Minimum Phase Systems Research Articles

A new tracking controller for discrete‐time SISO non minimum phase systems

AbstractA new tracking controller for discrete‐time Single Input Single Output (SISO) non‐minimum phase (NMP) systems is presented. In the proposed method, after cancelation of poles and cancelable zeros of the system, the controller adds some NMP zeros to compensate the effect of NMP zero (zeros) of the system. As a result, the phase of the overall transfer function will be almost linear and its magnitude approaches unity for all frequencies. The method can be applied even to the systems with complex conjugate NMP zeros. As well, it is applicable to the systems for which the conventional methods cannot properly be used. Furthermore, a generalization of method to continuous‐time systems is another given result. Several examples are provided to illustrate the effectiveness of the method. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

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.

Robust stabilization of nonlinear systems by quantized and ternary control

Results on the problem of stabilizing a nonlinear continuous-time minimum-phase system by a finite number of control or measurement values are presented. The basic tool is a discontinuous version of the so-called semi-global backstepping lemma. We derive robust practical stabilizability results by quantized and ternary controllers and apply them to some control problems. Estimates on the required bandwidth are also provided.

Adaptive backstepping control of uncertain systems with unknown input time-delay

In this paper, we establish the robustness of adaptive controllers designed using the standard backstepping technique with respect to unmodeled dynamics involving unknown input time delay. While noting that some results on robust stabilization of non-minimum phase systems using the backstepping technique are available, we realize that the standard adaptive backstepping technique has only been shown applicable to unknown minimum phase systems. Another significance of our result is to enable the class of systems stablizable by adaptive backstepping controllers to cross the boundary of minimum phase systems, since systems with input time delay belong to non-minimum phase systems. Moreover, the L 2 and L ∞ norms of the system output are also established as functions of design parameters. This implies that the transient system performance can be adjusted by choosing suitable design parameters.

Iterative Learning Control with Advanced Output Data Using Partially Known Impulse Response

This letter investigates an ADILC (Iterative Learning Control with Advanced Output Data) scheme for nonminimum phase systems using a partially known impulse response. ADILC has a simple learning structure that can be applied to both minimum phase and nonminimum phase systems. However, in the latter case, the overall control time horizon must be considered in the input update law, which makes the dimension of the matrices in the convergence condition very large. Also, this makes it difficult to find a proper learning gain matrix. In this letter, a new sufficient condition is derived from the convergence condition, which can be used to find the learning gain matrix for nonminimum phase systems if we know the first part of the impulse response up to a sufficient order. Based on this, an iterative learning control scheme is proposed using the estimation of the first part of the impulse response for nonminimum phase systems.

Design Method of Multi-Input Multi-Output Minimum-Phase State Control System with Decoupling and Feedforward Compensation

In this paper a design of the decoupling control of multi-input multi-output (MIMO) linear system is discussed. A new configuration of the prepositional tandem matrix is shown as a decoupling compensator, and the minimum-phase state control is applied to the resulting decoupled system. In general non minimum-phase characteristics is often accompanied to the decoupled systems. The feedforward compensation makes the non minimum-phase effect of each decoupled scalar system change to delay time. A numerical example is given for the MIMO linear system which conventionally results in non minimum-phase systems.

Robust Stabilization by Linear Output Delay Feedback

The main result of this paper establishes that if a controller $C$ (comprising of a linear feedback of the output and its derivatives) globally stabilizes a (nonlinear) plant $P$, then global stabilization of $P$ can also be achieved by an output feedback controller $C[h]$, where the output derivatives in $C$ are replaced by an Euler approximation with sufficiently small delay $h>0$. This is proved within the conceptual framework of the nonlinear gap metric approach to robust stability. The main result is then applied to finite dimensional linear minimum phase systems with unknown coefficients but known relative degree and known sign of the high frequency gain. Results are also given for systems with nonzero initial conditions.

Quasi-invariant control: Design and functional capabilities

The questions of physical realizability, stability, and control errors were considered by the example of a simplest mathematical model of the linear minimum phase system of quasi-invariant control. The problem of designing the quasi-invariant control was solved on this basis, thus providing the opportunity to extend control by designing a nonlinear system combining the quasi-invariant and classical control strategies.

On the control of non-linear processes: An IDA–PBC approach

A high performance non-linear controller that exploits the properties of port-Hamiltonian systems to precisely define the interconnections and energy dissipations of non-linear processes is presented for a large class of chemical processes. The controller is derived from classical interconnection and damping assignment–passivity based control theory but an additional degree of freedom is introduced in the controller design by the use of “non-exact matching” closed-loop storage functions. By a proper closed-loop interconnection assignment the proposed controller achieves total decoupling between outputs and since no inversion of the process dynamics is made in the design it is equally applicable to minimum phase and nonminimum phase system. The controller design methodology is presented and stability conditions are stated. As illustrative example the proposed method is used to design controllers for a multiple-input/multiple-output non-isothermal continued stirred tank reactor that exhibits nonminimum phase behavior.

Controlled functional differential equations: approximate and exact asymptotic tracking with prescribed transient performance

A tracking problem is considered in the context of a class S of multi-input, multi-output, nonlinear systems modelled by controlled functional differential equations. The class contains, as a prototype, all finite-dimensional, linear, m-input, m-output, minimum-phase systems with sign-definite high-frequency gain. The first control objective is tracking of reference signals r by the output y of any system in S :g ivenλ ≥ 0, construct a feedback strategy which ensures that, for every r (assumed bounded with essentially bounded derivative) and every system of class S, the tracking error e = y − r is such that, in the case λ> 0, lim supt→∞ � e(t)� <λ or, in the case λ = 0, limt→∞ � e(t)� =0 . The second objective is guaranteed output transient performance: the error is required to evolve within a prescribed performance funnel Fϕ (determined by a function ϕ). For suitably chosen functions α, ν and θ, both objectives are achieved via a control structure of the form u(t )= −ν(k(t))θ(e(t)) with k(t )= α(ϕ(t)� e(t)� ), whilst maintaining boundedness of the control and gain functions u and k .I n the case λ = 0, the feedback strategy may be discontinuous: to accommodate this feature, a unifying framework of differential inclusions is adopted in the analysis of the general case λ ≥ 0.

