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Showing papers on "Describing function published in 2001"


Journal ArticleDOI
TL;DR: The parameters of first and second order plant transfer functions, stable or unstable, with time delay can be found exactly, assuming no measurement errors, from measurements of the parameters of a single asymmetrical limit cycle in a relay controlled feedback loop.

116 citations


Journal ArticleDOI
TL;DR: In this paper, the authors examined tools for fitting low-complexity nonlinear models based on experimental data through the example problem of finding a reduced-order model suitable for control of a combustion instability operating in a limit cycle.

39 citations


Journal ArticleDOI
TL;DR: This article uses computer-aided design tools to develop a describing function analysis of a pendulum clock as a control system that allows the pendulum to provide the required time keeping and add enough energy to the pendula to overcome the damping caused by friction.
Abstract: This article uses computer-aided design tools to develop a describing function analysis of a pendulum clock. We design the escapement as a control system that allows the pendulum to provide the required time keeping and, at the same time, add enough energy to the pendulum to overcome the damping caused by friction. We use analysis tools in the MATLAB Control System Toolbox to accomplish the design and analysis. A by-product of our analysis is a simple MATLAB/Simulink model and a script that generates describing functions for any arbitrary nonlinear system (including systems with multiple nonlinearities and with frequency-dependent describing functions). We also develop a Simulink model of the clock to verify the results of the analysis. The analysis described here uses some object-oriented programming features of MATLAB.

36 citations


Proceedings ArticleDOI
30 Sep 2001
TL;DR: In this article, the authors present a design methodology, analysis, and practical considerations of self-oscillating drive circuit for electronic ballasts by considering the selfoscillated electronic ballast as a relay control system.
Abstract: This paper presents a design methodology, analysis, and practical considerations of self-oscillating drive circuit for electronic ballasts by considering the self-oscillating electronic ballast as a relay control system The first section of the paper analyzes the design of resonant elements, the nonlinearity present in the circuit and consideration to apply the methodology proposed The second section of the paper shows the design of self-oscillating electronic ballast using control tools such as: describing function, extended Nyquist criterion and block diagrams that allows an expression for designing the self-oscillating electronic ballast to be found

33 citations


Journal ArticleDOI
TL;DR: In this article, an indirect numerical approach is described which shows that, for a system of nonlinear differential equations, the eigenvalues of the quasi-linear system simply indicate all limit cycles and, additionally, yield stability regions for the linearized case.
Abstract: In the calculation of periodic oscillations of nonlinear systems –so-called limit cycles – approximative and systematic engineeringmethods of linear system analysis are known The techniques, working inthe frequency domain, perform a quasi-linearization of the nonlinear system,replacing nonlinearities by amplitude-dependent describing functionsFrequently, the resulting equations for the amplitude and frequency ofpresumed limit cycles are solved directly by a graphical procedure in aNyquist plane or by solving the nonlinear equations or a parameteroptimization problem In this paper, an indirect numerical approach isdescribed which shows that, for a system of nonlinear differentialequations, the eigenvalues of the quasi-linear system simply indicateall limit cycles and, additionally, yield stability regions for thelinearized case The method is applicable to systems with multiplenonlinearities which may be static or dynamic It is demonstrated foran example of aircraft nose gear shimmy dynamics in the presence ofdifferent nonlinearities and the results are compared with those fromsimulation

30 citations


Journal ArticleDOI
01 May 2001
TL;DR: In this article, the authors investigated the saturation phenomenon for a nonlinear inductance and showed that when the current load is high, the inductance always suffers from saturation or hysteresis.
Abstract: Passive filter design is traditionally considered to be very important for many power electronic circuits and power systems. It is is very critical in the design of these power electronic circuits and power systems to know how to obtain accurate corner frequencies. It is known that when the load is large, the inductance of an inductor is known to always change. This phenomenon is due to the fact that when the current load is high, the inductance always suffers from saturation or hysteresis. It is extremely difficult to estimate the actual value for the nonlinear inductance in large currents because the inductance is nonlinear. The paper investigates the saturation phenomenon for a nonlinear inductance. The describing function method can be used to linearise the nonlinear inductor and then estimate the inductance in large current situations. Hence, the corner frequency for the lowpass filter can also be calculated accurately. It is shown that, when the current is very large, the corner frequency drifts to a larger value in the low-pass filter. The drift value of the corner frequency can be easily calculated by the describing function. Simulation and experimental results verify this phenomenon. However, it should be stressed that the method used in the paper is restricted to low frequencies. The higher frequency effects are neglected throughout the paper.

