Minimum Phase Property Research Articles

Minimum Phase Property of Chebyshev-Sharpened Cosine Filters

We prove that the Chebyshev sharpening technique, recently introduced in literature, provides filters with a Minimum Phase (MP) characteristic when it is applied to cosine filters. Additionally, we demonstrate that cascaded expanded Chebyshev-Sharpened Cosine Filters (CSCFs) are also MP filters, and we show that they achieve a lower group delay for similar magnitude characteristics in comparison with traditional cascaded expanded cosine filters. The importance of the characteristics of cascaded expanded CSCFs is also elaborated. The developed examples show improvements in the group delay ranged from 23% to 47% at the cost of a slight increase of usage of hardware resources. For an application of a low-delay decimation filter, the proposed scheme exhibits a 24% lower group delay, with 35% less computational complexity (estimated in Additions per Output Sample) and slightly less usage of hardware elements.

Tracking control of a class of non-linear systems with applications to cruise control of air-breathing hypersonic vehicles

A new tracking-control method for general non-linear systems is proposed. A virtual controller and some command references are introduced to asymptotically stabilise the system of the tracking error dynamics. Then, the actual controller and command references are derived by solving a system of linear algebraic equations. Compared with other tracking-control methods in the literature, the tracking-controller design in this paper is simple because it needs only to solve a system of linear algebraic equations. The boundedness of the tracking controller and command references is guaranteed by the solvability of the terminal value problem (TVP) of an ordinary differential equation. For non-linear systems with minimum-phase properties, the TVP is automatically solvable. A numerical example shows that the tracking-control method is still available for some systems with non-minimum-phase properties. To enhance the robustness of the tracking controller, a non-linear disturbance observer (NDO) is introduced to estimate the disturbance. The combination of the tracking controller and the NDO is applied to the tracking control of an air-breathing hypersonic vehicle.

Characteristics analysis and stabilization of a planar 2R underactuated manipulator

SUMMARYThe weightless planar 2R underactuated manipulators with passive last joint are considered in this paper for investigating a feasible method to stabilize the system, which is a second-order nonholonomic-constraint mechanical system with drifts. The characteristics including the controllability of the linear approximation model, the minimum phase property, the Small Time Local Controllability (STLC), the differential flatness, and the exactly nilpotentizable properties, are analyzed. Unfortunately, these negative characteristics indicate that the simplest underactuated mechanical system is difficult to design a stable closed-loop control system. In this paper, nilpotent approximation and iterative steering methods are utilized to solve the problem. A globally effective nilpotent approximation model is developed and the parameterized polynomial input is adopted to stabilize the system to its non-singularity equilibrium configuration. In accordance with this scheme, it is shown that designing a stable closed-loop control system for the underactuated mechanical system can be ascribed to solving a set of nonlinear algebraic equations. If the nonlinear algebraic equations are solvable, then the controller is asymptotically stable. Some numerical simulations demonstrate the effectiveness of the presented approach.

A geometric approach to H∞ control of nonlinear Markovian jump systems

This paper first discusses the H∞ control problem for a class of general nonlinear Markovian jump systems from the viewpoint of geometric control theory. Following with the updating of the Markovian jump mode, the appropriate diffeomorphism can be adopted to transform the system into special structures, which establishes the basis for the geometric control of nonlinear Markovian jump systems. Through discussing the strongly minimum-phase property or the strongly γ-dissipativity of the zero-output dynamics, the H∞ control can be designed directly without solving the traditional coupled Hamilton–Jacobi inequalities. A numerical example is presented to illustrate the effectiveness of our results.

Minimalphasigkeit – keine relevante Eigenschaft für die Regelungstechnik!

Zusammenfassung Ursprünglich wurde die Minimalphasigkeit von H. W. Bode eingeführt, um den eineindeutigen Zusammenhang zwischen Amplitude und Phase des Frequenzgangs zu definieren. Beim Entwurf von digitalen Filtern wird eine andere Definition der Minimalphasigkeit verwendet und ein asymptotisch stabiles Filter als minimalphasig bezeichnet, wenn auch das inverse Filter asymptotisch stabil ist. Schließlich werden von C. I. Byrnes und A. Isidori Systeme mit einer asymptotisch stabilen Nulldynamik als minimalphasig bezeichnet. Wegen der nicht konsistenten Definitionen wird dafür plädiert, den Minimalphasigkeits-Begriff in der Regelungstechnik zu vermeiden und die Stabilität von Filtern und der Nulldynamik mit den wohl definierten Kriterien von Hurwitz oder Ljapunow zu beschreiben.

