Non-minimum Phase Model Research Articles

Tracking control of non-minimum phase systems: a kernel-based approach

Feedforward control with model inversion is a widely-used solution for high-precision output tracking. However, because inverting a non-minimum phase model generates unbounded control input, model-inversion only applies to limited types of systems. This paper presents a new non-parametric pseudo-inversion approach to design bounded optimal control input with desirable properties for arbitrary types of systems. Closed-form equations are presented for the batch (full preview) and recursive (limited preview) implementations of this approach, and its performance is compared against existing pseudo-inversion methods in benchmark numerical examples. Furthermore, the practical implementation of the proposed method is demonstrated by designing a feedforward controller for a commercial 3-Dimensional (3D) printer. The results show that the proposed approach effectively compensates for the structural vibrations of the printer, preventing layer-shifting errors that usually happen during high-speed printing.

Output redefinition-based active disturbance rejection control for nonminimum phase hypersonic vehicles

PurposeThis paper aims to propose and verify an improved cascade active disturbance rejection control (ADRC) scheme based on output redefinition for hypersonic vehicles (HSVs) with nonminimum phase characteristic and model uncertainties.Design/methodology/approachTo handle the nonminimum phase characteristic, a tuning factor stabilizing internal dynamics is introduced to redefine the system output states; its effective range is determined by analyzing Byrnes–Isidori normalized form of the redefined system. The extended state observers (ESOs) are used to estimate the uncertainties, which include matched and mismatched items in the system. The controller compensates observations in real time and appends integral terms to improve robustness against the estimation errors of ESOs.FindingsTheoretical and simulation results show that the stability of internal dynamics is guaranteed by the tuning factor and the tracking errors of external commands are globally asymptotically stable.Practical implicationsThe control scheme in this paper is expected to generate a reliable way for dealing with nonminimum phase characteristic and model uncertainties of HSVs.Originality/valueIn the framework of ADRC, a concise form of redefined outputs is proposed, in which the tuning factor performs a decisive role in stabilizing the internal dynamics of HSVs. By introducing an integral term into the cascade ADRC scheme, the compensation accuracy of matched and mismatched disturbances is improved.

Iterative learning control with discrete‐time nonlinear nonminimum phase models via stable inversion

AbstractOutput reference tracking can be improved by iteratively learning from past data to inform the design of feedforward control inputs for subsequent tracking attempts. This process is called iterative learning control (ILC). This article develops a method to apply ILC to systems with nonlinear discrete‐time dynamical models with unstable inverses (i.e., discrete‐time nonlinear nonminimum phase models). This class of systems includes piezoactuators, electric power converters, and manipulators with flexible links, which may be found in nanopositioning stages, rolling mills, and robotic arms, respectively. As these devices may be required to execute fine transient reference tracking tasks repetitively in contexts such as manufacturing, they may benefit from ILC. Specifically, this article facilitates ILC of such systems by presenting a new ILC synthesis framework that allows combination of the principles of Newton's root finding algorithm with stable inversion, a technique for generating stable trajectories from unstable models. The new framework, called invert‐linearize ILC (ILILC), is validated in simulation on a cart‐and‐pendulum system with model error, process noise, and measurement noise. Where preexisting Newton‐based ILC diverges, ILILC with stable inversion converges, and does so in less than one third the number of trials necessary for the convergence of a gradient‐descent‐based ILC technique used as a benchmark.

Cooperative optimization-based distributed model predictive control for constrained nonlinear large-scale systems with stability and feasibility guarantees

This paper proposes a cooperative distributed model predictive control (DMPC) to control the constrained interconnected nonlinear large-scale systems. The main contribution of this approach is its proposed novel cooperative optimization that improves the global cost function of any subsystem. Each subsystem calculates its optimal control by solving the corresponding global cost function. For each subsystem, the global cost function is defined based on a combination of cost functions of all subsystems. If the sampling time is selected appropriately, then the feasibility of the proposed approach will be guaranteed. Furthermore, the sufficient conditions for stability and consequently, for the convergence of the whole system states towards the neighborhood of the origin’s positive region are provided. The effectiveness and performance of the proposed approach are demonstrated via applying it to a nonlinear quadruple-tank system for both minimum-phase and nonminimum-phase models.