29 citations


Journal ArticleDOI
TL;DR: The "Describing Function" technique is applied to the analysis and design of oscillators and it is shown that, with some simplifications of the expressions involved, this technique allows one not only to quantify the amplitude but also to determine the degree of distortion of the generated sinusoidal signal.
Abstract: 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.

26 citations


Journal ArticleDOI
Chin Chang1, G.W. Bruning
TL;DR: In this article, the authors analyzed the self-oscillating LC parallel resonant inverter for electronic ballast applications from a control system point of view and found that the accuracy of the frequency prediction via the Hamel locus approach is high, while the accuracy via the describing function approach is circuit Q-value dependent.
Abstract: In this paper, the authors analyze the self-oscillating LC parallel resonant inverter for electronic ballast applications from a control system point of view. It is observed that the self-oscillating parallel resonant inverter with lamp loads can be naturally modeled as a relay system with negative hysteresis. Based on this formulation, the self-oscillating frequencies of the circuit are found via the time-domain-based Hamel locus. Also, the predicted self-oscillating frequencies via the describing function approach and the Hamel locus approach are compared with the prototype measurement results. It turns out that the accuracy of the frequency prediction via the Hamel locus approach is high, while the accuracy via the describing function approach is circuit Q-value dependent.

26 citations


Journal ArticleDOI
TL;DR: In this article, the analysis of pilot in-the-loop oscillations (PIO) of category II (with rate or position limiting) is presented, based on robust stability analysis, which assumes that PIOs are characterized by a limit cycle behavior.
Abstract: In this paper we deal with the analysis of pilot in-the-loop oscillations (PIO) of category II (with rate or position limiting). We propose an approach, based on robust stability analysis, which assumes that PIOs are characterized by a limit cycle behavior. In this way we obtain two different methods for the analysis of category II PIOs; the e rst method replaces the nonlinear element (contained in the model of the rate or position limited actuator ) by a linear time-invariantgainand isshownto beequivalenttotheclassicaldescribingfunction analysis,withtheadvantageof preventing thecomputational dife culties of the describing function approach. Theanalysis via time-invariant gain may be, however, optimistic in predicting PIOs for a given aircraft. Therefore a further method, based on stability analysisvia Lur’ eLyapunov functions, isprovided;such a method guaranteesasymptoticstability ofthenonlinear plant and can give conservative results. By the use of both methods, a complete analysis of the nonlinear system can be performed. The X-15 case study is considered to illustrate the effectiveness of the proposed methodology.

21 citations


Proceedings ArticleDOI
04 Dec 2001
TL;DR: The describing function method is employed in conjunction with some of the robustness tools for linear systems to develop algorithms for predicting the existence of limit cycles in uncertain systems with multiple nonlinearities connected in series.
Abstract: The describing function method can be used to analyze control systems with separable nonlinearities. Here, the method is employed in conjunction with some of the robustness tools for linear systems to develop algorithms for predicting the existence of limit cycles in uncertain systems with multiple nonlinearities connected in series. Uncertainty is assumed to exist in terms of parameter variations in both the linear and the nonlinear elements. Examples are provided to illustrate the results which are, of course, subject to the usual errors and restrictions of the describing function method.

16 citations


Journal ArticleDOI
TL;DR: Here, DFs are employed in an even more general setting, i.e. both types of uncertainty are allowed to enter the system, and computationally tractable stability criteria are derived.
Abstract: 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.

