Unstable Zeros Research Articles

Hidden dynamics and multi‐stability in an improved third‐order Chua's circuit

By utilising an active diode pair to substitute the classical Chua's diode, an improved third-order Chua's circuit is proposed in this study. This circuit has an unstable zero saddle point and two stable non-zero saddle-foci in a local parameter region, from which hidden dynamics and multi-stability are discovered. The parameter region of hidden dynamics is figured out and hidden attractors are captured numerically and experimentally. Furthermore, multi-stability in the proposed circuit are also exhibited by numerical and circuit simulations.

New interpolation solution and application in system modeling and optimal control with prescribed distance to instability

SummaryWe present a system theoretic interpretation of a two‐sided interpolation problem with a stable rational matrix U (interpolant) without constraints on its norm. It is known that all solutions U of that problem can be expressed as U = Uh+Up, where Uh ranges in the set of all solutions of the associated homogeneous problem, and Up is a particular solution. We present a new solution for Up and prove that it is actually the minimal ‐norm interpolant in the set of all interpolants.We apply these results in system modeling and in optimal control of one‐block plants, with a prescribed bound on the distance to instability of the closed‐loop system. The applications are illustrated by examples. Interesting connections to the augmented basic interpolation problem, to Nehari's problem, and to the stability of one‐block plants with multiple unstable invariant zeros are given. Copyright © 2015 John Wiley & Sons, Ltd.

Estimation of Unknown Disturbances in Gimbal Systems

This paper presents a method for the estimation of unknown disturbances for high precision gimbal systems. Alternative to the classical methods of model inversion and filtering, we employ an asymptotically stabilizing controller which achieves the estimation process even in the presence of unstable zeros. The architecture provides an input equivalent disturbance which can be thought to capture all real disturbances in the system as well as virtual ones such as unmodelled dynamics and nonlinearities. The proposed method is illustrated on a 2-axis gimbal system where system identification is followed by the design on an integral linear quadratic regulator as the stabilizing controller which forms the base of the disturbance observer. It is seen that a random unknown disturbance is estimated successfully in the presence of additional gyro noise.

Comparative study of discretization zero dynamics behaviors in two multirate cases

It is well known that the existence of unstable zero dynamics is recognized as a major barrier in many control systems. When the usual digital control with zero-order hold (ZOH) or fractional-order hold (FROH) input is used, unstable zero dynamics inevitably appear in the discrete-time model even though the continuous-time system with relative degree more than two is of minimum phase. This paper investigates the zero dynamics, as the sampling period tends to zero, of sampled-data models composed of a generalized sample hold function (GSHF), a continuous-time nonlinear plant and a sampler in cascade. More precisely, we show how an approximate sampled-data model can be obtained for nonlinear systems with two special GSHF cases such that sampled zero dynamics of the resulting model can be arbitrarily placed. Further, two GSHFs with appropriate parameters provide nonlinear zero dynamics as stable as possible, or with improved stability properties even when unstable, for a given continuous-time plant. It is also shown that the intersample behavior arising from the multirate input can be localised by appropriately selecting the design parameters based on the stability condition of the zero dynamics. The results presented here generalize well-known ideas from the linear to nonlinear cases.

Self-Excited and Hidden Attractors Found Simultaneously in a Modified Chua's Circuit

By replacing the Chua's diode in Chua's circuit with a first-order hybrid diode circuit, a fourth-order modified Chua's circuit is presented. The circuit has an unstable zero saddle point and two nonzero saddle-foci. By Routh–Hurwitz criterion, it is found that in a narrow parameter region, the two nonzero saddle-foci have a transition from unstable to stable saddle-foci, leading to generations of self-excited and hidden attractors in the modified Chua's circuit simultaneously, which have not been previously reported. Complex dynamical behaviors are investigated both numerically and experimentally. The results indicate that the proposed circuit exhibits complicated nonlinear phenomena including self-excited attractors, coexisting self-excited attractors, hidden attractors, and coexisting hidden attractors.

Disturbance rejection control for non-minimum phase systems with optimal disturbance observer

This paper investigates the disturbance rejection control for stable non-minimum phase (NMP) systems with time delay. A robust disturbance observer (DOB) based control structure is proposed. Specifically, the robust DOB is employed to compensate the uncertain plant into a nominal one, based on which a prefilter is adopted to acquire desired performance. Then, a novel DOB configuration strategy for stable NMP systems is proposed. This strategy synthesizes the internal and robust stability, relative order and mixed sensitivity design requirements together to establish the optimization function. The optimal solution is obtained by standard H∞ theory under the condition of guarantying the presented requirements. We also investigate how the DOB can compensate the uncertain plant into a nominal one. The specifical design procedure is presented for an uncertain plant with both unstable zeros and time delay.