Sampled-data extended state observer-based backstepping control of two-link flexible manipulator

Currently, space robots such as planetary robots and flexible-link manipulators (FLMs) are finding specific applications to reduce the cost of launching. However, the structural flexible nature of their arms and joints leads to errors in tip positioning owing to tip deflection. The internal model uncertainties and disturbance are the key challenges in the development of control strategies for tip-tracking of FLMs. To deal with these challenges, we design a tip-tracking controller for a two-link flexible manipulator (TLFM) by developing a sampled-data extended state observer (SD-ESO). It is designed to reconstruct uncertain parameters for accurate tip-tracking control of a TLFM. Finally, a backstepping (BS) controller is designed to attenuate the estimation error and other bounded disturbances. Convergence and stability of the proposed control system are investigated by using Lyapunov theory. The benefits (control performance and robustness) of the proposed SD-ESO-based BS controller are compared with other similar approaches by pursuing both simulation and experimental studies. It is observed from the results obtained that SD-ESO-based BS Controller effectively compensates the deviation in tip-tracking performance of TLFM due to non-minimum phase behavior and model uncertainties with an improved transient response.

Practical Implementation of a Factorized All Pass Filtering Technique for Non-minimum Phase Models

One of the problems of the inverse model-based control techniques is the stability of the identified inverse model after the estimation process by the recursive least square (RLS) method. One solution is to use the all pass filtering (APF) technique to transform non-minimum phase models into minimum phase models [9]. However, there are several cases not cured by the all pass filtering method when the all pass filter algorithm is implemented on the hardware. In this paper, an improved version of the all pass filtering technique is presented to deal with the non-minimum phase models. The factorized APF (fAPF) technique for non-minimum phase models is presented to suggest a simple solution. A simple method for avoiding the calculation of complex numbers is also presented for the easy implementation. Several examples are given to support the proposal.

PI controller tuning for load disturbance rejection using constrained optimization

In this paper, a simple and effective PI controller tuning method is presented. To take both performance requirements and robustness issues into consideration, the design technique is based on optimization of load disturbance rejection with a constraint either on the gain margin or phase margin. In addition, a simplified form of the resulting tuning formulae is obtained for first order plus dead time models. To demonstrate the ability of the proposed tuning technique in dealing with a wide range of plants, simulation results for several examples, including integrating, non-minimum phase and long dead time models, are provided.

Full Order Unknown Inputs Observer for Multiple Time-Delay Systems

Abstract In this paper, a design of a full-order observer for linear time-invariant (LTI) multivariable systems with multiple time-delays and unknown inputs (UI) is proposed. The main idea is to reduce the problem of the unknown input observer (UIO) for systems with multiple time-delays to that of a standard one. To that purpose, the orthogonal collocation method is used to transform the infinite dimensional model of the delayed system described by a set of linear partial differential equations (PDEs) to a finite dimensional one described by a set of linear ordinary differential equations (ODEs). Even using an approximation method, the asymptotic stability of the UIO is well proven. The efficiency of the proposed algorithm is shown using the quadruple-tank benchmark. The two cases of minimum and non-minimum phase models are considered.

Optimal acceleration autopilot design for non-minimum phase missiles using evolutionary algorithms

The flight control system is a key element to achieve required performance in missiles and aircrafts. First purpose of flight control system is to ensuring the stability of the system, then, it attempts to force it to track the guidance commands. This paper provides a straightforward method using evolutionary optimisation algorithms to design an appropriate autopilot for non-minimum phase missiles. In order to bring the results to the actual conditions, the missile non-minimum phase model and actuator dynamics with time delay is considered. Proper indices such as system speed, overshoot, undershoot, steady state error and control signal effort have been incorporated to propose an innovative cost function. Then, several applicable meta-heuristic techniques are employed to optimise this cost function. Genetic algorithm, particle swarm optimisation, artificial bee colony, imperialist competitive algorithm and cuckoo search techniques have been compared in this optimisation problem. Simulation results on two benchmark problem show that this method has acceptable speed and it can be used in gain scheduling control design method for non-minimum phase systems. This method can be a suitable replacement for the time consuming procedure of gain tuning in gain scheduling method. The superior advantage of this method compared to the other methods is automatic tuning of the autopilot gains.

Neural-network-based composite disturbance rejection control for a distillation column

Binary distillation columns are essentially multi-variable systems with couplings, non-minimum phase characteristics, model mismatches and various external disturbances. To get the desired top (distillate) and bottom product composition, a composite disturbance rejection control strategy using a radial basis function network (RBFN) is proposed in this paper. The composite controller includes neural network inverse controller (NNIC) and neural network disturbance observer (NNDOB) both using the inverse model of system which is identified by the RBFN. The stability of the identified inverse model is proved, and a rigorous analysis is also given to show why the NNDOB can effectively suppress the disturbances. Performances of the proposed scheme are compared with PID and NNIC without disturbance compensation in three cases by simulation studies. The simulations demonstrate the feasibility, effectiveness and disturbance rejection property of the proposed method in controlling the product composition of the binary distillation columns.