Automatic Generation of Feedforward Controllers using Dynamic Local Model Networks

A new approach for the automatic generation of a dynamic feedforward control law for nonlinear dynamic systems represented by discrete-time local model networks (LMN) is proposed. The generic model structure of LMNs offers the opportunity to apply such a general and automated approach for model inversion, even when the overall model complexity may be high. LMNs can represent nonlinear dynamic systems of almost arbitrary complexity. Their generic structure allows the generation of a feedback linearizing input transformation in a highly automated way. This paper proposes and discusses such an approach for the important class of LMNs with minimum-phase property. As a subclass of the class of minimum-phase LMNs, only those without numerator dynamics are considered in this manuscript. By using the input transformation, which results from feedback linearization, the feedforward control law is obtained. It can then be applied online for any reference trajectory without pre-planning. Thus, by representing a nonlinear dynamic system by the generic structure of an LMN and applying the proposed feedforward control law generation, a dynamic feedforward control for such a nonlinear system can be found automatically. Finally, the effectiveness of the method is shown on results for a Wiener model.

2B33 PFC design method based on frequency response fitting : Assurance of minimum phase property by stability theorem of descriptor system(The 12th International Conference on Motion and Vibration Control)

Simple adaptive control (SAC) is known as a control method that keeps control performance even if plant properties have changed. However, there is a problem such that the vibratory output occurs in the transient response when SAC is applied to a vibration system which includes anti-resonance modes. The reason why the vibratory output occurs is the structure of SAC with the output caused by the vibratory input corresponding to the anti-resonance frequency. In order to overcome the problem, it is shown that designing parallel feedforward compensator (PFC) appropriately is effective. One of the PFC design methods for continuous-time LTI SISO system has been proposed. In the method, the PFC is designed such that the gain of the augmented system is matched to the one of the desired model. Furthermore, the PFC must give the ASPR property the augmented system. These conditions are described using LMI/BMI conditions. The LMI/BMI conditions are solved by the iterative procedure. However, the leading coefficient of the PFC must be given preliminarily to guarantee the ASPR property. Thus, all coefficients of the PFC could not be design parameters in the conventional method. This paper proposes an alternative method to overcome the problem by using the stability theorem of the descriptor system. By this method, the value of the leading coefficient of the PFC is not required in advance. The effectiveness of the proposed method is verified by numerical simulations.

Adaptive Control of Singularly Perturbed Worm-Like Locomotion Systems

We consider a certain type of a worm-like locomotion system: a three mass point system. We assume that this worm is towing a payload, so the masses differ extremely, and we have to deal with a singularly perturbed system. The dynamics equations are coupled via a small parameter \({{\varepsilon}}\) , which is typical for singularly perturbed systems. In order to apply simple adaptive feedback mechanisms (to approach desired properties of the worm motion) we have to guarantee the minimum-phase property (among others) of such systems. We show that such properties are preserved under singular perturbation. This is done within a general system class via construction of a normal form to gain deeper insight and to analyze such systems. We present the general case and apply the basic transformations to the worm equations. We point out that we are able to classify all required assumptions and properties by system qualities as, e.g., zeros of the reduced model or boundary-layer system. This is in contrast to other authors.

Adaptive Control of MIMO T-S Fuzzy Systems with General Delay Matrices

AbstractThis paper proposes a new solution framework for adaptive control of a class of discrete-time nonlinear MIMO systems in an input-output form based on discrete-time MIMO T-S fuzzy models with general delay matrices and unknown parameters. A prediction fuzzy model is derived and its minimum phase property is clarified. Based the prediction fuzzy model, an adaptive control scheme is presented including parametrization and parameter estimation of the T-S fuzzy model with unknown parameters, parameter adaptation conditions for avoiding the singularity problem and the closed-loop stability proof. An illustrative example and simulation results are presented to demonstrate the studied new concepts and to verify the desired performance of the adaptive MIMO fuzzy control systems.

Optimal piecewise constant reference command for approximate output synthesis of scalar systems: an interpolation approach

AbstractA new method for the minimum-time output trajectory problem is proposed based on the parameterization of the input signal in terms of block-pulse functions. The method computes the optimal time interval and the projection of the input signal in the block-pulse functions basis such that the output system interpolates the desired trajectory in a finite number of points taking into account possible input saturation constraints. In fact it provides a piecewise constant reference signal due to the particular chosen basis functions. No assumptions are made on stability and/or minimum-phase properties and the case of time-varying systems is also discussed. The reliability of the method is extensively highlighted by the presented examples with smooth, triangular, trapezoidal and parabolic output signals.

