only be guaranteed for a limited range of actuator locations on the basis of the well-known circle criterion. The difference between local and global asymptotic stability in terms of the required values for the control parameters can be significant, and may result in large differences in control performance.

On the use of multi-loop circle criterion for saturating control synthesis

This paper is concerned with regional stabilization of linear time-invariant systems by dynamic output feedback controllers subject to known bounds on the magnitudes of the control inputs. Specifically, we consider the achievable region of attraction, i.e. the set of vectors with the following property: There exists a (nonlinear) controller such that any closed-loop state trajectory converges to the origin as long as the initial state belongs the set. Two subsets of such set are characterized. One is derived from the linear analysis that considers the behavior of the states in the linear (non-saturated) region only, while the other is based on the nonlinear analysis using the multi-loop circle criterion. The main result of this paper shows that the two sets are exactly the same. Thus we conclude that the circle criterion does not help, within our framework, to increase the size of the region of attraction in saturating control synthesis when compared with that resulting from the linear analysis.

Mapping of Nyquist/Popov Theta-Stability Margins Into Parameter Space

The application of the parameter space method has proven to be useful for robustness analysis of uncertain parametric systems and robust control synthesis for quite a number of applications. However, it has been restricted to linear systems and the consideration of eigenvalue criteria. This paper enhances the application of the parameter space method to include various locus criteria. This allows not only for incorporation of linear criteria (e. g. gain and phase margin) but for nonlinear criteria as well (e.g. Popov- or circle criterion and the dual locus method).

LMI representation of the shifted Popov criterion

This paper is concerned with a stability criterion of nonlinear feedback systems, called the shifted Popov criterion. It includes the Popov and circle criteria as special cases, and enables us to check stability in the frequency domain in a graphical manner. This paper derives an LMI representation of the shifted Popov criterion, both in the Lyapunov stability context and the input–output stability context, when the linear part of the nonlinear feedback system has a state–space realization.

Stability and robustness for discrete-time systems with control signal saturation

All practical control systems exhibit control signal saturation. The effect that this saturation has on the control system performance, especially stability and robustness, can be significant and must be understood at the controller design stage. This paper presents conditions for global asymptotic stability and measures of stability robustness for such systems. These are demonstrated through simulation examples, and it is shown how an understanding of the stability conditions can inform the controller design process. The off-axis circle criterion is used as the basis for a sufficient condition for stability, and it is argued that this is not overly restrictive in practice. The derivations are carried out in discrete time, and servo-system control is envisaged as an important application area for the techniques; however, the results are applicable more widely.

Stability of non-linear QFT designs based on robust absolute stability criteria

The paper is mainly devoted to the robust stability problem of non-linear QFT designs. The problem is first formulated for SISO non-linear systems, limited in practice to linear-memoryless sector-bounded non-linear interconnections, and in an I/O stability sense. The work investigates several possible robust adaptations of the Circle Criterion, depending on the type of resulting interconnection of linear and non-linear blocks, appearing in the feedback system. Also, the use of the Popov Criterion is investigated as an alternative less conservative than the Circle Criterion in some cases. The proposed techniques are given in usual QFT language, expressed as frequency conditions or boundaries in the Nichols Chart, allowing an easy integration with other design objectives. In addition, multivariable extensions using a conicity condition and the concept of maximal cone are adopted to give stability boundaries in interconnections of SIMO linear-MISO sector-bounded memoryless blocks. All the robust stability criteria are illustrated using significant examples to emphasize the practical application of the resulting techniques.

Robust stability of the systems with mixed uncertainties under the IQC descriptions

This paper studies the problem of robust stability checking for a single-input single-output uncertain feedback system which consists of a linear uncertain plant in the forward loop and a non-linear dynamic uncertain unit in the feedback loop. It is supposed that the linear part of the system is of parametric uncertainties described by a polytopic perturbation mode, and that the non-linear part of the system is of dynamic uncertainties characterized by an integral quadratic constraint (IQC). The problem of stability checking is discussed for different structures of the IQC multiplier based on the concepts of biconvex and convex-concave functions and their properties. For different uncertainty structures of the system, a finite vertex checking result and an edge checking result are presented. As an application of the above results, the problem of robust H performance checking is discussed for the case that the uncertain plant of a feedback system is parameterized by a polytopic perturbation mode, and an edge checking result is given under a fairly mild assumption. Also, the absolute stability of an interval system is investigated, and the vertex results of circle criterion and Popov criterion are obtained. Finally, a demonstrating example is included.