Coprime Factorization Research Articles

Robust nonlinear multivariable tracking control design to a manipulator with unknown uncertainties using operator-based robust right coprime factorization

In this paper, an operator-based robust nonlinear multivariable tracking control for a manipulator with uncertainties is proposed by using a robust right coprime factorization approach. In general, there exist unknown modelling errors in measuring structural parameters of the manipulator and external disturbances in real situations. In the present control system design, the effect of the modelling errors and disturbance on system performance is considered to be uncertainties in the manipulator dynamics. Considering the uncertainties, an operator-based robust nonlinear multivariable tracking control using robust right coprime factorization is studied. That is, firstly a robustly stable control based on robust right coprime factorization is designed. Secondly, a robust nonlinear multivariable tracking system is proposed for improving the trajectory of the manipulator, and the output tracking performance is evaluated based on the exponential iteration theorem. Finally, the effectiveness of the designed system is confirmed by simulation results.

Design of Robust Controller for STATCOM Applied to Large Induction Motor Using Normalized Coprime Factorization Approach

Between the different robust H-infinity methods to design the controller for FACTS, loop shaping is known to be one of the effective and feasible methods. This research presents an investigation on H-infinity loop shaping procedure through normalized coprime factorization (NCF) for STATCOM installed at terminals of a large induction motor. The dynamic behaviors of induction motor are analyzed while the uncertainty of system parameters and STATCOM parameters are considered and will be compared with conventional PI controller. Simulation results prove that NCF method has better dynamic response and is more robust than PI controller against variation of system parameters and STATCOM parameters, so in a specified operating point even the system goes to instability.

Right coprime factorizations of MISO fractional time-delay systems

Fractional order models have been more and more used as they offer better characterization of real systems of lumped as well as distributed nature. In this paper, we are interested in multiple-input and single-output (MISO) fractional systems of commensurate order with input or output delays, which have not been well understood so far. With the intent of using Youla parametrization for controller synthesis, we need coprime factorizations and Bézout factors of the systems. In a previous work, we have derived explicit expressions of left coprime factors. The right counterparts are determined in this paper, which allows us to obtain the set of all stabilizing controllers for some classes of delay fractional systems.

Survey of A New Nonlinear Control Technique: An Operator-based Approach

This paper introduces some concepts, theories and applications of operator-based robust right coprime factorization (RRCF) approach, which is a small new method in robust nonlinear control field. In details, the origin and basic concepts of this approach is introduced firstly. Then, the system designs on robust stability, tracking performance and fault detection using the operator-theory-based approach are discussed. Finally, the applications of this method in engineering are shown to validate its effectiveness.

Extended Self-tuning Generalized Predictive Control with Computation Reduction Focused on Closed-loop Characteristics

Abstract This paper proposes a design method of extended self-tuning generalized predictive control (GPC) with computation reduction focused on closed-loop characteristics. The authors have extended GPC by coprime factorization and proposed the extended controller for constructing a strongly stable system. Moreover, the proposed controller is able to be designed to make the same steady state output as pre-designed system's steady state output even if feedback loop is cut. Although self-tuning controller is one of the control methods for systems with uncertainty, there is a problem that the computation of self-tuning GPC increases as design engineer takes long prediction horizon in the design parameters. Therefore this paper considers computation reduction for extended self-tuning GPC focused on closed-loop characteristics. The validity of the proposed method is shown by numerical simulation.

Tracking control and error analysis for non-linear plants using robust right coprime factorisation

In this paper, robust non-linear tracking control problem is considered using robust right coprime factorisation. In previous studies, restrictive conditions that the controlled plant can be described by a monotone operator and an invertible operator were assumed. We propose a control system without these restrictive assumptions for plants described by operators in wider classes. Although the proposed control system does not guarantee perfect tracking, however, the upper bound of the norm of tracking error is estimated based on the generalised H.S. Black formula. The control system for a non-linear plant with uncertainty is also addressed and the upper bound of tracking error is estimated. A design algorithm for the proposed control system is shown. The effectiveness of the proposed control system is verified by numerical examples for nominal and uncertain non-linear plants.

Plasma vertical position, shape, and current control in T-15 tokamak

The paper is devoted to plasma shape, current, vertical position, and vertical speed control systems in T-15 tokamak. Brief definitions of the control problem statement and T-15 tokamak itself as a plant under control are given. A cascade plasma shape and current control system is designed using control channel decoupling matrixes in the inner loop and the H∞ controller in the outer loop obtained by means of H∞ optimization theory based on the normalized coprime factorization approach of a plant model. Proportional and H∞ controllers are used to stabilize the plasma vertical position as well as the plasma vertical speed. The first one was applied to plasma-physics DINA code. During simulations some essential difference between linear DINA-L model and plasma-physics DINA code itself was discovered which led to the conclusion to apply an identification procedure on DINA code to get more accurate linear models for the feedback controller design.