Pole-free vs. stable-pole designs of minimum variance control for nonsquare LTI MIMO systems

This paper takes advantage of nonuniqueness of the inverse problem for nonsquare transfer function matrices of multivariable systems in order to select such poles, if any, of a minimum variance control system that can either guarantee its closed-loop stability or provide (a sort of) robustness to the control system. As a result, new pole-free and stable-pole MVC designs are offered for nonsquare LTI MIMO systems, the most general of them utilizing the Smith-factorization approach and the so-called control zeros. The new designs contribute to an illustration and extension of the Davison’s theory of (nonsquare) minimum phase systems, in that the lack or presence of (appropriate) control zeros can provide a required performance to the MVC system. Simulation examples in the Matlab/Simulink environment confirm the potential of the control zeros and their impact on redefinition of the minimum phase property

Non-linear sliding mode surfaces for a class of underactuated mechanical systems

This study presents a new non-linear sliding surface to control a class of non-minimum phase underactuated mechanical systems, taking into account uncertainties in their physical parameters. The non-linear surface is designed through a fictitious output, which provides the minimum-phase property and allows to prove stability using Lyapunov theory. The non-linear surface is based on the fictitious output and augmented with a non-linear external controller designed using the Lyapunov theory. The present approach assures exponential stability of the equilibrium point and robust stability to parametric uncertainties, avoiding the appearance of non-desired phenomena, as limit cycles. Two pendulum-like examples inside the class are thoroughly analysed and solved, that is, the pendulum on a cart and the inertia wheel pendulum. Performance, time response and parametric robustness are shown through simulations.

Basilar-membrane responses to broadband noise modeled using linear filters with rational transfer functions.

Basilar-membrane responses to white Gaussian noise were recorded using laser velocimetry at basal sites of the chinchilla cochlea with characteristic frequencies near 10 kHz and first-order Wiener kernels were computed by cross correlation of the stimuli and the responses. The presence or absence of minimum-phase behavior was explored by fitting the kernels with discrete linear filters with rational transfer functions. Excellent fits to the kernels were obtained with filters with transfer functions including zeroes located outside the unit circle, implying nonminimum-phase behavior. These filters accurately predicted basilar-membrane responses to other noise stimuli presented at the same level as the stimulus for the kernel computation. Fits with all-pole and other minimum-phase discrete filters were inferior to fits with nonminimum-phase filters. Minimum-phase functions predicted from the amplitude functions of the Wiener kernels by Hilbert transforms were different from the measured phase curves. These results, which suggest that basilar-membrane responses do not have the minimum-phase property, challenge the validity of models of cochlear processing, which incorporate minimum-phase behavior.

Output Feedback Regulation of Stochastic Nonlinear Systems With Stochastic iISS Inverse Dynamics

This paper develops a unifying framework for output feedback regulation of stochastic nonlinear systems with more general stochastic inverse dynamics. The contributions of this work are characterized by the following novel features: (1) Motivated by the concept of integral input-to-state stability (iISS) in deterministic systems and stochastic input-to-state stability (SISS) using Lyapunov function in stochastic systems, a concept of stochastic integral input-to-state stability (SiISS) using Lyapunov function is first introduced, two important properties of SiISS are obtained: (i) SiISS is strictly weaker than SISS using Lyapunov function; (ii) SiISS is stronger than the minimum-phase property. However, only under the minimum-phase assumption, there is no dynamic output feedback control law for global stabilization in probability. (2) Almost sure boundedness, a reasonable and stronger concept than boundedness in probability, is introduced. The purpose of introducing the concept is to prove the boundedness and convergence of some signals in the closed-loop control system. (3) Some important mathematical tools which play an essential role in the boundedness and convergence analysis of the closed-loop system are established. (4) A unifying framework is proposed to design a dynamic output feedback control law, which drives the states to the origin almost surely while maintaining all the closed-loop signals bounded almost surely.

