Describing Function Technique Research Articles

On the Application of the Describing Function Technique to the Bifurcation Analysis of Nonlinear Systems

Spectral techniques, like harmonic balance, are classical numerical tools for designing nonlinear oscillators and microwave circuits. Recently these techniques have been exploited for investigating complex dynamics in nonlinear systems. In this manuscript we firstly show that limit cycle Floquet's multipliers and the related bifurcation phenomena can be estimated through a spectral approach, entirely based on the describing function technique. Then we consider some significant case studies, and we show that our method yields more accurate results than the other describing function-based approaches proposed in the literature

Intelligent mobile robot navigation, Federico Cuesta and Aníbal Ollero, Springer Tracts in Advanced Robotics, vol. 16, Springer: Berlin–Heidelberg, pages: XII + 204. Price $99.00, ISBN: 3‐540‐23956‐1 2005

Intelligent mobile robot navigation, Federico Cuesta and Aníbal Ollero, Springer Tracts in Advanced Robotics, vol. 16, Springer: Berlin–Heidelberg, pages: XII + 204. Price $99.00, ISBN: 3‐540‐23956‐1 2005

DEAD ZONE OPTIMISATION IN A HEIGHT CONTROL SYSTEM OF A FIELD SPRAYER

Simple on-off control algorithms are often found back in practice. These algorithms compare the target value with a reference value and based on that, there is switched to one or the other direction. Such type of controller drives the system in limit cycle behaviour. Therefore, a dead band is placed around the reference value in order to stop the oscillations. A disadvantage of the dead band is that the accuracy is reduced. As the amplitude of the oscillations is a function of the system parameters, this amplitude can be minimised with respect to system parameters. A methodology based on the describing function technique is developed, estimating the amplitude of the oscillation during limit cycle behaviour and applied to the height control system of field sprayer. By tuning the filter constant of the ultrasonic sensor, measuring the height above the crop, the amplitude of the oscillations during limit cycle behaviour can be reduced, minimising the required dead zone in the control, such that the accuracy of the control system is improved.

Nonlinear controller synthesis based on inverse describing function technique in the MATLAB environment

In this research, a new software based on a systematic nonlinear controller design technique for nonlinear systems is developed. This software is in the MATLAB environment, and it is based on inverse describing functions in order to arrive at nonlinear gains of the controller. The software may be used to synthesize nonlinear controllers for nonlinear systems without restriction on system order, the number, type, and arrangement of nonlinear terms. The goal is to design an output feedback system that is a solution to a robust design such that sensitivity of the feedback control system with respect to amplitude of the excitation signal and the operating regimes would be at a minimum.

ON GLOBAL DYNAMIC BEHAVIOR OF WEAKLY CONNECTED OSCILLATORY NETWORKS

The global dynamics of weakly connected oscillatory networks is investigated: as a case study, one-dimensional arrays of third-order oscillators are considered. Through the joint application of the describing function technique and Malkin's Theorem a very accurate analytical expression of the phase deviation equation (i.e. the equation that describes the phase deviation due to the weak coupling) is derived. The total number of limit cycles and their stability properties are estimated via the analytical study of the phase deviation equation. The proposed technique significantly extends the results available in the literature and can be applied to almost all complex networks of oscillators. In particular two-dimensional, space variant and fully connected networks can be dealt with here.

CONTACT INSTABILITY IN RATE-CONTROL TELEMANIPULATION

Telemanipulation is usually provided with the force-reflective feature to give human operators information about the remote site. However, oscillations can always been found in contact with rigid environment. This paper presents theoretical analysis about the phenomenon of the contact oscillation in rate-control manipulation using the describing function technique. Occurrence and characteristics of limit cycles due to contact motion are verified by results of computer simulations.

STABLE 2-WAY FUZZY ADAPTIVE CONTROL OF A FLEXIBLE-JOINT ROBOT ARM

We analyze the stability of our novel 2-way fuzzy adaptive controller using describing function technique. We use additivity property of fuzzy systems to develop a systematic analytical approach for the design of a multi-input single-output fuzzy controller that we use for the control of a flexible-joint robot arm.

Periodic Oscillations and Bifurcations in Cellular Nonlinear Networks

A spectral technique is proposed for investigating periodic oscillations and bifurcations in cellular nonlinear networks. The method consists of the following three fundamental steps: 1) an accurate estimation of the whole set of stable and unstable limit cycles is provided through the application of the describing function technique; 2) a detailed characterization of each limit cycle is obtained via the harmonic balance (HB) technique, by exploiting as input parameter the single harmonic approximation, provided by the describing function technique; 3) limit cycle stability and bifurcations are studied by computing the Floquet's multipliers and exploiting both a time-domain and an HB-based technique.