Proceedings ArticleDOI
01 Sep 2001
TL;DR: A nonlinear feedback system is designed and analyzed for a surface micromachined resonant accelerometer, and a simple phase shifted relay with finite slope is found for the nonlinearity implementation.
Abstract: A nonlinear feedback system is designed and analyzed for a surface micromachined resonant accelerometer. For this, a brief illustration of the plant dynamics is given. In the analysis, the periodic signal in the nonlinear feedback loop is obtained by the limit cycle point, which is best approximated via the describing function method. This approach is reasonable in that the describing function method gives the optimal quasi-linearization for a sinusoidal input under an adequate filtering condition. In the sequel, it is possible that the nonlinearity form in the loop is determined by the graphical analysis in complex plane. Considering the characteristic feature of plant dynamics, a simple phase shifted relay with finite slope is found for the nonlinearity implementation. Also is concisely illustrated the stability of limit cycle. Finally, simulation and experimental result is given to show the properness of the designed loop.

Journal ArticleDOI
TL;DR: In this paper, the problem of robust control of an uncertain nonlinear plant is reduced to a linear equivalent problem decoupled from the linear QFT formalism, and the problem is then reduced to linear equivalent control for the linear equivalent family of linear plants.
Abstract: SUMMARY Nonlinear QFT (quantitative feedback theory) is a technique for solving the problem of robust control of an uncertain nonlinear plant by replacing the uncertain nonlinear plant with an &equivalent’ family of linear plants. The problem is then "nding a linear QFT controller for this family of linear plants. While this approach is clearly limited, it follows in a long tradition of linearization approaches to nonlinear control (describing functions, extended linearization, etc.) which have been found to be quite e!ective in a wide range of applications. In recent work, the authors have developed an alternative function space method for the derivation and validation of nonlinear QFT that has clari"ed and simpli"ed several important features of this approach. In particular, single validation conditions are identi"ed for evaluating the linear equivalent family, and as a result, the nonlinear QFT problem is reduced to a linear equivalent problem decoupled from the linear QFT formalism. In this paper, we review this earlier work and use it in the development of (1) new results on the existence of nonlinear QFT solutions to robust control problems, and (2) new techniques for the circumvention of problems encountered in the application of this approach. Copyright ( 2001 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this article, the authors analyzed the fully-developed pilot-induced oscillation (PIO) as a worst case for the safety of piloted airplanes, including actuator rate limiting, feedback control loop, and pilot delay by using describing function method.
Abstract: Fully-developed pilot-induced oscillation (PIO) is an important issue to be solved in the development of modern fly-by-wire flight control systems. In this paper, the fully-developed PIO is analyzed as a worst case for the safety of piloted airplanes, including actuator rate limiting, feedback control loop, and pilot delay by using describing function method. It is shown that the predictions obtained with this method closely match results of the simulation in the frequency and the amplitude of the PIO limit cycle. And it demonstrates that the feedback control loop has a positive effect on PIO and decreases amplitude of the oscillation.

Dissertation
21 Sep 2001
TL;DR: In this paper, a physically-based, reduced-order, nonlinear model was developed based on the proper orthogonal decomposition technique and generalized Galerkin method for a laminar premixed flame.
Abstract: Lean-premixed combustion has the advantage of low emissions for modern gas turbines, but it is susceptible to thermoacoustic instabilities, which can result in large amplitude pressure oscillations in the combustion chamber. The thermoacoustic limit cycle is generated by the unsteady heat release dynamics coupled to the combustor acoustics. In this dissertation, we focused on reduced-order modeling of the dynamics of a laminar premixed flame. From first principles of combustion dynamics, a physically-based, reduced-order, nonlinear model was developed based on the proper orthogonal decomposition technique and generalized Galerkin method. In addition, the describing function for the flame was measured experimentally and used to identify an empirical nonlinear flame model. Furthermore, a linear acoustic model was developed and identified for the Rijke tube experiment. Closed-loop thermoacoustic modeling using the first principles flame model coupled to the linear acoustics successfully reproduced the linear instability and predicted the thermoacoustic limit cycle amplitude. With the measured experimental flame data and the modeled linear acoustics, the describing function technique was applied for limit cycle analysis. The thermoacoustic limit cycle amplitude was predicted with reasonable accuracy, and the closed-loop model also predicted the performance for a phase shift controller. Some problems found in the predictions for high heat release cases were documented.