Contributions to non-identifier based adaptive control in mechatronics

Funnel-Control (FC)–an adaptive (time-varying) MIMO/SISO control strategy–is re-introduced and its applicability introductory for position control of nonlinear, coupled (rigid) robotic systems and for speed control of nonlinear two-mass flexible servo systems is shown. Additionally Error Reference Control (ERC)–a direct derivative of FC–is established. ERC is specially designed with asymmetric boundaries and auxiliary reference, ensuring that the control error evolves within a prespecified tube (a shrinked funnel region, achieving even better tracking performance). FC and hence ERC are based on the high-gain stabilizability of minimum-phase systems with relative degree one and known sign of the high-frequency gain. Although the plant only needs to be known in structure, both concepts assure prescribed transient behavior without identification and/or parameter estimation. As most industrial applications, also the considered robotic and two-mass flexible servo systems, exhibit relative degrees greater than one, FC and ERC cannot directly be applied. Therefore a special state feedback is introduced, reducing the relative degree and retaining the minimum-phase property. The additional implementation of a nominal PI-like extension guarantees good disturbance rejection and asymptotic tracking of (constant) velocity and position reference trajectories. Simulation results for a 6-DOF Manutec r3 robot underpin and compare the achievable position control performance of one overall MIMO Funnel Controller for all joints and one SISO Funnel Controller for each joint (6 controllers). For nonlinear two-mass flexible servo systems, measurement results demonstrate the achievable load speed control performance of FC and ERC in comparison to optimal LQ state feedback.

A Dissipation Inequality for the Minimum Phase Property

The minimum phase property is an important notion in systems and control theory. In this paper, a characterization of the minimum phase property of nonlinear control systems in terms of a dissipation inequality is derived. It is shown that this dissipation inequality is equivalent to the classical definition of the minimum phase property in the sense of Byrnes and Isidori, if the control system is affine in the input and the so-called input-output normal form exists.

Minimum phase properties of finite-interval stochastic realization

A finite-interval stochastically balanced realization is analyzed based on the idealized assumption that an exact finite covariance sequence is available. It is proved that a finite-interval balanced realization algorithm [Maciejowski, J. M. (1996). Parameter estimation of multivariable systems using balanced realizations. In S. Bittanti, & G. Picci (Eds.), Identification, adaptation, learning (pp. 70–119). Berlin: Springer] provides stable minimum phase models, if the size of the interval is at least two times larger than the order of a minimal realization. New algorithms for finite-interval stochastic realization and stochastic subspace identification are moreover derived by means of block LQ decomposition, and the stability and minimum phase properties of models obtained by these algorithms are considered. Numerical simulation results are also included.

Asymptotic stabilization by PID control: Stability analysis based on minimum phase and high-gain feedback

This paper is concerned with the regulator problem for an SISO system using a PID controller. The controlled object is an SISO system which is controllable. Our purpose is to analyze the stability of the PID control system and to determine the PID parameters so that the closed system is asymptotically stable. The main idea is based on the minimum phase property and Lyapunov's stability theorem. The quasi pole-placement method is used to asymptotically stabilize zero dynamics whose poles depend on the PID parameters. The effectiveness of the proposed method is confirmed by simulation results for various plants. © 2006 Wiley Periodicals, Inc. Electr Eng Jpn, 156(1): 44–53, 2006; Published online in Wiley InterScience (www. interscience.wiley.com). DOI 10.1002/eej.20336

STABILITY ANALYSIS OF THE BANDLIMITED AND BANDPASS LINEAR PREDICTION MODELS

A constructive method of proof is presented for the minimum-phase property of two important linear prediction models, i.e., the bandlimited and bandpass linear prediction models. A generic functional symmetric matrix is first constructed. It is shown that the matrix is positive definite under a mild condition. The result is used, along with a special eigen-structure of the associated companion matrix, to establish the stability of the two linear prediction models.

Robust spectral factor approximation of discrete-time frequency domain power spectras

This paper presents a subspace-based identification algorithm for estimating the state-space quadruple [ A , B , C , D ] of a minimum-phase spectral factor from matrix-valued power spectrum data. For a given pair [ A , C ] with A stable, the minimum-phase property is guaranteed via the solution of a conic linear programming (CLP) problem. In comparison with the classical LMI-based solution, this results in a more efficient way to minimize the weighted 2-norm of the error between the estimated and given power spectrum. The conic linear programming problem can be solved in a globally optimal sense. This property is exploited in the derivation of a separable least-squares procedure for the (local) minimization of the above 2-norm with respect to the parameters of the minimal phase spectral factor. The advantages of the derived subspace algorithm and the iterative local minimization procedure are illustrated in a brief simulation study. In this study, the effect of dealing with short length data sets for computing the power spectrum, on the estimated spectral factor, is illustrated.