Reduction of Limit Cycle Amplitude in the Presence of Backlash

1999 ◽  
Vol 121 (2) ◽  
pp. 278-284 ◽  
Author(s):  
Ronen Boneh ◽  
Oded Yaniv
Keyword(s):  
Nonlinear Control ◽  
Limit Cycle ◽  
Closed Loop ◽  
Controller Design ◽  
Linear Time ◽  
Design Technique ◽  
Feedback Systems ◽  
Mass System ◽  
Cycle Amplitude ◽  
Limit Cycle Amplitude

The majority of feedback systems driven by an electric motor can be represented by a two-mass system connected via a flexible drive element. Owing to the presence of backlash, the closed-loop performance such as precision speed, position and force control that can be achieved using a linear time invariant controller is limited, and it is expected that a nonlinear control would be superior. In this paper a nonlinear control structure is proposed and a systematic design technique presented. The advantages of the proposed design technique are: (i) It is robust to plant and backlash uncertainty; (ii) it is quantitative to specifications on the maximum limit cycle amplitude; and (iii) the closed loop is superior to a linear controller design both in lower bandwidth and in lower limit cycle amplitude. A design example is included.

scholarly journals A Graphical Design Approach for Two-Input Single-Output Systems Exploiting Plant/Controller Alignment: Design and Application

2019 ◽  
Vol 142 (1) ◽  
Author(s):  
Nathan A. Weir ◽  
Andrew G. Alleyne
Keyword(s):  
Closed Loop ◽  
Controller Design ◽  
Linear Time ◽  
Design Approach ◽  
Analysis Tool ◽  
Feedback Systems ◽  
Linear Time Invariant ◽  
Single Output ◽  
Graphical Design ◽  
Time Invariant

Abstract Due to the unique structure of two-input single-output (TISO) feedback systems, several closed-loop properties can be characterized using the concepts of plant and controller “directions” and “alignment.” Poor plant/controller alignment indicates significant limitations in terms of closed-loop performance. In general, it is desirable to design a controller that is well aligned with the plant in order to minimize the size of the closed-loop sensitivity functions and closed-loop interactions. Although the concept of alignment can be a useful analysis tool for a given plant/controller pair, it is not obvious how a controller should be designed to achieve good alignment. We present a new controller design approach, based on the PQ method (Schroeck et al., 2001, “On Compensator Design for Linear Time invariant Dual-Input Single-Output Systems,” IEEE/ASME Trans. Mechatronics, 6(1), pp. 50–57), which explicitly incorporates knowledge of alignment into the design process. This is accomplished by providing graphical information about the alignment angle on the Bode plot of the PQ frequency response. We show the utility of this approach through a design example.

scholarly journals Design of active disturbance rejection controller for compass-like biped walking

2018 ◽  
Vol 15 (3) ◽  
pp. 172988141877684 ◽  
Author(s):  
Sumian Song ◽  
Chong Tang ◽  
Zidong Wang ◽  
Yinan Wang ◽  
Gangfeng Yan
Keyword(s):  
Limit Cycle ◽  
Disturbance Rejection ◽  
Controller Design ◽  
Linear Time ◽  
Biped Robot ◽  
Transverse Coordinate ◽  
Linear Time Invariant ◽  
Active Disturbance Rejection ◽  
Unstable Limit Cycle ◽  
Coordinate Control

This article proposes an active disturbance rejection controller design scheme to stabilize the unstable limit cycle of a compass-like biped robot. The idea of transverse coordinate transformation is applied to form the control system based on angular momentum. With the linearization approximation, the limit cycle stabilization problem is simplified into the stabilization of an linear time-invariant system, which is known as transverse coordinate control. In order to solve the problem of poor adaptability caused by linearization approximation, we design an active disturbance rejection controller in the form of a serial system. With the active disturbance rejection controller, the system error can be estimated by extended state observer and compensated by nonlinear state error feedback, and the unstable limit cycle can be stabilized. The numerical simulations show that the control law enhances the performance of transverse coordinate control.

Nonlinear Phenomena in Thermoacoustic Systems With Premixed Flames

2012 ◽  
Author(s):  
Karthik Kashinath ◽  
Santosh Hemchandra ◽  
Matthew P. Juniper
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycle ◽  
Release Rate ◽  
Time Domain ◽  
Limit Cycles ◽  
Integral Relation ◽  
Single Mode ◽  
Rate Of Change ◽  
Cycle Amplitude ◽  
Limit Cycle Amplitude