Digital Adaptive Control of Uncertain Continuous Linear Systems with Quantized Measurements

Digital control algorithm of continuous linear time-invariant uncertain systems with quantized measurements is presented. The control algorithm is derived by the introduction of the State and Parameters Observability Canonical form – SPOC. This representation of linear time-invariant systems enables application of tools from the existing theory of control and estimation of linear systems. The controller applies the certainty equivalence principle for the state estimation and parameters identification. This controller needs only the knowledge of a bound on the number of modes within the required bandwidth of the close loop system. The scheme is BIBO stable for sufficiently adequate quantization level. As an example, the proposed algorithm is applied to an unstable nonminimum phase model of a dynamic vehicle. Simulations demonstrate the performance of the algorithm for different values of quantization input and output levels. Robustness issues with respect to unmodelled dynamics and nonlinearities are briefly addressed and simulated.

Adaptive restricted trajectory tracking for a non-minimum phase hypersonic vehicle model

The design of a nonlinear robust controller for a non-minimum phase model of an air-breathing hypersonic vehicle is presented in this work. When flight-path angle is selected as a regulated output and the elevator is the only control surface available for the pitch dynamics, longitudinal models of the rigid-body dynamics of air-breathing hypersonic vehicles exhibit unstable zero-dynamics that prevent the applicability of standard inversion methods for control design. The approach proposed in this paper uses a combination of small-gain arguments and adaptive control techniques for the design of a state-feedback controller that achieves asymptotic tracking of a family of velocity and flight-path angle reference trajectories belonging to a given class of vehicle maneuvers, in spite of model uncertainties. The method reposes upon a suitable redefinition of the internal dynamics of a control-oriented model of the vehicle dynamics, and uses a time-scale separation between the controlled variables to manage the peaking phenomenon occurring in the system. Simulation results on a full nonlinear vehicle model that includes structural flexibility illustrate the effectiveness of the methodology.

CONTINUOUS-TIME ADAPTIVE LQG/LTR CONTROL

An adaptive continuous-time LQG control with loop transfer recovery is considered. The control problem is analyzed using state-space model and the parameter estimation problem is implemented for corresponding transfer function model. The direct identification of continuous-time model parameters is possible by means of IDTOOL procedure. Computer simulations of third-order systems modeled by a second-order minimum-phase and nonminimum-phase models are given to illustrate the robustness and performance properties of the adaptive LQG/LTR controller.

Non-Minimum Phase Model of Vertical Position Electro-Hydraulic Cylinder for Trajectory Zpetc

Abstract Hydraulic actuator has been widely used in industrial applications due to its fast response and capability of moving heavy load. The nonlinear properties of hydraulic cylinder had challenged researchers to design a suitable controller for position control, motion control and tracking control. Based on these problems, we had done a real-time digital tracking control studies on electro-hydraulic cylinder using trajectory zero phase error tracking control (ZPETC) without factorization of zeros polynomial algorithm. With the proposed strategy, the controller parameters are determined using comparing coefficients methods. The electro-hydraulic system mathematical model is approximated using system identification technique with non-minimum phase system being considered. The real-time experimental result will be compared with simulation result using a model from a real plant.

Robust Adaptive Control Based on Specified Region Pole Assignment for Flexible Hypersonic Vehicle

This paper describes a robust adaptive controller based on specified region pole assignment for flexible hypersonic vehicle. The dynamic model of air-breathing hypersonic vehicle retains features including flexible effects, non-minimum phase behavior, model uncertainties, and strong couplings between flight dynamic and engine. To track velocity and altitude commands, robust controller based on specified region pole assignment is used to make unstable modes of open-loop system stable and guarantee dynamic performance of attitude. Meanwhile adaptive controller is proposed to solve tracking problems when existing control failures or saturation. The simulation results demonstrate that the proposed controller achieves excellent dynamic performance while the engine operates normally.

Flexible Links Manipulators: from Modelling to Control

This paper relates recent results obtained in the field of modelling and control of flexible link manipulators and proposes an investigation of the problem raised by this type of systems (at least in the planar case). First, adopting the modal floating frame approach and the Newton–Euler formalism, we propose an extension of the models for control to the case of fast dynamics and finite deformations. This dynamic model is based on a nonlinear generalisation of the standard Euler–Bernoulli kinematics. Then, based on the models recalled we treat the end-effector tracking problem for the one-link case as well as for the planar multi-link case. For the one-link system, we propose two methods, the first one is based on causal stable inversion of linear non-minimum phase model via output trajectory planning. The other one is an algebraic scheme, based on the parametrization of linear differential operators. For the planar multi-link case the control law proposed is based on causal stable inversion over a bounded time domain of nonlinear non-minimum phase systems. Numerical tests are presented together with experimental results, displaying the well behaved of these approaches.