Robust non-linear control design to an ionic polymer metal composite with hysteresis using operator-based approach

In this article, robust non-linear control design to an ionic polymer metal composite (IPMC) with hysteresis, uncertainties and input constraints is studied. The IPMC is a novel smart polymer material, and many potential applications for low mass high displacement actuators in biomedical and robotic systems have been shown. In general, the IPMC has highly non-linear property and hysteretic behaviour, and the control input is subject to some constraints to ensure safety and longer service life of IPMC. Moreover, there exist uncertainties caused by identifying some physical parameters and approximate calculation in dynamic model. As a result, considering measurement error of parameters and model error, a practical non-linear model is obtained, and a non-linear robust control design with hysteresis, uncertainties and input constraints using operator-based robust right coprime factorisation is proposed for an IPMC setup, where, the Prandtl–Ishlinskii (PI) model is used to describe the hysteresis, and which is identified using experimental data. The effectiveness of the proposed control method based on obtained non-linear model is confirmed by simulation and experimental results.

Based on the PS/T Method Control Design of Flexible Systems

The lightly damped resonant poles of flexible system increased norm of coprime factor perturbation, which reduced robust of system after loop shaping design. This paper proposes approaches considering the performance of constraints, to select performance weight function through disturbance attenuation and to adjust the complementary sensitivity weight function through design index. The flexible satellite's attitude control demonstrates that the two approaches are able to rapidly select weight function under ensuring the system's robust and have the ability to depress disturbance.

Robust Control for Nonlinear Systems Using Passivity-Based Robust Right Coprime Factorization

The robust control for nonlinear feedback system with unknown perturbations is discussed based on the robust right coprime factorization and passivity property. Firstly, based on passivity property, the right factorization of the perturbed plant is realized. Secondly, a robust control scheme is proposed considering the tracking property for the nonlinear system with nominal plant and perturbed plant. Finally, we show the connection between the passivity and robust right coprime factorization for the obtained nonlinear system. A numerical example is given to show the effectiveness of the proposed design scheme.

Robust control oriented identification of errors-in-variables models based on normalised coprime factors

A robust control oriented identification approach is proposed to deal with the identification of errors-in-variables models (EIVMs), which are corrupted with input and output noises. Based on normalised coprime factor model (NCFM) representations, a frequency-domain perturbed NCFM for an EIVM is derived according to a geometrical explanation for the v-gap metric. As a result, identification of the EIVM is converted into that of the NCFM. Besides an identified nominal NCFM, its worst case error has to be quantified. Unlike other traditional control-oriented identification methods, the v-gap metric is employed to measure the uncertainties including a priori information on the disturbing noises and the worst case error for the resulting nominal NCFM. Since this metric is also used as an optimisation criterion, the associate parameter estimation problem can be effectively solved by linear matrix inequalities. Finally, a numerical simulation shows the effectiveness of the proposed method.

Which subnormal Toeplitz operators are either normal or analytic?

We study subnormal Toeplitz operators on the vector-valued Hardy space of the unit circle, along with an appropriate reformulation of P.R. Halmosʼs Problem 5: Which subnormal block Toeplitz operators are either normal or analytic? We extend and prove Abrahamseʼs theorem to the case of matrix-valued symbols; that is, we show that every subnormal block Toeplitz operator with bounded type symbol (i.e., a quotient of two bounded analytic functions), whose analytic and co-analytic parts have the “left coprime factorization”, is normal or analytic. We also prove that the left coprime factorization condition is essential. Finally, we examine a well-known conjecture, of whether every subnormal Toeplitz operator with finite rank self-commutator is normal or analytic.

A unified dual-stage actuator control scheme for track seeking and following in hard disk drives

This paper presents a unified approach to track seeking and following control for a dual-stage actuator (DSA) hard disk drive (HDD) system. Based on a doubly coprime factorisation (DCF) method, the DSA controller and the closed-loop dynamics are expressed explicitly in terms of two design parameters. This greatly simplifies the optimisation of design parameters in meeting desired specifications. We then address how to use the design parameters to deal with specific problems in the DSA, that is, control allocation for disturbance rejection and trajectory planning for track seeking. Simulated results are also presented to verify the effectiveness of the proposed DSA controller. Compared with previous works, the proposed approach can fulfil track seeking and following tasks by a single controller without switching. Moreover, the unified controller can achieve desirable performances for both tasks that are equivalent to that by two separate conventional controllers.

System identification for achieving robust performance

The performance of robust controllers hinges on the underlying model set. The aim of the present paper is to develop a system identification procedure that enables the design of a controller that achieves optimal robust performance. Hereto, the complex interrelation between system identification and robust control is thoroughly analyzed and novel connections are established between (i) control-relevant and coprime factor identification and (ii) model uncertainty size and the control criterion. The key technical results include new robust-control-relevant and (Wu,Wy)-normalized coprime factorizations. The results enable the identification of multivariable model sets that achieve high robust performance in a subsequent robust control synthesis. Superiority of the proposed results compared to existing approaches is shown by means of an example.