Nonlinear analysis of thermoacoustic instability is essential for prediction of frequencies, amplitudes and stability of limit cycles. Limit cycles in thermoacoustic systems are reached when the energy input from driving processes and energy losses from damping processes balance each other over a cycle of the oscillation. In this paper an integral relation for the rate of change of energy of a thermoacoustic system is derived. This relation is analogous to the well-known Rayleigh criterion in thermoacoustics, but can be used to calculate the amplitudes of limit cycles, as well as their stability. The relation is applied to a thermoacoustic system of a ducted slot-stabilized 2-D premixed flame. The flame is modelled using a nonlinear kinematic model based on the G-equation, while the acoustics of planar waves in the tube are governed by linearised momentum and energy equations. Using open-loop forced simulations, the flame describing function (FDF) is calculated. The gain and phase information from the FDF is used with the integral relation to construct a cyclic integral rate of change of energy (CIRCE) diagram that indicates the amplitude and stability of limit cycles. This diagram is also used to identify the types of bifurcation the system exhibits and to find the minimum amplitude of excitation needed to reach a stable limit cycle from another linearly stable state, for single-mode thermoacoustic systems. Furthermore, this diagram shows precisely how the choice of velocity model and the amplitude-dependence of the gain and the phase of the FDF influence the nonlinear dynamics of the system. Time domain simulations of the coupled thermoacoustic system are performed with a Galerkin discretization for acoustic pressure and velocity. Limit cycle calculations using a single mode, as well as twenty modes, are compared against predictions from the CIRCE diagram. For the single mode system, the time domain calculations agree well with the frequency domain predictions. The heat release rate is highly nonlinear but, because there is only a single acoustic mode, this does not affect the limit cycle amplitude. For the twenty-mode system, however, the higher harmonics of the heat release rate and acoustic velocity interact resulting in a larger limit cycle amplitude. Multi-mode simulations show that in some situations the contribution from higher harmonics to the nonlinear dynamics can be significant and must be considered for an accurate and comprehensive analysis of thermoacoustic systems.

Limit Cycle Behavior of a Flexible Truck

1987 ◽  
Vol 54 (4) ◽  
pp. 930-934 ◽  
Author(s):  
A. M. Whitman ◽  
J. E. Molyneux
Keyword(s):  
Limit Cycle ◽  
Critical Speed ◽  
Perturbation Procedure ◽  
Cycle Amplitude ◽  
Creep Force ◽  
Limit Cycle Amplitude ◽  
Limit Cycle Behavior

We calculate the variation in critical speed of a flexible truck as a function of limit cycle amplitude and truck parameters (i.e., shear and bending stiffnesses, and truck geometry), by means of a perturbation procedure. We find that the creep force nonlinearity is dominant, and that it can cause the nonlinear critical speed to be either lower or higher than the linear critical speed, depending on the values of the two stiffnesses.

Sensitivity Enhancement of Cantilever-Based Sensors Using Feedback Delays

2010 ◽  
Vol 5 (4) ◽  
Author(s):  
Calvin Bradley ◽  
Mohammed F. Daqaq ◽  
Amin Bibo ◽  
Nader Jalili
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Stable Limit Cycle ◽  
Sensitivity Enhancement ◽  
Design Parameters ◽  
Feedback Signal ◽  
Stable Limit ◽  
Theoretical Understanding ◽  
Cycle Amplitude ◽  
Limit Cycle Amplitude

This paper entails a novel sensitivity-enhancement mechanism for cantilever-based sensors. The enhancement scheme is based on exciting the sensor at the clamped end using a delayed-feedback signal obtained by measuring the tip deflection of the sensor. The gain and delay of the feedback signal are chosen such that the base excitations set the beam into stable limit-cycle oscillations as a result of a supercritical Hopf bifurcation of the trivial fixed points. The amplitude of these limit-cycles is shown to be ultrasensitive to parameter variations and, hence, can be utilized for the detection of minute changes in the resonant frequency of the sensor. The first part of the manuscript delves into the theoretical understanding of the proposed mechanism and the operation concept. Using the method of multiple scales, an approximate analytical solution for the steady-state limit-cycle amplitude near the stability boundaries is obtained. This solution is then utilized to provide a comprehensive understanding of the effect of small frequency variations on the limit-cycle amplitude and the sensitivity of these limit-cycles to different design parameters. Once a deep theoretical understanding is established, the manuscript provides an experimental study to investigate the proposed concept. Experimental results demonstrate orders of magnitude sensitivity enhancement over the traditional frequency-shift method.

Bounded-Input-Bounded-Output Stability of Nonlinear Feedback Systems in a Volterra Representation

2000 ◽  
Author(s):  
John W. Glass ◽  
Matthew A. Franchek
Keyword(s):  
Stability Condition ◽  
Closed Loop ◽  
Controller Design ◽  
External Disturbance ◽  
Design Procedure ◽  
Nonlinear Feedback ◽  
Feedback Systems ◽  
Volterra Kernels ◽  
The Stability ◽  
Nonlinear Feedback Systems

Abstract Presented in this paper is a stability condition for a class of nonlinear feedback systems where the plant dynamics can be represented by a finite series of Volterra kernels. The class of Volterra kernels are limited to p-linear stable operators and may contain pure delays. The stability condition requires that the linear kernel is nonzero and that the closed loop characteristic equation associated with the linearized system is stable. Next, a sufficient condition is developed to upper bound the infinity-norm of an external disturbance signal thereby guaranteeing that the internal and output signals of the closed loop nonlinear system are contained in L∞. These results are then demonstrated on a design example. A frequency domain controller design procedure is also developed using these results where the trade-off between performance and stability are considered for this class of nonlinear feedback systems.

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