Proceedings ArticleDOI
29 Nov 2001
TL;DR: In this paper, a systematic parameter design method for two-inertia two-mass systems is proposed, where a load variation, Coulomb friction, and fast and precise control are considered.
Abstract: Two-mass systems are shown as a model of two masses connected by a spring and often seen in mechanical systems. PID control, resonance ratio control, H/sup /spl infin// control, and etc. have been applied to the two-mass systems. Conventional controllers that use a disturbance observer for two-inertia systems are adopted. A systematic parameter design method for the systems is proposed, where a load variation, Coulomb friction, and fast and precise control are considered. For limit cycle due to friction, a suppression condition of limit cycle is derived by an analysis using a describing function method. For load variation, robust stability for time varying system with structured uncertainty is guaranteed by the quadratic stability. Moreover, nominal performance is improved by maximizing the smallest eigen value of control system under the above restrictions. Effectiveness is confirmed by some simulations and experiments.

Proceedings ArticleDOI
01 Jan 2001
TL;DR: A feedback system design methodology for a plant subject to input rate saturation, or both amplitude and rate saturation is presented and results in a 4DOF feedback system with two extra DOFs: H/sub a/(s) and H/ sub r/(s).
Abstract: In this paper, a feedback system design methodology for a plant subject to input rate saturation, or both amplitude and rate saturation is presented. Based on Horowitz's original 3DOF design (Horowitz, 1983) and extensions developed in (Wu and Jayasuriya, 1999; 2000; Wu, 2000), the structure of the additional loop transmission around the rate saturating element is first proposed for designing the 3rd DOF, H/sub r/(s), using the 3DOF non-interfering loop structure. Robust stability and performance are investigated. The circle criterion, describing function, and non-overshooting conditions are utilized to obtain design constraints. In the case of having both amplitude and rate saturation, approaches developed for amplitude and rate saturation respectively are combined into one method. It results in a 4DOF feedback system with two extra DOFs: H/sub a/(s) and H/sub r/(s). Finally, all these design constraints are expressed as frequency domain bounds and synthesis follows from loop shaping methods such as QFT. This methodology applies to type I or higher stable and neutrally stable plants.

Proceedings Article
01 Jan 2001
TL;DR: A technique for generating symbolic expressions for the distortion in weakly nonlinear analog integrated circuits is presented, which uses some acceptable assumptions to reduce the task of analyzing the nonlinear circuit to a repeated analysis of derived linear circuits.
Abstract: A technique for generating symbolic expressions for the distortion in weakly nonlinear analog integrated circuits is presented. This technique uses some acceptable assumptions to reduce the task of analyzing the nonlinear circuit to a repeated analysis of derived linear circuits. This repetitive algorithm has been implemented and it is demonstrated on an example circuit. In the analysis of analog integrated circuits, distortion and intermodulation are important factors. Either they are unwanted, as is the case in linear building blocks like opamps, or they are explicitly wanted to obtain a signal shifted in frequency, as is the case with mixers. Distortion and intermodulation need to be assessed ac- curately in both cases. Classical numerical simulation techniques using it- erative algorithms for solving the differential equations are slow and inaccurate due to the large difference be- tween the time constants normally present in the circuits of interest. Several numerical methods have been devel- oped to overcome this problem, e.g. the harmonic bal- ance technique (1), multitime analysis (2) and the use of describing functions in circuits with feedback (3). How- ever, the numerical nature of these techniques implies that no symbolic results can be derived, so that re-use of results — in the form of design equations — is not possible. An analysis technique that does yield symbolic re- sults is described in this paper. Based on a set of as- sumptions, the analysis of a weakly nonlinear circuit is reduced to a number of analyses of linear circuits. A lin- ear symbolic analysis core is used for these individual analysis steps, and its results are combined and manip- ulated to get a closed-form symbolic end result. This result can be used as a design equation, or the impact of the circuit nonlinearities on distortion and intermodula- tion can be derived from it. Before explaining this technique, it is to be noted that similar approaches have been followed in the past to obtain symbolic expressions for the distortion in spe- cific classes of circuits. E.g. the distortion in sampling mixers is analyzed in (4), and a method for analyzing the distortion in analog building blocks is presented in (5). All symbolic approaches are intrinsically limited someway, and these publications are no exceptions. The scope of the algorithm presented in this paper is limited to weakly nonlinear circuits. This means that the circuit