Ultimate Data Based Robust PID Design for Performance

A novel PID tuning method is presented to achieve prescribed performance (in terms of maximum overshoot and settling time) for uncertain systems with unstructured uncertainties. In the first step of the design procedure the plant is identified using a harmonic excitation signal with frequency ©n. In the second step, two recently developed PID controller design approaches are applicable: 1. the approach based on guaranteed phase margin ϕM suitable for systems without integral behaviour and with/without time delay, and for systems with integral behaviour; 2. the approach based on guaranteed gain margin GM, suitable for systems without integral behaviours with unstable zero. Both approaches are based on shifting a selected point of the frequency response of the unknown system into the gain crossover frequency (if using the HM approach) or the phase crossover frequency (if using the GM approach). The developed algorithm has been extended to robust PID controller design for plants with unstructured uncertainties.

On Stable Controller Design For Robust Stabilization of Time Delay Systems

This paper studies the problem of robust stabilization of an infinite dimensional plant by a stable and possibly low order controller. The plant of interest is assumed to have only finitely many simple unstable zeros, however, may have infinitely many unstable poles. In the literature, it has been shown that the problem can be reduced to an interpolation problem and it is possible to obtain lower and upper bounds of the multiplicative uncertainty under which an infinite dimensional stable controller can be generated by a modified Nevanlinna-Pick formulation. We propose that the same interpolation problem can be solved approximately by a finite dimensional approach and present a finite dimensional interpolation function which can be used to find a stable controller. We illustrate this idea by a numerical example and additionally show the effects of the free design parameters of the rational interpolating outer function approach on the numerical example.

基于脉冲计数的零相位差跳变补偿外差干涉信号处理方法

基于脉冲计数的零相位差跳变补偿外差干涉信号处理方法

A simple existence condition of one-degree-of-freedom block decoupling controllers

A block decoupling problem in linear multivariable systems is treated for one-degree-of-freedom controller configuration with unity output feedback. The plant transfer matrix, which may be non-square, is assumed to have unstable simple poles and zeros that may coincide. A simple existence condition of a block decoupling controller is obtained by directional interpolation approaches.

Accurate output tracking for nonminimum phase nonhyperbolic and near nonhyperbolic systems

A new method to achieve an accurate output tracking for nonminimum phase linear systems with nonhyperbolic and near nonhyperbolic internal dynamics is presented. For the classical methods, developed in the framework of the preview based stable inversion, the nonhyperbolicity represents a rather serious inconvenience because the required preactuation time tends to be unacceptably large. Different stable inversion techniques, developed for SISO systems, are based on a proper redefinition of the desired output trajectory with the aim of canceling the undesired effects of unstable zeros.The main purpose of the new approach is to alleviate some theoretical and practical limitations inherent in the above methods.The desired output is partitioned into the transient and steady-state components. The transient input is “a priori” assumed to be given by a spline function. Once the desired output trajectory has been specified, this allows the computation of the unknown transient input as the approximate least-squares solution of Fredholm׳s integral equation corresponding to the explicit formula of the output forced response. The steady-state input is analytically computed exploiting the steady-state output response expressions for inputs belonging to the same set of the desired steady-state output.The main advantage of the resulting technique is that its generality does not require “ad hoc” procedures depending on the particular plant to be controlled. This allows the designer to freely specify the desired output trajectory without requiring it to depend on the unstable zeros of the plant or it to be null over an initial time interval.

Problem of tracking in linear systems with parametric uncertainties under unstable zero dynamics

Proposed was a solution to the problem of tracking in linear SISO systems with unstable zero dynamics in the conditions of parametric uncertainty of the models of plant and reference generator under incomplete information about the state vectors. The approach developed employs the methods of the sliding mode theory in the problems of real-time feedback design, observation, and parameter identification.

Stable controller design for mixed sensitivity reduction of infinite-dimensional systems

In this paper we study the design of stable controllers for mixed sensitivity reduction for systems that have input/output delays and infinitely many unstable zeros. The proposed method is based on interpolation with a unit in H∞, which gives a necessary and sufficient condition for stabilization by stable controllers. Using the Youla parameterization, we introduce a two-block problem for the stable H∞ controller design. The two-block problem is then transformed to a one-block problem solvable by the skew Toeplitz approach.

Performance limitations for networked control systems with plant uncertainty

There has recently been significant interest in performance study for networked control systems with communication constraints. But the existing work mainly assumes that the plant has an exact model. The goal of this paper is to investigate the optimal tracking performance for networked control system in the presence of plant uncertainty. The plant under consideration is assumed to be non-minimum phase and unstable, while the two-parameter controller is employed and the integral square criterion is adopted to measure the tracking error. And we formulate the uncertainty by utilising stochastic embedding. The explicit expression of the tracking performance has been obtained. The results show that the network communication noise and the model uncertainty, as well as the unstable poles and non-minimum phase zeros, can worsen the tracking performance.

