Minimum Phase Systems Research Articles

Evaluation of the relationship between gain and phase using extrapolation techniques

The relationships between the magnitude and phase of the frequency response of a stable linear minimum phase system have been examined by several authors, including Bode, and they prove valuable in the design of a wide range of systems. It is shown that direct application of the expressions to estimate either the phase response from a known gain response, or the gain response from a known phase response, can lead to rather inaccurate results. It is also shown that the accuracy can be significantly improved using appropriate extrapolation of the relevant expressions and functions at high and low frequencies. A range of numerical examples demonstrates the improved accuracy as the extrapolation process is refined, and provides a comparison with the situation where the initial given data are not limited in bandwidth.

Direct Adaptive Dynamic Compensation for Minimum Phase Systems With Unknown Relative Degree

We consider parameter-monotonic direct adaptive control for single-input-single-output minimum-phase linear time-invariant systems with knowledge of the sign of the high-frequency gain (first nonzero Markov parameter) and an upper bound on the magnitude of the high-frequency gain. The first part of the paper is devoted to fixed-gain analysis of single-parameter high-gain-stabilizing controllers. Two novel fixed-gain dynamic compensators are presented for stabilizing minimum-phase systems. One compensator stabilizes systems with arbitrary-but-known relative degree, while the other utilizes a Fibonacci series construction to stabilize systems with unknown-but-bounded relative degree. Next, we provide a general treatment of parameter-monotonic adaptive control, including a result that guarantees state convergence to zero. This result is then combined with the high-gain-stabilizing controllers to yield parameter-monotonic direct adaptive dynamic compensation for minimum-phase systems with either arbitrary-but-known or unknown-but-bounded relative degree

On a stability of minimum variance control

AbstractSelf‐tuning control (STC) is a control technique to adjust the parameter of the on‐line controller which adapts itself when plant parameter changes. Effective ways to design STC include minimum‐variance control (MVC), generalized minimum‐variance control (GMVC), and generalized predicted control (GPC). The structure of MVC and GMVC is especially easy and manageable. It is necessary to use MVC so that a plant is a minimum phase system and stable in the discrete time‐domain. But, in general the actual controlled system is in continuous time. Then, when digitizing is performed, it sometimes becomes a non‐minimum phase system. If we set the sampling time very low, an MVC controller has unstable poles. In this paper, we propose an MVC design for a plant which is stable and a minimum phase system in a continuous time domain by introducing a delta operator and weight adjustment of the cost function. Then, we evaluate its stability. © 2007 Wiley Periodicals, Inc. Electr Eng Jpn, 159(1): 17–25, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20472

Neural network based feedback linearization control of an unmanned aerial vehicle

This paper presents a flight control design for an unmanned aerial vehicle (UAV) using a nonlinear autoregressive moving average (NARMA-L2) neural network based feedback linearization and output redefinition technique. The UAV investigated is non-minimum phase. The output redefinition technique is used in such a way that the resulting system to be inverted is a minimum phase system. The NARMA-L2 neural network is trained off-line for forward dynamics of the UAV model with redefined output and is then inverted to force the real output to approximately track a command input. Simulation results show that the proposed approaches have good performance.

拡張逆関数をもちいたフィードフォワードによる非最小位相系制御

In principle feedback control can not change the non minimum-phase characteristics with undershooting phenomenon. An extended inverse function is introduced to realize the stable input-output dynamic inversion of all-pass functions involved in non minimum-phase characteristics. Responses of the non minimum-phase systems can be improved by the combination of the feedforward based on the extended inverse function and the feedback from the minimum-phase state of the object. Numerical example reveals that the feedforward control suppresses the undershooting phenomenon acconmpanied by the responses.

SELF-TUNING CONTROL BASED ON GENERALIZED MINIMUM VARIANCE CRITERION.

