Non-minimum Phase Zeros Research Articles

On Overshoot and Nonminimum Phase Zeros

It is widely known that real nonminimum phase zeros lead to step response undershoot, and that the size of the undershoot necessarily tends to infinity as the settling time tends to zero. In this note, we show that the presence of two or more real nonminimum phase zeros can lead to step response overshoot in addition to undershoot. A lower bound on the overshoot is derived, and it is shown that the overshoot, like the undershoot, necessarily tends to infinity as the settling time tends to zero. The results are derived for single-input-single-output linear time-invariant continuous-time systems, and apply to both open-loop control and general two degree-of-freedom closed-loop control

On Performance Limitation in Tracking a Sinusoid

This note studies the performance limitation of a feedback system with a given linear time-invariant (LTI) plant in tracking a sinusoidal signal. It continues and goes beyond some recent studies in the same topic in which it is assumed that the controller can access all the past and future values of the reference signal. In this note, we consider the more realistic (and more difficult) situation where the controller only accesses the current and past values of the reference. An explicit formula of the best attainable performance is obtained for a single-input-single-output (SISO) system which depends on the nonminimum phase zeros of the plant and the frequency of the reference sinusoid. Compared to the previously studied case when the future of the reference is available, this formula clearly shows the extra effort one has to pay due to the lack of the reference information. A partial result for a multiple-input-multiple-output (MIMO) system is also given.

MIMO Linear Equalization With an$H^infty$Criterion

In this paper, we study the problem of linearly equalizing the multiple-input multiple-output (MIMO) communications channels from an $H^infty$ point of view. $H^infty$ estimation theory has been recently introduced as a method for designing filters that have acceptable performance in the face of model uncertainty and lack of statistical information on the exogenous signals. In this paper, we obtain a closed-form solution to the square MIMO linear $H^infty$ equalization problem and parameterize all possible $H^infty$ -optimal equalizers. In particular, we show that, for minimum phase channels, a scaled version of the zero-forcing equalizer is $H^infty$ -optimal. The results also indicate an interesting dichotomy between minimum phase and nonminimum phase channels: for minimum phase channels the best causal equalizer performs as well as the best noncausal equalizer, whereas for nonminimum phase channels, causal equalizers cannot reduce the estimation error bounds from their a priori values. Our analysis also suggests certain remedies in the nonminimum phase case, namely, allowing for finite delay, oversampling, or using multiple sensors. For example, we show that $H^infty$ equalization of nonminimum phase channels requires a time delay of at least $l$ units, where $l$ is the number of nonminimum phase zeros of the channel.

Adaptive output feedback control methodology applicable to non-minimum phase nonlinear systems

An adaptive output feedback control methodology is developed for a class of uncertain multi-input multi-output nonlinear systems using linearly parameterized neural networks. The methodology can be applied to non-minimum phase systems if the non-minimum phase zeros are modeled to a sufficient accuracy. The control architecture is comprised of a linear controller and a neural network. The neural network operates over a tapped delay line of memory units, comprised of the system's input/output signals. The adaptive laws for the neural-network weights employ a linear observer of the nominal system's error dynamics. Ultimate boundedness of the error signals is shown through Lyapunov's direct method. Simulations of an inverted pendulum on a cart illustrate the theoretical results.

１入力多出力線形離散時間系のＨ２制御性能限界

This paper is concerned with intrinsic control performance limitations achievable by feedback for single-input and multi-output (SIMO) possibley unstable and non-minimum phase linear discrete-time systems. We investigate two typical H2 optimal control problems, namely optimal tracking control problem and minmal energy control problem. We derive closed-form analytical expressions of the best achievable tracking error and minmal energy H2 norms. The former consists of plant gain as well as unstable poles and non-minimum phase zeros of the plant, while the latter is only given by unstable poles and non-minimum phase zeros of the plant. Those results are confirmed by several numerical examples, and they characterize easiness and difficulty of plants to be controlled.

DISCRETE-TIME ADAPTIVE CONTROL FOR CONTINUOUS-TIME SYSTEMS USING 2-DELAY LIMITING-ZERO MODEL

A new method for designing a discrete-time adaptive control system for continuous-time system is presented. Using a new discrete-time model of the continuous-time system, called the 2-delay limiting-zero model, this method avoids the problem that the non-minimum phase zeros which arise in the case of rapid sampling of the continuous-time systems having the relative degree greater than one. In this paper, first, the motivation and principle of the 2-delay limiting-zero model of the continuous-time system is discussed. Next, the procedure of constructing the adaptive control system based on the 2-delay limiting-zero model is presented. Moreover, in order to illustrate the effectiveness of the proposed method, computer simulations are carried out for a detailed model of the electro-hydraulic servo system.

