Showing papers on "Describing function published in 2016"

Detection of replay attacks in cyber-physical systems

Abstract: A new approach for detecting replay attacks on cyber-physical systems is presented in this paper. The diagnosis system under a replay attack will not trigger an alarm as long as the received sensor readings are plausible. Since an adversary is not able to react on system changes due to the simplicity of the attack and a lack of system knowledge, our goal is to excite the system in non-regular time intervals. Subsequently we can check by means of signal processing whether the received sensor readings are valid. To excite harmonic oscillations, we insert a nonlinear element in the control loop and exploit the theory of describing functions. Since the nonlinear function is a design parameter, we are able to design robust harmonic oscillations. Another advantage of the proposed approach is the possibility of elegant implementation into existing control systems.

Back-stepping robust trajectory linearization control for hypersonic reentry vehicle via novel tracking differentiator

Abstract: This paper proposes a back-stepping robust trajectory linearization control (TLC) design for hypersonic reentry vehicle (HRV) attitude tracking problem from a novel tracking differentiator perspective. First, the attitude kinematics and dynamics for HRV is formulated and rewritten in feedback form with mismatched and matched uncertainties introduced by variations of various aerodynamic coefficients. Second, a sigmoid function based novel tracking differentiator (STD) with global fast convergence property, simple structure and chattering-free in differential estimation is developed to handle the “explosion of term” problem in back-stepping TLC design. In addition, dynamical performance and noise-attenuation ability of STD are analyzed in frequency domain by describing function method. Third, how to convert between sigmoid function based disturbance observer (SDO) and STD is given, and based on the estimates of uncertainties provided by SDO in attitude and angular rate loop, the back-stepping robust TLC is synthesized to track the respective commands in dual-loop. Then, the stability of the composite SDO-enhanced back-stepping TLC approach is established. Finally, extensive simulation results are presented to demonstrate the effectiveness of the proposed control strategy in improving disturbance attenuation ability and performance robustness against multiple uncertainties.

Design of Ship Course-Keeping Autopilot using a Sine Function-Based Nonlinear Feedback Technique

Abstract: Course keeping for ships is the core of automatic navigation in sea transportation. Much work has concentrated on developing novel control strategies for closed loop systems. We have turned our attention the other way to improve the control performance of marine autopilots in this work by “modulating” the control error using a sine function while the construction of the controller is not changed. The nonlinear feedback signal thus obtained is sent to the controller to replace the control error itself, which used to be the deviation between the output response and the reference input of the system. Such a control scheme is called “nonlinear feedback control” hereafter. Theoretical analysis by using a describing function and robust control theory shows that the same control quality is guaranteed with minor control actions for the nonlinear feedback scheme. Simulation experiments were carried out for the ship Yulong of Dalian Maritime University. It is shown that the method postulated in this paper has advantages of safety and energy saving in navigation; the maximum initial rudder angle is reduced by 31·2% with satisfactory control effect.

Second-order reset elements for stage control design

Abstract: In dealing with inherent limitations during stage control design, the possibilities of a second-order reset element (SORE) are studied, in particular, a second-order low-pass filter with reset. Inducing significantly less phase lag, which follows from describing function analysis, SOREs allow for a significant increase of bandwidth in comparison to linear second-order elements. Being part of a reset control design procedure, loop-shaping of the linear feedback loop with a reset element will be based on a describing function description of this element. For the reset control system, closed-loop stability will be verified by solving linear matrix inequalities. The validity and predictive value of the control design procedure (both in terms of stability and performance) will be demonstrated by means of (measurement) results obtained from an industrial wafer stage system.

Chattering analysis of conventional and super twisting sliding mode control algorithm

Abstract: In this paper, the describing function method is used to analyze the behavior of the two methods, the conventional and the super twisting sliding mode control regarding chattering phenomena. The cases are demonstrated where the first order sliding mode control is more efficient then second order one. This cases are shown based on analysis and simulation.