The stability of adaptive control systems has been studied extensively for minimum phase systems, mainly for model reference adaptive systems, but complete stability proof for non-minimum phase systems have not been given. In this paper, the stability of two types of self-tuning controllers for discrete time minimum and non-minimum phase plants is studied, namely: recursive estimation of the implicit self-tuning controller parameters based on generalized minimum variance criterion (REGMVC), and another based on generalized minimum variance criterion - β equivalent control approach (REGMVC-β). Stability of the algorithms are proved by the Lyapunov theory.

Equivalent Sliding Mode Control Based On Nonlinear Observer For Nonlinear Non-Minimum-Phase Systems

A continuous sliding mode controller based on nonlinear robust observer for a class of nonlinear non-minimum-phase systems is presented, which utilizes synthetic output that is statically equivalent to the original process output and make the system minimum-phase. The designed continuous sliding mode controller can avoid chattering phenomenon.

Direct adaptive command following and disturbance rejection for minimum phase systems with unknown relative degree

AbstractThis paper considers parameter‐monotonic direct adaptive command following and disturbance rejection for single‐input single‐output minimum‐phase linear time‐invariant systems with knowledge of the sign of the high‐frequency gain (first non‐zero Markov parameter) and an upper bound on the magnitude of the high‐frequency gain. We assume that the command and disturbance signals are generated by a linear system with known characteristic polynomial. Furthermore, we assume that the command signal is measured, but the disturbance signal is unmeasured. The first part of the paper is devoted to a fixed‐gain analysis of a high‐gain‐stabilizing dynamic compensator for command following and disturbance rejection. The compensator utilizes a Fibonacci series construction to control systems with unknown‐but‐bounded relative degree. We then introduce a parameter‐monotonic adaptive law and guarantee asymptotic command following and disturbance rejection. Copyright © 2006 John Wiley & Sons, Ltd.

PERFORMANCE LIMITATION OF SYSTEMS UNDER SIGNAL-TO-NOISE CONSTRAINED CHANNEL

This paper re-discusses [1] and [11], where the prob- lems of feedback stabilization over a signal-to-noise ratio (SNR) constrained channel are studied. The first paper considers both continuous and discrete-time minimum phase systems, while the second extends the results to non-minimum phase ones and pro- poses a linear time-varying feedback strategies to eliminate the effect of non-minimum phase zeros in SNR limited stabilization. In general, the limitations on the ability to stabilize a plant over an SNR constrained channel are imposed mainly by unstable poles and non-minimum phase zeros of the plant.

Flaw impulse response estimation in ultrasonic non-destructive evaluation using bi-cepstra

Pulse-echo measurements in ultrasonic non-destructive evaluation (NDE) are masked by the characteristics of the measuring instruments, the propagation paths taken by the ultrasonic pulses, and are considered to be corrupted by additive noise. It is assumed that the measured pulse echo is obtained by linearly convolving the defect impulse response (IR) with the measurement system response. Deconvolution operation, therefore, seeks to undo the effect of this convolution and extract the IR which is essential for defect identification. Autocorrelation (or power spectrum) deconvolution techniques are limited to the identification of minimum-phase systems. In this work, we show that higher-order cepstra can be used to deconvolve the IR of the flaw from ultrasonic echo measurements (synthetic as well as real). We also demonstrate that a deconvolution based on the use of the bi-cepstrum gives lower identification error variance when compared with the results of Wiener filtering deconvolution, specially at low signal-to-noise ratios (SNR).

Mitigation of symmetry condition in positive realness for adaptive control

Feasibility of nonlinear and adaptive control methodologies in multivariable linear time-invariant systems with state-space realization { A , B , C } is apparently limited by the standard strictly positive realness conditions that imply that the product CB must be positive definite symmetric. This paper expands the applicability of the strictly positive realness conditions used for the proofs of stability of adaptive control or control with uncertainty by showing that the not necessarily symmetric CB is only required to have a diagonal Jordan form and positive eigenvalues. The paper also shows that under the new condition any minimum-phase systems can be made strictly positive real via constant output feedback. The paper illustrates the usefulness of these extended properties with an adaptive control example.