Multi-rate suboptimal digital redesign of a cascaded continuous-time input time-delay system

This paper presents a new multi-rate control scheme for optimal digital-redesign approach of the input time-delay continuous-time cascaded system. It shows that the states of the multi-rate sampled-data system with the digitally redesigned control law closely match those of the original cascaded continuous-time closed-loop system with analogue control law at any instant between sampling periods. Moreover, the developed digital control law can be implemented using low-cost microcomputers with a relatively longer sampling period. Also, it utilizes the innovative one-duration description of input time-delay sampled-data system which is suitable for any value of input time delay, even larger than one sampling period. As a digitized continuous-time system with a very small sampling period may become an unstable and/or non-minimum phase model, even if the original system is stable and/or minimum phase. A new method, with different sampling rates, suboptimal digital-redesign of a cascaded continuous-time system with different input time delays is accessible in this paper. An illustrative example is presented to demonstrate the effectiveness of the proposed design methodology.

Fourier series based nonminimum phase model for statistical signal processing

In this paper, a parametric Fourier series based model (FSBM) for or as an approximation to an arbitrary nonminimum-phase linear time-invariant (LTI) system is proposed for statistical signal processing applications where a model for LTI systems is needed. Based on the FSBM, a (minimum-phase) linear prediction error (LPE) filter for amplitude estimation of the unknown LTI system together with the Cramer-Rao (CR) bounds is presented. Then, an iterative algorithm for obtaining the optimum LPE filter with finite data is presented that is also an approximate maximum-likelihood algorithm when data are Gaussian. Then three iterative algorithms using higher order statistics (HOS) with finite non-Gaussian data are presented to estimate parameters of the FSBM followed by some simulation results as well as some experimental results with real speech data to support the efficacy of the proposed algorithms using the FSBM. Finally, we draw some conclusions.

Estimation of glottal waves based on nonminimum-phase models

Since the characteristics of the glottal sound source due to vibration of the vocal cords have a great effect on the quality of synthesized speech, there have been intensive studies on glottal waves. It is known experimentally that the waveform is a rounded asymmetrical triangular wave. Many voiced source models have been proposed in which the glottal waves are parametrically represented as possible approaches toward a more natural synthesized speech. There are many unsolved problems, however, since the characteristics of the glottal source must be known for various speech utterances in order to construct the source model. Method of estimating the glottal wave from the observed speech signal include the inverse filtering method, where a filter with the inverse characteristic to the transfer function of the vocal tract is used. In this method, however, there remains the problem of how the essential error due to the separate estimations of the vocal-tract transfer function and the glottal wave can be eliminated. This paper proposes a new estimation algorithm for glottal waves, where the characteristics of the glottal waves and the vocal tract are estimated simultaneously by considering the vocal-tract transfer function, including the characteristics of the glottal source. In the proposed method, the speech generation process is represented by a nonminimum-phase model including the characteristics of the glottal source, and the glottal wave is estimated by estimating the parameters of the transfer function. In the estimation of the glottal wave, the unknown driving input signal must be estimated in parallel to the estimation of the transfer function parameters. An approximate inverse system is introduced in the proposed method, since the inverse system for the transfer function of the nonminimum-phase model is unstable. Using the proposed model, the glottal wave can be directly estimated when the vocal-tract characteristic can be represented by an all-pole model. It is also possible to use the nonminimum-phase ARMA model in this method for the analysis/synthesis of speech that includes glottal waves. The glottal wave is estimated by simulation as well as by observation of actual vowels, and satisfactory results are obtained, indicating the usefulness of the proposed estimation algorithm. © 1998 Scripta Technica, Electron Comm Jpn Pt 3, 81(11): 56–66, 1998

Gain-scheduling trajectory control of a continuous stirred tank reactor

The control of continuous stirred tank reactors is often a challenging problem because of the strong pronounced nonlinearity of the process dynamics. Exact feedback linearization and gain-scheduling are two well-known approaches to the design of nonlinear process control systems. The basic idea in this paper is to combine these techniques to obtain a control structure which preserves the advantages and overcomes some of the problems of the two concepts. In a first step, a nonlinear state feedback controller is computed by exact linearization of the process model to shape the nominal closed-loop system. The required unmeasurable state variables are obtained by simulation of the process model. This part of the controller thus is a pure nonlinear feedforward compensator for the nominal plant. To act against disturbances and model uncertainty, a nonlinear gain-scheduled controller is designed by approximately linearizing the process model not for a number of operating points as in the standard gainscheduling approach but around the nominal trajectory generated by the nonlinear feedforward controller. The design approach is applied to a non-trivial concentration control problem in a continuous stirred tank reactor with nonminimum phase behaviour, unmeasurable states, and model uncertainties as well as unknown disturbances. The nonlinear control structure is compared to a linear controller and to a pure gain-scheduling controller and shows excellent performance even for worst case disturbances and model uncertainties.