REJECTION OF UNKNOWN SINUSOIDAL DISTURBANCES IN NON-MINIMUM-PHASE NONLINEAR SYSTEMS

This paper deals with disturbance rejection with stability based on the estimated state and disturbances. The unknown disturbances are 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 in this paper consists of nonlinear systems in the output feedback form and the systems are nonminimum phase, ie, with unstable zero dynamics. Based on the adaptively estimated disturbances, a new control design is proposed for stabilization and disturbance rejection of the nonlinear system in output feedback form which has a nonminimum phase zero.

CLOSED LOOP IDENTIFICATION OF UNSTABLE POLES AND NON-MINIMUM PHASE ZEROS

This paper addresses estimation of poles and zeros in closed loop systems. For many quantities of interest, e.g. frequency function estimates, overparameterization results in a large increase of the variance but this is not the case for estimates of non-minimum phase zeros and unstable poles. Variance expressions that are asymptotic in model order and sample size are derived and for some systems it is found that open loop and closed loop experiments give the same accuracy.

INPUT DESIGN FOR IDENTIFICATION OF ZEROS

The objective of this contribution is input design for accurate identification of non-minimum phase zeros in linear systems. Recently, several variance results regarding estimation of non-minimum phase zeros have been presented. Based on these results, we will show how to design the input that has the least energy content required to keep the variance of an estimated zero below a certain limit. Both analytical and numerical results are presented. A striking fact of the analytical results is that the variance of an estimated zero is independent of the model order when the optimal input is applied.We will also quantify the benefits of using the optimal design compared to using a white input signal or a square-wave. Robustness issues will also be covered in this presentation. The optimal design depends on the location of the true unknown zero and is therefore infeasible. This is typically circumvented by replacing the true zero by an estimate. The sensitivity of the solution to this estimate is investigated.

A KALMAN FILTER-BASED STABLE DYNAMIC INVERSION FOR DISCRETE-TIME, LINEAR, TIME-VARYING SYSTEMS

In this paper the problem of estimating an unknown input for discrete-time, non-minimum phase, multivariable, linear time-varying systems (LTV) is considered. The initial condition of the plant may be unknown and stochastic process and measurement noise are included. The input signal is modelled as a random walk with drifts. Then it is estimated using a Kalman filter for a uniformly detectable augmented system. A necessary and sufficient condition for the detectability of the augmented system is provided. A Kalman filter-based stable dynamic inversion (SDI) for LTV systems is obtained as a consequence of our solution to the proposed problem. The inversion technique can be applied to achieve output tracking for LTV systems in the presence of non-minimum phase zeros and measurement and/or system noise. We are mainly motivated by typical need to replicate time signals in the automobile industry. Similar problems appear also in other fields as machine tool applications, aeronautic industry a.o.

Electromagnetic Coupling in a dc Motor and Tachometer Assembly

This paper presents an enhanced tachometer model that takes into account the effect of electromagnetic coupling that can exist between the actuator and sensor in an integrated dc motor-tachometer assembly, where the conventional model is found to be inadequate. The tachometer dynamics identified in this paper is experimentally verified, and incorporated in the modeling and parameter identification of a motion system that has multiple flexible elements. It is shown that the tachometer dynamics contributes additional nonminimum phase zeros that degrade the servo system performance in terms of closed-loop bandwidth, disturbance rejection and sensitivity to modeling uncertainty. The zeros of the open loop system are found to vary with the geometric parameters of the motor-tachometer assembly. Based on the insight gained by modeling the electromagnetic coupling, methods for eliminating it and its resulting detrimental effects are also suggested.

Handling State and Output Constraints in MPC Using Time-dependent Weights

A popular method for handling state and output constraints in a model predictive control (MPC) algorithm is to use 'soft constraints', in which penalty terms are added directly to the objective function. Improved closed loop performance can be obtained for plants with nonminimum phase zeros by modifying the MPC formulation to include suitably-designed time-dependent weights on the penalty terms associated with the state and output constraints. When the penalty terms are written in terms of the 'worst-case' l∞-norm, incorporating the appropriate time dependence into the weights provides much better closed loop performance. The approach is illustrated using two multivariable plants with nonminimum phase transmission zeros, where the time-dependent weights cause the open loop predictions to coincide with closed loop predictions, which results in a reduction of output constraint violations.

Identification of performance limitations in control using general SISO models

Previously it has been shown for FIR and ARX models that the variance of identified non-minimum phase zeros depends very little on the model order. In this paper that result is extended to more general SISO model structures. An asymptotic, in the model order and the number of data, expression for the variance of non-minimum phase zeros is derived, The relevance of this expression for finite model orders and number of data is examined.