Stabilization of Mechanical Systems with Backlash by PI Loop Shaping

Abstract: Backlash is one of several discontinuities found in different kinds of systems, it can be found in actuators of different types, such as mechanical and hydraulic, giving way to unwanted effects in the system behavior. PI loop shaping control design implementing a describing function to find the limit cycle oscillations and the appropriate control gain is developed. Therefore a frequency domain approach is implemented for the control of nonlinear system of any kind such as robotics, mechatronics, other kind of mechanisms, electrical motors etc. Finally, in order to corroborate the theoretical background explained in this article, the stabilization of a cart-pendulum system with the proposed control strategy is shown.

A Three-Terminal Switch Model of Constant On-Time Current Mode With External Ramp Compensation

Abstract: Multiphase constant on-time current-mode control based on pulse distribution structure is widely used in voltage regulator application for microprocessor. To minimize ripple cancellation effect, external ramp compensation is used in commercial products. However, external ramp will introduce dynamic to the system and stability margin will be suffered without considering its effect. This paper first studies the effect of external ramp by deriving small-signal transfer function based on describing function method. It is found that external ramp brings additional dynamic, with time constant related with switching period. Then, a simple equivalent circuit model based on three-terminal switch concept is proposed, which considers the effect of external ramp by adding an additional R–L branch. The equivalent circuit model can be reduced to previous unified three-terminal switch model when external ramp is zero and can be reduced to model of constant on-time voltage mode control when external ramp is much larger than inductor current ramp. The proposed three-terminal switch model is a complete model, which can be used to examine all transfer functions and is accurate up to half of switching frequency. The analytical transfer functions are provided for easy reference. The model is verified by SIMPLIS simulation and experimental measurement.

Feedback control of combustion instabilities from within limit cycle oscillations using H∞ loop-shaping and the ν-gap metric

Abstract: Combustion instabilities arise owing to a two-way coupling between acoustic waves and unsteady heat release. Oscillation amplitudes successively grow, until nonlinear effects cause saturation into limit cycle oscillations. Feedback control, in which an actuator modifies some combustor input in response to a sensor measurement, can suppress combustion instabilities. Linear feedback controllers are typically designed, using linear combustor models. However, when activated from within limit cycle, the linear model is invalid, and such controllers are not guaranteed to stabilize. This work develops a feedback control strategy guaranteed to stabilize from within limit cycle oscillations. A low-order model of a simple combustor, exhibiting the essential features of more complex systems, is presented. Linear plane acoustic wave modelling is combined with a weakly nonlinear describing function for the flame. The latter is determined numerically using a level set approach. Its implication is that the open-loop transfer function (OLTF) needed for controller design varies with oscillation level. The difference between the mean and the rest of the OLTFs is characterized using the ν-gap metric, providing the minimum required 'robustness margin' for an [Formula: see text] loop-shaping controller. Such controllers are designed and achieve stability both for linear fluctuations and from within limit cycle oscillations.

A new method for tuning PI controllers with symmetric send-on-delta sampling strategy.

Abstract: In this paper we present a new method for tuning PI controllers with symmetric send-on-delta (SSOD) sampling strategy. First we analyze the conditions that produce oscillations in event based systems considering SSOD sampling strategy. The Describing Function is the tool used to address the problem. Once the conditions for oscillations are established, a new robustness to oscillation performance measure is introduced which entails with the concept of phase margin, one of the most traditional measures of relative stability in closed-loop control systems. Therefore, the application of the proposed robustness measure is easy and intuitive. The method is tested by both simulations and experiments. Additionally, a Java application has been developed to aid in the design according to the results presented in the paper.

Analysis of limit cycle mechanism for two-mass system with backlash nonlinearity

Abstract: The main objective of this paper is to focus on the study of limit cycle characteristics due to backlash nonlinearity in drive system. For a two-mass system, the existence of gear reduction mechanism or ball screw will introduce transmission backlash inevitably, which can aggravate mechanical resonance. Therefore, it is essential to study the limit cycle characteristics, so as to provide a good theoretical basis for limit cycle suppression. Firstly, based on describing function analysis, the system is separated into linear part and nonlinear part in order to facilitate the analysis. Then the stability and oscillation frequency of limit cycle are predicted. Through the study, it is found that the increasing of backlash amplitude does not affect the limit cycle frequency, yet it will aggravate the oscillation amplitude. Moreover, the limit cycle frequency and amplitude will increase with the increasing of system control stiffness. Finally, the theoretical analyses are confirmed by the experimental results.