Adaptive estimation and rejection of unknown sinusoidal disturbances in a class of non-minimum-phase nonlinear systems

The author deals with adaptive estimation of unknown disturbances in a class of non-minimum phase nonlinear systems, and the stabilisation and disturbance rejection based on the estimated disturbances. The unknown disturbances are the combination of sinusoidal disturbances with unknown frequencies, unknown phases and amplitudes. The only information of the unknown disturbances is the number of distinctive frequencies inside. The class of nonlinear systems considered by the author consists of nonlinear systems in the output feedback form and the systems may be non-minimum phase, that is with unstable zero dynamics. An adaptive estimation algorithm is developed to give exponentially convergent estimates of the unknown disturbance and the system states. The asymptotic convergent estimates of unknown frequencies are also obtained. The proposed estimation algorithm works for both minimum phase and non-minimum phase nonlinear systems in output feedback form. Disturbance rejection with stabilisation is achieved by combining the control designed for the stabilisation of the disturbance-free system and the exponentially convergent estimate of the disturbance. Under the conditions specified for the control design of the system with no disturbance, the overall stability and complete rejection of the unknown disturbance are guaranteed by the proposed control algorithm.

Adaptive aeroelastic vibration suppression of a supersonic airfoil with flap

This paper concerns the flutter, post-flutter and adaptive control of a non-linear 2-D wing-flap system operating in supersonic/hypersonic flight speed regimes. An output feedback control law is implemented and its performance toward suppressing flutter and limit cycle oscillations (LCOs) as well as reducing the vibrational level in the subcritical flight speed range is demonstrated. This control law is applicable to minimum phase systems and we provide conditions for stability of the zero dynamics. The control objective is to design a control strategy to stabilize the pitch angle while adaptively compensating for uncertainties in all the aeroelastic model parameters. It is shown that all the states of the closed-loop system are asymptotically stable.

Nonlinear Adaptive Control of an Aeroelastic Two-Dimensional Lifting Surface

Adaptive control of a nonlinear two-dimensional wing-flap system operating in an incompressible flowfield is studied. An output feedback control law is implemented, and its performance toward suppressing flutter and limit cycle oscillations, as well as reducing the vibrational level in the subcritical flight speed range, is demonstrated. The control law proposed here is applicable to minimum phase systems, and conditions for stability of the zero dynamics are provided. The control objective is to design a control strategy to drive the pitch angle to a setpoint while adaptively compensating for uncertainties in all of the aeroelastic model parameters. It is shown that all of the states of the closed-loop system are asymptotically stable. Furthermore, an extension is presented to include flap actuator dynamics. Simulations have been presented to validate the efficacy of the proposed strategy. Pertinent conclusions have been outlined.

最小分散制御の安定性に関する一考察

Self tuning control (STC) is a control technique to adjust the parameter of the on-line controller which adapts itself when plant parameter changes. There are minimum-variance control (MVC), generalized minimum-variance control (GMVC), generalized predicted control (GPC), and so on as a effective way of design STC. The structure of MVC and GMVC is especially easy and manageable.It is necessary to use MVC that plant is minimum phase system and stable in the discrete time-domain. But, generally actual controlled system is continuous time. Then, when digitizing is given, it sometimes becomes non-minimum phase system. And if we set up sampling time very small, MVC controller has unstable poles.In this paper, we propose that how to design MVC for plant which is stable and minimum phase system in continuous time domain by introducing a delta-operator and weight adjustment of the cost function. Then, we study about stability.

