Bezout Identity Research Articles

Cyclicity of the shift operator through Bezout identities

Cyclicity of the shift operator through Bezout identities

On transforming commensurate time-delay systems with multi-inputs to fully actuated systems via differential flatness

The main purpose of the paper is to transforming the commensurate time-delay systems (TDSs) with multi-inputs to fully actuated systems via differential flatness. The flat output of commensurate TDSs with multi-inputs can be constructed explicitly on the basis of the solution to the Bezout identity. By taking the flat output of TDSs as the generalised state, the commensurate TDSs with multi-inputs can be equivalently transformed into a higher-order fully actuated systems (HOFAS). Then the finite spectrum assignment (FSA) problem of the commensurate TDSs with multi-inputs can be addressed immediately by the utilisation of the HOFAS approach. A numerical example illustrates the effectiveness of the results presented in this paper.

Robustness Analysis for Sundry Disturbed Open Loop Dynamics Using Robust Right Coprime Factorization

In this paper, the robustness of a system with sundry disturbed open loop dynamics is investigated by employing robust right coprime factorization (RRCF). These sundry disturbed open loop dynamics are present not only in the feed forward path, but also within the feedback loop. In such a control framework, the nominal plant is firstly right coprime factorized and a feed forward and a feedback controllers are designed based on Bezout identity to ensure the overall stability. Subsequently, considering the sundry disturbed open loop dynamics, a new condition formulated as a disturbed Bezout identity is put forward to achieve the closed loop stability of the system, even in the presence of disturbances existing in sundry open loops, where in the feedback loop a disturbed identity operator is defined. This approach guarantees the system robustness if a specific inequality condition is satisfied. And, it should be noted that the proposed approach is applicable to both linear and nonlinear systems with sundry disturbed open loop dynamics. Simulations demonstrate the effectiveness of our methodology.

Operator-based adaptive robust control for uncertain nonlinear systems by combining coprime factorization and fuzzy control method1

In this paper, the robust stability of nonlinear system with unknown perturbation is considered combining operator-based right coprime factorization and fuzzy control method from the input-output view of point. In detail, fuzzy logic system is firstly combined with operator-based right coprime factorization method to study the uncertain nonlinear system. By using the operator-based fuzzy controller, the unknown perturbation is formulated, and a sufficient condition of guaranteeing robust stability is given by systematic calculation, which reduces difficulties in designing controller and calculating inverse of Bezout identity. Implications of the results related to former results are briefly compared and discussed. Finally, a simulation example is shown to confirm effectiveness of the proposed design scheme of this paper.

Nonlinear Control System Design of an Underactuated Robot Based on Operator Theory and Isomorphism Scheme

The number of actuators of an underactuated robot is less than its degree of freedom. In other words, underactuated robots can be designed with fewer actuators than fully actuated ones. Although an underactuated robot is more complex than a fully actuated robot, it has many advantages, such as energy, material, and space saving. Therefore, it has high research value in both control theory and practical applications. Swing-up is a mechanism with two links, which mimics a gymnast performing a horizontal bar movement. Over the past few decades, many sufficiently robust control techniques have been developed for a fully actuated robot but almost none of them can be directly applicable to an underactuated robot system. The reason is that such control techniques require certain assumptions that are valid only for fully actuated robot systems but not for underactuated ones. In this paper, a control system design method for underactuated robots based on operator theory and an isomorphism scheme is first proposed. Bezout identity is designed using isomorphism. The effectiveness of the design method is confirmed by simulation. The simulation results show that the performances, such as robust stability and response time, of an underactuated robot control system are improved.

Operator-based robust stability for nonlinear systems with multiple uncertainties using coprime factorisation method

In this paper, the robust stability of nonlinear systems with multiple uncertainties is considered by using a composite operator-based coprime factorisation method. Firstly, as for the exogenous external disturbance, the disturbance model is considered and the disturbance output is guaranteed to be bounded. Meantime, the adverse effect resulting from it is transformed to an equivalent effect on the stable part of the systems. By using operator-based coprime factorisation method, a feasible framework on multiple uncertainties is obtained. Then, the obtained equivalent effects resulting from the modelled disturbance and adverse effect from the internal perturbation are unified and addressed. Thirdly, sufficient conditions on guaranteeing robust stability of the considered nonlinear systems are considered by using robust coprime factorisation method, for relaxing computation burden on Bezout identity and avoiding requirement on knowing the perturbation signal. Based on the proposed conditions, two controllers are designed and the robust stability of the considered systems is proved. Finally, simulation results are shown to explain the proposed design scheme and confirm its effectiveness of this paper.