High-Sampling Rate Dynamic Inversion—Filter Realization and Applications in Digital Control

This paper presents a stable inversion of nonminimum phase systems with highly efficient computation for high-sampling rate applications. The stable filter that inverts the dynamics of a nonminimum system is based on cascading a stable pole-zero cancellation infinite impulse response (IIR) filter with a high-order finite impulse response (FIR) filter which inverts the unstable zero dynamics. The high-order FIR filter is realized based on efficient IIR filter implementation first introduced by Powell and Chau then later modified by Kurosu. As a demonstrative example, the inversion filters are applied to feedforward tracking and repetitive control algorithms and realized by a field programmable gate array. The controllers are implemented at 100-kHz sampling rate to control the motion of a 4 degrees-of-freedom magnetically levitated shaft in experiment.

Performance-guaranteed Robust PID Controller Design for Systems with Unstable Zero

The paper deals with the development of a new robust PID controller design method that guarantees designer-specified maximum overshoot and settling time for non-minimum phase processes with unstable zero. The PID controller design provides guaranteed gain margin GM. The parameter of the tuning rules is a suitably chosen point of the plant frequency response obtained by a sine-wave signal with excitation frequency ωn. Then, the designed controller moves this point into the phase crossover with the required gain margin GM. The couple (ωn;GM) is specified with respect to closed-loop performance requirements in terms of emax (maximum overshoot) and ts (settling time) according to developed parabolic dependences. The new approach has been verified on a vast batch of benchmark examples; subsequently, the developed algorithm has been extended to robust PID controller design for plants with unstable zero and unstructured uncertainties.

Real-Time Walking Pattern Generation Method for Humanoid Robots by Combining Feedback and Feedforward Controller

This paper focuses on real-time walking pattern generation for humanoid robots with linear inverted pendulum model (LIPM). In general, there are many issues in generating proper walking patterns of center of mass and zero moment point (ZMP) with the LIPM since the LIPM has two drawbacks such as instability and non-minimum phase property. For resolving these difficulties, the paper proposes a new real-time approach by combining a feedback and a feedforward controller. The feedback controller employs a pole placement method which shifts the poles of the LIPM in order to improve system stability. The feedforward controller utilizes advanced pole-zero cancelation by series approximation method (APZCSA) for reducing non-minimum phase property which occurs by an unstable zero and is not able to be dealt with by the feedback controller. In addition, the APZCSA improves the tracking error induced by finite series approximation. Using the two controllers, the proposed method makes the transfer function of overall walking pattern generation system approximately unity and consequently generates a stable walking pattern which follows a desired ZMP according to walking path. The efficiency of the proposed method is verified by walking pattern planning examples and experiments with the humanoid robot MAHRU-R.

Moving horizon least-squares input estimation for linear discrete-time

This paper presents a novel moving horizon least-squares input estimation method for linear discrete-time stochastic systems. For systems with completely unknown initial state and no unstable zeros, some existing work showed that asymptotic input reconstruction is possible in the absence of noises. However, under the same condition but with stochastic noises, most existing input estimators, which are designed to optimally deal with noises, fail to ensure asymptotic unbiasedness. In order to address this limitation for linear discrete-time stochastic systems, we characterize necessary and sufficient conditions for input observability and detectability, and propose a moving horizon least-squares input estimator. Based on the conditions for input observability and detectability, it is proved that our proposed input estimator gives an asymptotically unbiased estimate and has minimal estimation error variance over all linear asymptotically unbiased input estimators. Its effectiveness is illustrated by simulation examples involving aircraft sensor and actuator faults.

A Novel Design of Unknown Input Observer for Fault Diagnosis in Non-minimum Phase Systems

Fault diagnosis has become a significant issue due to rising demands for reliability as well as higher system performance and product quality. There are various approaches for the goal of fault detection and isolation among which we focus on the unknown input observer since observer-based schemes have been successfully adopted in a variety of application fields. This paper presents a novel design of Unknown Input Observer (UIO) for fault detection and isolation which can be applied to non-minimum phase systems. Unlike many previous UIO methods developed in the literature, the proposed scheme can carry out real-time reconstruction of unknown input in the presence of unstable invariant zero (with respect to the relation between the output and the unknown input). We extend the system by augmenting low pass filter(s) in the input(s)-output(s) path to tackle the non-minimum phase problem and generate robust residuals. Finally, the effectiveness of the proposed design method is confirmed through demonstrating a simulation-base example.

Improvement of the Asymptotic Properties of Zero Dynamics for Sampled-Data Systems in the Case of a Time Delay

It is well known that the existence of unstable zero dynamics is recognized as a major barrier in many control systems, and deeply limits the achievable control performance. When a continuous-time system with relative degree greater than or equal to three is discretized using a zero-order hold (ZOH), at least one of the zero dynamics of the resulting sampled-data model is obviously unstable for sufficiently small sampling periods, irrespective of whether they involve time delay or not. Thus, attention is here focused on continuous-time systems with time delay and relative degree two. This paper analyzes the asymptotic behavior of zero dynamics for the sampled-data models corresponding to the continuous-time systems mentioned above, and further gives an approximate expression of the zero dynamics in the form of a power series expansion up to the third order term of sampling period. Meanwhile, the stability of the zero dynamics is discussed for sufficiently small sampling periods and a new stability condition is also derived. The ideas presented here generalize well-known results from the delay-free control system to time-delay case.