Stabilization of Delay Systems with Nonlinearly Correlated Perturbations by a Constant Controller

In this paper we prove that the same design method of constant controllers for a delay systems given in the literature can be used to stabilize a minimum phase systems with nonlinearly correlated perturbations in the coefficients. This type of perturbation is induced by the composition with strictly positive real functions. This result can also be interpreted on the context of simultaneous and robust stabilization for single-input single-output, linear time invariant delay systems.

Stability of self‐tuning control based on Lyapunov function

AbstractThe overall stability of a self‐tuning controller for discrete‐time systems is proved by using the Lyapunov function in this paper, for minimum and a class of non‐minimum phase systems. The self‐tuning controller utilizes a recursive estimate algorithm for the controller parameters based on the generalized minimum variance criterion. Previously, the stability had been proved in the case of minimum phase systems. A new type of self‐tuning controller for the discrete‐time system with delay in control input is also studied which is named the generalized minimum variance criterion‐β equivalent control, and its stability is proved. The simulation is done to evaluate the performance of the proposed self‐tuning controller. Copyright © 2008 John Wiley &amp; Sons, Ltd.

The Transmission Zero at s Radius and the Minimum Phase Radius of LTI Systems

Abstract In this paper, the transmission zero at s radius and the minimum phase radius of a linear time-invariant (LTI) system are introduced. The former radius gives a measure of how “near” a LTI system is from having a transmission zero at a specified point s ∈ ℂ in the complex plane, and the latter radius measures how “near” a minimum phase system is to a nonminimum phase system. Formulas for computing both radii are presented, along with the procedures for constructing the minimum norm perturbations that achieve the respective radius. Some properties of the two radii and numerical examples are also given.

Adaptive Backstepping Control of Uncertain Systems with Unknown Input Time-Delay

In this paper, we show that adaptive controllers designed using the standard backstepping technique proposed in Krstic [1995] globally stabilize a class of uncertain systems with unknown input time delay and unmodeled dynamics. As such systems belong to non-minimum phase systems and thus our result extends the class of systems stabilizable from minimum phase systems when the standard backstepping technique is employed. Moreover, it is shown that the transient system performance can be improved by adjusting the design parameters.

Robustness of the Inner–Outer Factorization and of the Spectral Factorization for FIR Data

The present paper analyzes the construction of outer functions, the factorization of finite-impulse response (FIR) systems into a minimum phase system and an all-pass part, and the spectral factorization. It investigates the behavior of these operations with respect to errors in the given data. It shows that the error in the constructed outer function grows at least and at most proportional with the logarithm of the degree of the given FIR system and proportional to the error in the given data. For the other two operations, it turns out that they show the same behavior with respect to errors in the given FIR data as the construction of outer functions.

Nonovershooting linear multivariable state feedback tracking controllers

We consider the use of linear multivariable feedback control to achieve a nonover-shooting step response. A method is given for designing an LTI state feedback controller to asymptotically track a constant step reference with no overshoot and arbitrarily small rise time. Results are given for both minimum phase and nonminimum phase multivariable LTI systems.

Adaptive Regulator for Uncertain Linear Minimum Phase Systems with Unknown Undermodeled Exosystems

The design of regulators is addressed for uncertain minimum phase linear systems with known bounds, known upper bound on system order, known relative degree, known high frequency gain sign and for exosystems with unknown order and unknown frequencies with known upper bound. A new adaptive output error feedback control algorithm is proposed which guarantees exponential convergence of the output error into a region which decreases with the order of the unmodeled exosystem dynamics. Exponential regulation is obtained when the regulator exactly models all of the exosystem excited frequencies, while asymptotic regulation is achieved when the regulator overmodels the actual exosystem.

Output Regulation for Linear Minimum Phase Systems With Unknown Order Exosystem

Under the assumption that the regulation error is the only feedback signal, it is shown how to design a global regulator capable of rejecting and/or tracking signals made of at most m sinusoidal terms with unknown frequencies, magnitudes, and phases for linear observable minimum phase systems. The asymptotic convergence of the tracking error to zero is guaranteed provided that an upper bound is known on the number of sinusoidal terms, even though the values of the unknown frequencies may not be recovered. Exponential convergence is achieved if all the exosystem frequencies are excited.

Investigation of inverse simulation for design of feedforward controllers

This paper describes the use of inverse simulation to develop feedforward controllers for model-based output-tracking control system structures, thus avoiding the more complicated techniques of model inversion. Similarities and shortcomings of the inverse simulation and model inversion approaches are explored. It is found that, with suitable values of discretized time interval, the method based on inverse simulation may be preferable for minimum-phase systems. Depending upon zero redistribution within the process of inverse simulation, non-minimum-phase problems for linear systems can also be handled. The conclusions are demonstrated using a non-linear HS125 aircraft model, a linearised Lynx helicopter model and a container ship model for ship steering control and roll stabilization.

Minimum-Phase Infinite-Dimensional Second-Order Systems

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In general, better performance can be achieved with a controlled minimum-phase system than a controlled nonminimum-phase system. We show that a wide class of second-order infinite-dimensional systems with either velocity or position measurements are minimum-phase. The results are illustrated by two examples. </para>