Robust stabilization control for uncertain nonlinear systems based on two-step coprime factorization

In this paper, nonlinear control for dealing with a class of uncertain nonlinear systems is discussed according to operator-based coprime factorization method. In detail, firstly, for guaranteeing internal-output stability of nominal nonlinear systems with left factorization, a simple and effective design scheme is proposed with a weak assumption. Simultaneously, based on the proposed scheme, the given nominal nonlinear systems are proved to be coprime. Secondly, tracking performance of the considered systems is obtained by designing a suitable Bezout identity without complicated calculation on the obtained left factorization. Thirdly, with respect to the considered nonlinear system, when it comes to appear perturbation, different adverse effects resulting from different perturbation are considered to guarantee robust stability. For the two different cases, two suffcient conditions using inherent characteristics are proposed for guaranteeing robust stability. Finally, two examples are shown to confirm effectiveness of the proposed design scheme of this paper.

Rational forms for block lower-triangular matrices

Linear time invariant (LTI) systems are often represented by rational forms or rational matrix functions. Such forms exhibit important properties of a system, but these properties are often thought to be due to time invariance. Are there such forms for linear time-variant (LTV) systems with comparable properties, generalizing the LTI case? One major obstacle to derive them has been the lack of a divisibility theory for general block lower-triangular matrices, in particular the lack of a Euclidean algorithm or, in its wake, Smith forms and Smith-McMillan forms. But do rational forms for matrices, generalizing the time-invariant properties, exist at all? It turns out they do. The theory presented here produces representations for block lower-triangular matrices as ratios of (block) staircase (or echelon) matrices, and shows how central properties such as co-prime factorization or Bezout identities hold. Instead of the Euclidean algorithm, the theory uses a technique derived from a paper of Paul Van Dooren, namely ‘dead-beat control’. The theory finds good applications in various domains (computational efficiency, determining pre-conditioners, control theory and model reduction via generalized forms of interpolation).

Robust control for uncertain nonlinear feedback system using operator-based right coprime factorization

The robust control issue for uncertain nonlinear system is discussed by using the method of right coprime factorization. As it is difficult to obtain the inverse of the right factor due to the high nonlinearity, the proving of the Bezout identity becomes troublesome. Therefore, two sufficient conditions are derived to manage this problem with the nonlinear feedback system as well as that with the uncertain nonlinear feedback system under the definition of Lipschitz norm. A simulation of temperature control is given to demonstrate the validity of the proposed method.

Nonlinear Control Method for Nonlinear Systems with Unknown Perturbations by Combining Left and Right Factorization

AbstractIn this paper, nonlinear control design scheme for a class of nonlinear systems is proposed based on operator coprime factorization theory. In detail, two stable controllers are provided to design a Bezout identity by combining left factorization (not coprime) with right factorization. Based on the proposed design method, a realization approach to left coprime factorization for the nonlinear system is obtained, which provides an effective framework for constructing left coprime factorization. Meanwhile, internal‐output stability of the nonlinear system is guaranteed. After that, based on the obtained left coprime factorization, the cases of the nonlinear systems with perturbations are discussed for guaranteeing robust stability for the perturbed systems. For the perturbations, two different cases, known bounded perturbations and unknown bounded perturbations, are investigated from different viewpoints to analyze robust stability issue for the perturbed systems. Finally, a simulation example is given to confirm the effectiveness of the proposed design method.

Divisor Function and Bounds in Domains with Enough Primes

In this note, first we show that there is no uniform divisor bound for the Bezout identity using Dirichlet's theorem on arithmetic progressions. Then, we discuss for which rings the absolute value bound for the Bezout identity is not trivial and the answer depends on the number of small primes in the ring.