ROBOTIC UNMANNED AERIAL VEHICLE TRAJECTORY TRACKING CONTROL

Control of Unmanned Aerial Vehicles is a difficult task. One of the most challenging problems are nonlinear nature of the vehicle dynamics and the second is related to the nonminimum phase system. There are several methods to design nonlinear controller. One of them is so-called feedback linearization. Even, after successful design of nonlinear controller several important issues like nonminimum phase problem remain. The system with nonminimum phase dynamics needs to have outputs redefined. The output redefinition technique is used in a way such that the resulting system to be inverted is a minimum phase system. The complex calculation related to the system dynamics redefinition make a real-time computation very difficult. The real-time control system needs to be fast and reliable. In order to make a real-time control possible, a neural networks method has been developed and presented. The NARMA-L2 Neural Network is trained off-line to identify the forward dynamics of the UAV model with the redefined output, which is subsequently inverted to force the real output to approximately track a command input. Simulation results show that the proposed neural network method has overall good performance.

Comparision of Multivariable Controllers for Non-Minimum Phase Systems

A system with one or more right half plane transmission (RHPT) zero is generally called a non-minimum phase system. In this paper, the authors compare two methods of designing multivariable controllers for systems with RHPT zeros. The methods used are decoupled internal model controller and a simple tuning method. In the decoupled IMC method, controller design procedure is developed with the help of a model reduction theorem. Davison has proposed a simple tuning method, which is based on inverse of the steady-state gain matrix [G(0)]. Previously the method was applied to multivariable minimum phase systems only. In the present work the Davison method is applied to systems with multivariable right half plane zero. The decoupled IMC method involves some complex calculations in designing controllers, whereas the Davison method is simple to apply. Simulation results are given for the following 2 × 2 transfer function matrix of the systems: (1) a four-tank system, a benchmark example of multivariable system; (2) a system with individual and multivariable transmission zero at s = 1; and (3) binary distillation column. The performance comparisons are given by the sum of IAE values based on response and interaction both for servo and regulatory problems. The presented simple tuning method gives an improved performance.

The linear periodic output regulation problem

The problem of asymptotic output regulation for linear systems driven by time-varying, T-periodic exosystems is considered in this paper. Necessary and sufficient condition for its solvability based on the existence of periodic solutions of differential Sylvester equations are derived. These conditions constitute a generalization to the periodic case of the celebrated algebraic regulator equations of Francis. A general algorithm for the synthesis of an error-feedback regulator is given. For the case of minimum-phase systems, it is shown that the regulator design can be carried out without the knowledge of the Floquet decomposition of the exosystem, thus extending significantly the applicability of the general result. The more challenging issue of robust regulation by error feedback is also addressed, and solved under a stronger observability condition.

Internal model control structure using adaptive inverse control strategy

In this paper, we propose a new adaptive internal model control scheme based on adaptive finite impulse response filters. This scheme provides the same design procedure for both minimum and nonminimum phase plants. The plants under consideration may contain time delay. The tracking objective is accomplished for both invertible and noninvert-ible stable plants. The internal model of the plant and its inverse are estimated by recursive least-squares and least-mean-squares algorithms, respectively. The closed loop is designed such that the system from the reference input to the plant output can be approximately represented by a pure delay. The effect of the process zeros on the output is compensated by using adaptive finite impulse response filters. This avoids the cancellation of noncancellable zeros of the plant and forces the plant output to track the reference input with a delay. The stability of the closed loop for both minimum and nonminimum phase systems is guaranteed. Computer simulation and laboratory scale experimental results are included in the paper to demonstrate the effectiveness of the proposed method.

Design and Experiments of Anticipatory Learning Control: Frequency-Domain Approach

A frequency-domain design is presented for the anticipatory learning control. Convergence conditions are derived in terms of two design parameters, the lead-time and the learning gain. For minimum phase systems, the design of the anticipatory learning control in the frequency domain is decoupled into a two-step procedure. The design is robust against uncertainties in system modeling. The effectiveness of the anticipatory learning control is demonstrated by an example and experiments. Comparisons of the anticipatory learning control with the conventional P-type, D-type, and PD-type learning control highlight the differences between these close yet distinctive approaches.