Dynamical system design from a control perspective: finite frequency positive-realness approach

Nonminimum phase zeros are well known to limit the best achievable control performance when the control gain is allowed to be arbitrarily high. On the other hand, the phase crossover appears to be a limiting factor for performance when high-gain controllers are not allowed. In particular, the positive-realness in a finite frequency range seems crucial for achieving good performance in the presence of control constraints. This paper will first give multiple reasons to support this conjecture, and then develop a systematic method for designing mechanical systems to achieve the finite frequency positive-real (FFPR) property. Specifically, we present a state-space characterization of the FFPR property by generalizing the well known Kalman-Yakubovich-Popov lemma to deal with a class of frequency domain inequalities that are required to hold within a finite frequency interval. The result is further extended for uncertain systems to give a sufficient condition for satisfaction of a robust FFPR property. The (nominal) FFPR result is interpreted in the time-domain in terms of input/output signals. Finally, we show that certain sensor/actuator placement problems to achieve the FFPR property can be reduced to finite dimensional convex problems involving linear matrix inequalities. The method is applied to the shape design of a swing-arm for magnetic storage devices with the objective of maximizing the control bandwidth achievable with a limited actuator power.

Best tracking and regulation performance under control energy constraint

This paper studies optimal tracking and regulation control problems, in which objective functions of tracking error and regulated response, defined by integral square measures, are to be minimized jointly with the control effort, where the latter is measured by the plant input energy. Our primary objective in this work is to search for fundamental design limitations beyond those known to be imposed by nonminimum phase zeros, unstable poles, and time delays. For this purpose, we solve the problems explicitly by deriving analytical expressions for the best achievable performance. It is found that this performance limit depends not only on the plant nonminimum phase zeros, time delays, and unstable poles-a fact known previously-but also on the plant gain in the entire frequency range. The results thus reveal and quantify another source of fundamental performance limitations beyond those already known, which are nonexistent when only conventional performance objectives such as tracking and regulation are addressed without taking into account the control energy constraint. Among other things, they exhibit how the lightly damped poles, the anti-resonant zeros, as well as the bandwidth of the plant may affect the performance.

Undershoot and settling time tradeoffs for nonminimum phase systems

It has been known for some time that real nonminimum phase zeros imply undershoot in the step response of linear systems. Bounds on such undershoot depend on the settling time demanded and the zero locations. In this note, we review such constraints for linear time invariant systems and provide new stronger bounds that consider simultaneously the effect of two real nonminimum phase zeros. Using the concept of zero dynamics, we extend these results to a class of nonlinear systems.

Fundamental performance limitations in tracking sinusoidal signals

This paper attempts to give a thorough treatment of the performance limitation of a linear time invariant multivariable system in tracking a reference signal which is a linear combination of a step signal and several sinusoids with different frequencies. The tracking performance is measured by an integral square error between the output of the plant and the reference signal. Our purpose is to find the fundamental limit for the attainable tracking performance, under any control structure and parameters, in terms of the characteristics and structural parameters of the given plant, as well as those of the reference signal under consideration. It is shown that this fundamental limit depends on the interaction between the reference signal and the nonminimum phase zeros of the plant and their frequency-dependent directional information.

Finite-time behavior of inner systems

In this paper, we investigate how nonminimum phase characteristics of a dynamical system affect its controllability and tracking properties. For the class of linear time-invariant dynamical systems, these characteristics are determined by transmission zeros of the inner factor of the system transfer function. The relation between nonminimum phase zeros and Hankel singular values of inner systems is studied and it is shown how the singular value structure of a suitably defined operator provides relevant insight about system invertibility and achievable tracking performance. The results are used to solve various tracking problems both on finite as well as on infinite time horizons. A typical receding horizon control scheme is considered and new conditions are derived to guarantee stabilizability of a receding horizon controller.

Convergence analysis of the Filtered-U LMS algorithm for active noise control in case perfect cancellation is not possible

The Filtered-U LMS algorithm, proposed by Eriksson for active noise control applications, adapts the coefficients of an infinite-impulse response controller. Conditions for global convergence of the Filtered-U LMS algorithm were presented by Wang and Ren (Signal Processing, 73 (1999) 3) and Mosquera and Pérez-González (Signal Processing, 80 (2000) 5) for the case where perfect noise cancellation is achievable, which means only measurement noise remains. This paper shows that the assumption of perfect cancellation is not necessary. In real situations perfect cancellation is often not achievable due to delays and non-minimum phase zeros. The conclusion is derived by analysis of the structure of the Wiener optimal solution. This also leads to the suggestion of preconditioning filters in the Filtered-U LMS updating. The preconditioning has shown considerable increase of the convergence rate in a realistic simulation study.

A model-based sliding mode control methodology applied to the HDA-plant

Abstract A sliding mode control methodology using output information is demonstrated in application to the HDA-plant, a plant for production of benzene. This process is a highly integrated, non-linear large scale process with non-minimum phase and relative degree zero characteristics. The non-linear control law is designed on the basis of a linear observer-based control system. The non-linear control law uses the states of the linear observer. The performance in the sliding mode is determined by a linear stable sub-manifold of the linear closed loop control system chosen via a robust pole selection scheme. The sliding mode control is optimized to operate in a wide operating region.