Linearization and Control of Series-Series Compensated Inductive Power Transfer System Based on Extended Describing Function Concept

Abstract: The extended describing function (EDF) is a well-known method for modelling resonant converters due to its high accuracy. However, it requires complex mathematical formulation effort. This paper presents a simplified non-linear mathematical model of series-series (SS) compensated inductive power transfer (IPT) system, considering zero-voltage switching in the inverter. This simplified mathematical model permits the user to derive the small-signal model using the EDF method, with less computational effort, while maintaining the accuracy of an actual physical model. The derived model has been verified using a frequency sweep method in PLECS. The small-signal model has been used to design the voltage loop controller for a SS compensated IPT system. The designed controller was implemented on a 3.6 kW experimental setup, to test its robustness.

Finding hidden oscillations in the operation of nonlinear electronic circuits

Abstract: Several dynamic behaviours can simultaneously coexist in the operation of a nonlinear electronic circuit. However, some of these dynamics are difficult to be observed due to their sensibility to initial conditions. The use of describing functions to determine the coexistence of multiple dynamics in the operation of an electronic circuit with nonlinear feedback is proposed.

Small signal modeling of the hysteretic modulator with a current ripple synthesizer

Abstract: In this paper, the synthetic ripple modulator for synchronous buck converter is modeled by the describing function approach. A detailed circuit configuration of the PWM modulator is introduced, including the hysteretic band control and the ripple synthesizer. The transfer function of the modulator is derived. By comparing the Bode plots for the open-loop hysteretic modulator and the closed-loop system, a good match is obtained between the analytical model and the SIMPLIS simulation results. The proposed model can be used in the design of the current ripple synthesizer to achieve fast transient response, and in the prediction of the system stability.

Simplified mathematical modelling of phase-shift controlled series-series compensated inductive power transfer system

Abstract: In this paper small-signal model of a symmetrical clamped mode controlled series-series compensated inductive power transfer (IPT) system has been derived using the extended describing function (EDF) method. The derived model has duty cycle, input voltage and switching frequency as control variables and an output voltage as an output variable. A voltage controller has been designed for symmetrical clamped mode control using the SISO Design Tool from MATLAB®. To test the designed controller, a prototype of the series-series compensated IPT system has been developed in lab. Simulation results and experimental results are presented in this paper for the validation of the designed controller.

Influence of jitter on limit cycles in bang-bang clock and data recovery circuits

Abstract: In bang-bang (BB) clock and data recovery circuits (CDR) limit cycles can occur, but these limit cycles are undesired for a good operation of the BB-CDR. Surprisingly, however, a little bit of noise in the system is beneficial, because it will quench the limit cycles. Until now, authors have always assumed that there is enough noise in a BB-CDR such that no limit cycle occurs. In this work, a pseudo-linear analysis based on describing functions is used to investigate this. In particular, the relationship between the input noise and the amplitude of eventual limit cycles is investigated. An important result of the theory is that it allows to quantify the influence of the different loop parameters on the minimal amount of input jitter needed to destroy the limit cycle. Additionally, for the case that there is not enough noise, the worst case amplitude of the limit cycle (which is unavoidable in this case) is quantified as well. The presented analysis exhibits excellent matching with time domain simulations and leads to very simple analytical expressions.

Particle Swarm Optimization of Matsuoka's oscillator parameters in human-like control of rhythmic movements

Abstract: In the field of neuroscience, the Matsuoka's nonlinear neural oscillator is commonly used to model Central Pattern Generator (CPG) in humans/animals. How the parameters of such structure should be selected is not always clear. It was generally done in past studies thanks to a trial-and-error method that needs to be reiterated each time the task changes. Recent studies using a Describing Function Analysis (DFA) of this CPG model provide interesting analytical tuning methods. Nevertheless, as they are based on a linear approximation, they might have a limited efficiency in the particular case of timing-sensitive task, such as the ball-bouncing task considered in this study. A Particle Swarm Optimization (PSO) is thus proposed to select the parameters of a novel neural oscillator-based human-like control architecture able to face disturbances and to adapt to new reference set-points during the ball-bouncing task. The general method presented in the present paper can also be used for other Matsuoka's oscillator tunings and other tasks.