Development and test of MEMS accelerometer with self-sustatined oscillation loop

In this paper, presented are summarized results of the design, analysis, and experiment of a self-sustained oscillation loop for a surface micromachined resonant accelerometer, ACRC-RXL. After introduction of the operating principle, the micromachined device is illustrated including the fabrication process. In designing the oscillation loop, a describing function technique based on the nonlinear system analysis approach is applied. Using the analytic results, a particular loop parameter is designed and loop performance is characterized. Then a real accelerometer system is implemented using mechanical structure and analog electronics. Experimental results show that the designed oscillation loop works well in real environments and the closed loop system has good performances of bias stability about 0.7 mg and dynamic range over 10 g.

New approach to assessing the effects of parametric variations in feedback loops

A method for the computation of the magnitude and phase envelopes of uncertain transfer functions is presented. The idea is to factor the transfer function into its real and complex pair roots and find the maximum and minimum magnitudes of the gain and phase of each factor. The Bode envelopes of the given uncertain system are then found from those of the individual factors. This approach, which is different from those based on the interval polynomial method of Kharitonov, has the major advantage that the representation is more applicable to practical situations where typically the coefficients of the various factored terms relate to physical parameters of a mathematical model. Further, the method results in the narrowest Bode envelopes and therefore can yield improved controller designs. The describing function analysis and the absolute stability problem of nonlinear systems with variable plant parameters are also studied. An approach which enables one to predict the existence of limit cycles in a control system which simultaneously contains nonlinearities and parametric uncertainties is given. The proposed method makes use of the popular describing function technique and these narrowest possible Bode envelopes of linear uncertain transfer functions in factored form. The technique can be used to cover the cases of linear elements, which have a multilinear or nonlinear uncertainty structure, and a nonlinear element with or without memory. New formulations of the circle and off-axis circle criteria are given for use with Bode diagrams so that the absolute stability of nonlinear systems with variable plant parameters can be studied. Examples are given to show how the proposed method can be used to assess the effects of parametric variations in feedback loops.

Design and performance test of an oscillation loop for a MEMS resonant accelerometer

In this paper, the design, analysis and experimental results of the self-sustained oscillation loop for a tunable surface micromachined resonant accelerometer, ACRC-RXL are presented. The fabrication process of the mechanical structure is also illustrated. For the oscillation loop analysis, an operator-theoretical approach is applied based on the describing function technique. Using the analytical results, feedback parameters are designed and the expected loop performance is characterized. Then the accelerometer system is practically implemented using the mechanical structure and signal processing electronics. The experimental results show that the developed accelerometer has a performance of bias stability of about 0.7 mg and a dynamic range over 10 g, which satisfies the navigation-graded sensor performance.

Limitations of the describing function for limit cycle prediction

We consider comparator-based nonlinear feedback systems, and use Tsypkin's method to develop a strategy with which to find systems with low-pass linear part for which the describing function technique erroneously predicts limit cycles. We produce an infinite set of examples of systems with very low-pass linear parts for which the describing function technique predicts spurious limit cycles, and also provide a more practical example in which limit cycles are erroneously predicted.

On the sliding mode control for DC servo mechanisms in the presence of unmodeled dynamics

In this paper, via describing function techniques, sliding mode control of DC servo mechanisms is analyzed in the presence of unmodeled dynamics. We first show that, a conventional sliding mode controller with signum function will inevitably generate a limit cycle where a second or higher order unmodeled dynamics exists. A fractional interpolation based smoothing scheme is then proposed to eliminate the limit cycle, and maintain a reasonable tracking precision bound. In particular, a DC servo motor with unmodeled stator and sensor dynamics is taken into consideration. Both theoretical analysis and simulation results verify the effectiveness of the proposed new fractional interpolation control scheme.

ON THE SLIDING MODE CONTROL FOR DC SERVO MECHANISM IN THE PRESENCE OF UNMODELED DYNAMICS

In this paper, via describing function techniques, sliding mode control of DC servo mechanisms is analyzed in the presence of unmodeled dynamics. We first show that, a conventional sliding mode controller with signum function will inevitably generate a limit cycle where a second or higher order unmodeled dynamics exists. A fractional interpolation based smoothing scheme is then proposed to eliminate the limit cycle, and maintain a reasonable tracking precision bound. In particular, a DC servo motor with unmodeled stator and sensor dynamics is taken into consideration. Both theoretical analysis and simulation results verify the effectiveness of the proposed new fractional interpolation control scheme.

Robust Nonlinear H2/H.INF. Control for a Parallel Inverted Pendulum with Dry Friction.