Proceedings ArticleDOI
04 Dec 2001
TL;DR: This paper proposes a computationally attractive algorithm for identifying static nonlinearity in a thermoacoustic feedback loop which is either in a limit cycle or is being, driven by Gaussian noise.
Abstract: Many models of systems, important in practice have the form of an interconnection of a known linear model and an unknown nonlinear function. One example of such a system is a model of thermoacoustic instability affecting gas turbine engines and rockets (so-called thermoacoustic feedback loop). In this paper, we propose a computationally attractive algorithm for identifying static nonlinearity in a thermoacoustic feedback loop which is either in a limit cycle or is being, driven by Gaussian noise. The algorithm is based upon functional analytic treatment of the describing function method and lends itself nicely to a class of limit cycling or noise driven feedback systems where the nonlinearity is of a special type. We present examples as well as a simulations with the thermoacoustic feedback loop as an application of the identification algorithm.

Journal Article
TL;DR: In this article, a nonlinear compensator with proper use of the saturation of the operational amplifier is introduced to solve this stability problem, and the stability of the system with multiple nonlinearities is proved by using the combination of the phase plane and describing function method.
Abstract: It is necessary for high precision servo systems to use phase lag compensation with large attenuation to make the system stably conditioned Because there are saturation elements in real systems, they may cause the stably conditioned system become unstable for large offset errors A nonlinear compensator with proper use of the saturation of the operational amplifier is introduced to solve this stability problem The design principle is discussed, and the stability of the system with multiple nonlinearities is proved by using the combination of the phase plane and the describing function method

Proceedings ArticleDOI
01 Sep 2001
TL;DR: The describing function method has been applied to a nonlinear plant with two stable poles and a rate-limiter nonlinearity in the actuator and allows the detection of a saddle-node bifurcation of limit cycles.
Abstract: Limit cycles analysis of feedback systems with rate limiters can be implemented by a classical method in the frequency domain, the harmonic balance method. In this paper, the rate limiter describing function is obtained and applied to the search of limit cycles. Finally, the describing function method has been applied to a nonlinear plant with two stable poles and a rate-limiter nonlinearity in the actuator. This study allows the detection of a saddle-node bifurcation of limit cycles.

Journal ArticleDOI
TL;DR: In this paper, the authors present a method to analyze limit cycle phenomenon for underwater vehicle control systems subject to inherent nonlinearities, characterized by using their corresponding describing functions, and the stability equations decomposed from characteristic equation establish a necessary condition to sustain a limit cycle with respect to the selection of controller coefficients.

Proceedings ArticleDOI
10 Oct 2001
TL;DR: In this article, a combined nonlinearity consisting of resolution (quantifier) and mechanical backlash has been assumed to operate on a linear plant subject to input discretization under a sampling and zero-order hold device.
Abstract: The main objective in the project of a last year Master Engineering course on discrete control where students develop projects based on discrete control at a Spanish university has been to emphasize that there are several points of view when focusing on discretization and the associate mathematical developments depending on the particular analysis technique and problem at hand. In particular, a combined nonlinearity consisting of resolution (quantifier) and mechanical backlash has been assumed to operate on a linear plant subject to input discretization under a sampling and zero-order hold device. It has been proved through the project that because of sampling the system possess nonlinear inertia with the frequency of the first harmonics of nonlinear steady-state sustained oscillations being dependent on the sampling period and the critical locus being depending on frequency.

Journal ArticleDOI
TL;DR: In this paper, a nonlinear quadratic Gaussian/H-infinity/loop transfer recovery (NQG/H∞/LTR) controller was proposed for nonlinear servo systems with hard nonlinearities such as Coulomb friction, dead-zone.
Abstract: 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.