Robust control for nonlinear systems with unknown perturbations using simplified robust right coprime factorisation

This article is concerned with operator-based robust control for nonlinear feedback system with unknown perturbations with simplified robust right coprime factorisation method. By the designed simplified robust controllers, the Bezout identity for the nominal nonlinear system and the perturbed nonlinear system are guaranteed to be simply satisfied, which means that not only the nonlinear system with nominal plant but also that with perturbed plant can be stabilised. Moreover, the plant output can be guaranteed to asymptotically track the reference output. The effectiveness of the proposed design scheme is verified by the simulation results.

Hyponormality and subnormality of block Toeplitz operators

In this paper, we are concerned with hyponormality and subnormality of block Toeplitz operators acting on the vector-valued Hardy space HCn2 of the unit circle.First, we establish a tractable and explicit criterion on the hyponormality of block Toeplitz operators having bounded type symbols via the triangularization theorem for compressions of the shift operator.Second, we consider the gap between hyponormality and subnormality for block Toeplitz operators. This is closely related to Halmos’s Problem 5: Is every subnormal Toeplitz operator either normal or analytic? We show that if Φ is a matrix-valued rational function whose co-analytic part has a coprime factorization then every hyponormal Toeplitz operator TΦ whose square is also hyponormal must be either normal or analytic.Third, using the subnormal theory of block Toeplitz operators, we give an answer to the following “Toeplitz completion” problem: find the unspecified Toeplitz entries of the partial block Toeplitz matrix A≔[U∗??U∗] so that A becomes subnormal, where U is the unilateral shift on H2.

The dual channel sinewave model: Co-prime sparse sampling, parameter estimation, and the Cramér–Rao Bound

The seven-parameter sine fit algorithm for a dual channel signal model with common frequency has been considered in a prior work as an accurate procedure for parameter estimation. In this paper, we investigate an approach in the dual channel sine fit where the signals from both channels are uniformly subsampled with coprime pair of sparse samplers and present analysis of the frequency estimation problem from undersampled measurements thereby relaxing the demand on devices and instruments by lowering the required sampling frequency as well as bandwidth. The Cramér–Rao Bound based on the coprime subsampled data set is derived and numerical illustrations are given showing the relation between the cost in performance based on the mean squared error and the employed coprime factors for a given measurement time.

GIMC architecture for linear systems with a single I/O delay

A new feedback controller architecture is presented for linear systems with a single I/O delay in the generalized internal model control (GIMC) framework. According to the doubly coprime factorization of these systems, traditional GIMC strategy is extended to linear systems with a single I/O delay. The distinguished feature of the control system architecture is that high tracking performance and good external disturbance rejection could be done separately by a nominal Smith predictor part and a finite dimensional conditional controller. First, a nominal Smith predictor part could be designed to deal with command tracking performance. Second, according to the left coprime factorization of the nominal controller, a finite dimensional conditional controller could be considered for external disturbance rejection, when the controlled plant should be assumed to be a square one. Finally, a simple design example is illustrated the effectiveness of the presented method.Finally, a simple design example is illustrated the effectiveness of the presented method.

Sun–Earth L2 point formation control using polynomial eigenstructure assignment

A nonlinear controller based on polynomial eigenstructure assignment (PEA) is presented for the control of Sun–Earth L2 point formation flying. The relative motion dynamics is formulated as a nonlinear equation and rewritten as a Quasi-Linear Time-Varying (QLTV) model. Using a coprime factorization of the desired closed-loop transfer function, the PEA controller structure is calculated by representing the controller gains as polynomials. During the implementation of spacecraft formation flying, the PEA method is extended from Linear Time-Invariant (LTI) and Linear Parameter-Varying (LPV) models to a QLTV model to produce a closed-loop system with invariant performance over a wide range of conditions. To ensure system performance, the analytic stability analysis of the closed-loop system is developed and a position keeping controller for MIMO formation flying is designed using a decoupling method to achieve the desired performance. Finally, a simulation is carried out to validate the controller performance for the formation flying.

Operator-Based Robust Right Coprime Factorization and a Nonlinear Control Scheme for Nonlinear Plant with Unknown Perturbations

This paper is concerned with operator-based robust right coprime factorization for nonlinear plants with unknown perturbations. Firstly, the right factorization for by nonlinear plant with unknown perturbations is realized using isomorphism and some sufficient conditions are proposed to guarantee the robust stability of the obtained nonlinear feedback control system as well as the plant output asymptotically tracking to the reference input. Secondly, a quantitative control scheme is designed to satisfy the sufficient conditions. Finally, the effectiveness of the proposal is demonstrated by the simulation results.