A robust nonlinear H2/H∞ control method is presented for a parallel inverted pendulum system with uncertain parameters and dry friction. The dry friction is quasi-linearized into a describing function model by the random input describing function technique, and the uncertain system under consideration is described by state equations that depend on time invariant, unknown-but-bounded, uncertain parameters. Then, the robust nonlinear H2/H∞ control system that establishes the stability of the closed loop system in face of the nonlinear dry friction and uncertain parameters was constructed. However, it is difficult to solve coupled Riccati equations due to the nonlinear correction term contained in the Riccati equations. Thus, it is shown that the Riccati equations can be solved by some algebraic transformations of the coupled Riccati equations. This result enables robust nonlinear controllers to be designed easily. Hence, if the parameters of the parallel inverted pendulum vary as certain bounds, the proposed control has the robustness to both the system parameter variations and the nonlinear dry friction.

Describing functions in non-linear systems with structured and unstructured uncertainties

The describing function (DF) method offers a useful a way to analyse the stability of control systems with separable nonlinearities and recently it has been applied to systems with either parametric or norm-bounded perturbations. Here, DFs are employed in an even more general setting, i.e. both types of uncertainty are allowed to enter the system. Furthermore, the high frequency dynamics neglected by the DF technique are also incorporated into the system description. Exploiting some well known results from the robust control literature, computationally tractable stability criteria are derived. Two numerical examples are provided to illustrate the various aspects of these new tools which are, of course, subject to the usual errors and the restrictions of the DF method.

NQG/H.INF./LTR Control for a Hard Nonlinear Servo System.

In this paper we propose a new nonlinear controller design method, called nonlinear quadratic Gaussian/H-infinity/loop transfer recovery (NQG/H∞/LTR), for nonlinear servo systems with hard nonlinearities such as Coulomb friction, dead-zone. We consider a H∞ performance constraint for the optimization of statistically linearized systems, by replacing a covariance Lyapunov equation into a modified Riccati equation of which solution leads to an upper bound of the nonlinear quadratic Gaussian (NQG) performance. As a result, the nonlinear correction term is included in coupled Riccati equation, which is generally very difficult to have a numerical solution. To solve this problem, we use the modified loop shaping technique and show some analytic proofs on LTR condition. Finally, the NQG/H∞/LTR controller is synthesized by inverse random input describing function techniques (IRIDF). It is shown that the proposed design method has a better performance robustness to the hard nonlinearity than the LQG/H∞/LTR method via simulations and experiments for the timing-belt driving servo system that contains the Coulomb friction and dead-zone.

Describing functions and oscillators

In this article the Describing Function technique is applied to the analysis and design of oscillators. It is shown that, with some simplifications of the expressions involved in the analysis, this technique allows one not only to quantify the amplitude but also to determine the degree of distortion of the generated sinusoidal signal. The advantage of the describing function is that it allows the inclusion of the nonlinear behavior of a system while maintaining the simplicity often associated with the linear system analysis. This will be demonstrated through an example presenting the analysis and design of a sinusoidal oscillator in the frequency domain. The method as such allows formulation of closed-form expressions for the amplitude and distortion levels in much the same way classical linear techniques are used.

A New QFT Design Methodology for Feedback Systems Under Input Saturation

Presented in this paper is a design methodology for robust stability and robust performance for a large class of plants subject to an input saturation nonlinearity. Based on Horowitz’s 3-DOF linear design method, a more general form for the additional loop transmission around the saturating element is proposed to design the third degree of freedom Hs, for Type n plants, n⩾1. Robust stability is established with the circle criterion and the describing function technique, and robust performance is improved by imposing nonovershooting conditions. All stability and performance requirements are then transferred into frequency domain design bounds which are easily obtained using QFT design techniques. This procedure applies to all SISO, Type n, stable plants with uncertainties (parametric or nonparametric), and to marginally stable and conditionally stable plants as well.

Frequency-based controller design for a class of nonlinear systems

Presented in this paper is a frequency-based controller design methodology for a class of nonlinear systems that can be characterized by a class of Hammerstein models. The controller design is directed toward the disturbance rejection problem where the output performance specification appears as a time domain tolerance. The controller design process is executed in three stages. First, the system nonlinearities are quasilinearized via describing function techniques. Second, the design methodology enforces a time domain tolerance ∣y(t)∣⩽α via frequency domain constraints on the quasilinearized closed-loop system. Finally, the controller is designed via classical loop shaping of the quasilinearized system on the open-loop gain-phase plane. The design process is illustrated on the idle speed control of a Ford 4.6L-2 valve V-8 fuel injected engine. Copyright © 1999 John Wiley & Sons, Ltd.