Dynamic model identification of a real-time simple level control system

Abstract: Using a relay feedback approach, mathematical expressions for identification of real-time level control system in transfer function form are proposed in this paper. An ideal relay with hysteresis is feedback to a simple level control system for the generation of sustained oscillations. Subsequently, limit cycle information is utilised in deduced expressions for the estimation of unknown system parameters. Using describing function analysis, the gain of an ideal relay with hysteresis is approximated to yield simple mathematical expressions. Typical examples from literature are considered to show the effectiveness of the proposed identification scheme. Yokogawa distributed control system: CENTUM CS 3000 is considered as a platform for the realisation of both relay feedback and Ziegler–Nichols (ZN) method. The accuracy of derived transfer function models is compared with a model derived from ZN method and thereafter demonstrated through frequency response plots which illustrate the experimental validation of...

Analysis and Compensation of Oscillations Induced by Control Valve Stiction

Abstract: A time-domain approach (TDA) is formulated to analyze oscillations that are induced by control valve stiction in feedback control loops. Analytical relationships are established between the proportional-integral controller parameters, and the oscillation amplitude and period of the process output. Based on the relationships and the robustness to model uncertainties, a new compensation method by tuning controller parameters is proposed to reduce the oscillation amplitude to a desired value. Compared with the describing function approach, the TDA achieves a significant improvement in accuracy in calculating the oscillation amplitude and period. By contrast to the counterparts in the literature, the proposed compensation method is quantitative and avoids tuning the controller parameters in a trial-and-error manner to compensate oscillations. Experimental examples illustrate the effectiveness of the obtained results.

Solved Problems in Dynamical Systems and Control

Abstract: This book presents a collection of exercises on dynamical systems, modelling and control. Each topic covered includes a summary of the theoretical background, problems with solutions, and further exercises. Topics covered include: block diagram algebra and system transfer functions; mathematical models; analysis of continuous systems in the time domain; root locus analysis; frequency domain analysis; PID controller synthesis; state space analysis of continuous systems; controller synthesis by pole placement; discrete time systems and the z transform; analysis of non-linear systems with the describing function method; analysis of nonlinear systems with the phase plane method; and fractional order systems and controllers. Based on tried-and-tested problems and solutions that the authors use in teaching over 500 students each year, this book is essential reading for advanced students with courses in modelling and control in engineering, applied mathematics, biomathematics and physics.

A novel nonlinear feedback control and its application to course-keeping autopilot

Abstract: Over the past years, more and more concentration has been attended on developing novel control algorithms to stabilize or regulate the practical plant. In this note, a novel technique is presented to improve the control performance by modulating the output error using a sine function. This nonlinear feedback signal is sent to the original control law instead of the output error itself, which is the derivation between the system output and the reference signal. That is the so-called nonlinear feedback control technique. By virtue of the describing function and the robust control theory, the theoretical analysis shows that the minor control efforts are required to obtain the same control performance due to the merit of the nonlinear feedback scheme. Simulation experiment based on 〞YULONG〞 vessel is presented to illustrate the effectiveness of the proposed algorithm.

A new view of anti‐windup design for uncertain linear systems in the frequency domain

Abstract: Summary This paper presents a new perspective on the stability problem for uncertain LTI feedback systems with actuator input amplitude saturation. The solution is obtained using the quantitative feedback theory and a 3 DoF non-interfering control structure. Describing function (DF) analysis is used as a criterion for closed loop stability and limit cycle avoidance, but the circle or Popov criteria could also be employed. The novelty is the combination of a controller parameterization from the literature and describing function-based limit cycle avoidance with margins for uncertain plants. Two examples are given. The first is a benchmark problem and a comparison is made with other proposed solutions. The second is an example that was implemented and tested on an X-Y linear stage used for nano-positioning applications. Design and implementation considerations are given. An example is given on how the method can be extended to amplitude and rate saturation with the help of the generalized describing function, and a novel anti-windup compensation structure inspired by previous contributions. Copyright © 2015 John Wiley & Sons, Ltd.

A Modal Superposition Method for the Analysis of Nonlinear Systems

Abstract: In the determination of response of nonlinear structures, computational burden is always a major problem even if frequency domain methods are used. One of the methods used to decrease the computational effort is the modal superposition method for nonlinear systems where the modes of the linear system are used in the calculation. However, depending on the type of the nonlinearity, in order to obtain an accurate response, the number of modes retained in the response calculations needs to be increased, which increases the number of nonlinear equations to be solved. In this study, a method is proposed to decrease the number of modes used for systems having nonlinearities where the equivalent stiffness varies between two limiting values. For such systems, one can define different linear systems for each value of the limiting equivalent stiffness. In this study, it is proposed to use a combination of these linear mode shapes in the modal superposition method. It is shown that proper combination of mode shapes of different linear systems provides satisfactory results by keeping the number of modes used at a minimum. The method is demonstrated on case studies where describing function method is used in the analysis of the nonlinear system.

Sampled describing function analysis of second order sliding modes

Abstract: The describing function method is applied to characterize the steady state behavior of the linear sampled data systems stabilized by a second-order sliding mode algorithm. A describing function of the nonlinearity including a sample and hold element is derived. The proposed method allows to predict the amplitude, frequency and phase of occurring limit cycles caused by the discrete time implementation of discontinuous control algorithms. It is shown that the derived describing function is closely related to its classical continuous time counterpart. The proposed approach is demonstrated using a numerical simulation example as well as a real world experiment.

Reset control of pressure-drop oscillations in boiling micro-channel systems

Abstract: A micro-channel has become a popular scheme in high heat flux electronics cooling. But boiling flow instabilities can occur in these cooling systems with micro-channel heat sinks. These pressure-drop oscillations are limit cycles. We model the system by a van der Pol equation with a control input, which is represented by a control system with a nonlinear feedback element. Using a describing function, we approximate the pressure-drop oscillation in the boiling micro-channel system controlled by a reset controller. Then we obtain a control parameter region where the pressure-drop oscillation is eliminated by the controller. By simulation, we also show that the pressure-drop oscillation is eliminated by the reset controller.

Limit cycle based identification of second order processes with time delay

Abstract: General equations for the identification of stable and unstable second order plus dead time (SOPDT) processes are proposed in this paper. During feedback test, a nonlinear device (relay) is connected to process with a phase lag of n radians yielding sustained oscillations known as limit cycle. With the help of describing function (DF) approximation, equivalent gain of an asymmetrical relay is derived and thereafter used in the derivation of mathematical equations for the identification of process dynamics. When a process output is subjected to measurement noise, relay output shows spurious switching which brings chattering in the limit cycle output. In order to retrieve a clean limit cycle output, Fourier series based curve fitting method is implemented. Using frequency response plots, the efficacy of the proposed methodology is drawn through comparisons between actual and estimated process models.

Fractional order PID controller design for an SOPDT model by online tuning method

Abstract: A relay based online tuning of a series fractional order proportional integral and derivative (FOPID) controller is proposed in this paper. A relay with hysteresis is connected in parallel to the FOPID controller inorder to obtain the limit cycle oscillation. The information provided by this limit cycle can be expressed by using a describing function which helps in determining the process parameters of the unknown plant which is further modeled into an equivalent second order plus dead time (SOPDT) plant. The identification of these parameters is a much needed necessity because the re-tuning of the FOPID controller is based on these model parameters. The proposed method highlights a simpler yet potent identification method which in turn leads to the design of a powerful controller. Also the controller always stays in loop which is not only cost-effective but also negates the harmful effects to the on-off replacement of the controller with the relay and vice versa. This method is then verified through the simulations of certain process models.