Design of controllers by indices of precision and speed. II. Nonminimal-phase plants

A method to design controllers of one-dimensional nonminimal-phase plants under unknown bounded external perturbations was proposed. It is based on determining the parameters of the Bezout identity. The attainable indices of precision and speed were determined.

Controller design in precision and speed. I. Minimal phase one-dimensional plants

A method to design the controllers of one-dimensional minimum phase plants under unknown bounded exogenous disturbances was proposed. It relies on determining the parameters of the Bezout identity providing the desired control precision and speed.

Improving Neural Network based Vibration Control for Smart Structures by Adding Repetitive Control

Neural networks (NN) has been a popular vibration control method because of its robustness and practicability to reject broad band disturbances for complex systems such as smart structures. However, the benign characteristic of NN, suppressing a wide range frequency of disturbances, may also limit its control performance at specific frequencies and inevitably cause no n-minimum output responses in particular under persistent excitatio n. To alleviate this limitation and improve the performance of NN based control methods, this paper presents a hybrid control strategy comprising a neural controller and a repetitive controller for a ctive vibration control of smart structures. The neural controller is a fund amental controller which applies back-propagation networks for performance evaluation. To add repetitive control into the existing NN control system, the work transforms a feedback controller to a feedforward control problem with the solutions of a bezout identity embedded with known internal models of injecting disturbances. The presented hybrid control provides a synergetic effect and aims for better suppression performance subject to complicated disturbances in stringent environments. Experimental results on a flexible beam demon strate the effectiveness of the proposed control method.

Adaptive Control With Internal Model for High-Performance Precision Motion Control and Its Application to a Fast-Acting Piezoelectric Actuator

This paper proposes an adaptive control scheme that minimizes the least-mean-square (LMS) value of the plant output while meeting the constraints of canceling deterministic exogenous signals generated by a priori dynamics. This scheme may be applied to a broad range of applications in which the exogenous input signals to the plant contain both deterministic and stochastic components. The adaptive control includes both feedback and previewed feedforward actions. In both actions, the deterministic signal model is included as a constraint of the dynamics from the external input to the plant output by determining solutions for a Bezout identity. The proposed scheme is applied to a fast-acting piezoelectric actuator (PZT) to generate precise dynamic motion profiles. This paper presents the experimental results to demonstrate the effectiveness of the proposed adaptive control scheme.

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.

Hybrid proportional derivative/repetitive control for active vibration control of smart piezoelectric structures

This paper presents a hybrid control method for active vibration control of smart piezoelectric structures. The proposed controller consists of a proportional derivative (PD) controller for broadband disturbance rejection and a repetitive controller for narrowband disturbance rejection. The proposed design embeds the repetitive controller by solving a Bezout identity satisfying deterministic constraint into an existing PD feedback control loop. The applied method has two main advantages: (1) each controller design is modularized with an “add-on” property for performance improvement; (2) it achieves superior vibration suppression for both impulse and persistent excitation. Experimental results on a piezoelectric cantilever beam demonstrate the effectiveness of the proposed hybrid controller design method.

Isomorphism-based robust right coprime factorization realization for non-linear feedback systems

The realization of the system design problem for a non-linear feedback control system is considered by robust right coprime factorization and isomorphism. First, a feedback system with a series compensator and a feedback compensator is designed to construct a stable isomorphic subspace of the controlled plant’s input space, and then the right factorization of the plant can be realized. Moreover, the right factorization can be guaranteed to be a right coprime factorization by satisfying a Bezout identity. Further, based on asymptotic stability, a quantitative design scheme is proposed to guarantee the robust stability of the non-linear feedback system and the plant output tracking to the reference input. The effectiveness of the proposed design scheme is confirmed by the simulation results.

Synthesis of decoupling controller for non-minimum phase plants of different pole numbers on RHP within uncertainties

This article mainly studies the decoupling controller design for non-minimum phase plants of different pole numbers on RHP within uncertainties. The normalised coprime factorisation is considered to achieve the robustness requirements. The pole-zero cancellations on RHP should be averted for the sake of robustness. For convenience, the H ∞ sub-optimal controller is utilised to meet the robust criterion of the plant. Some necessary state space formulae are also provided to facilitate the synthesis of the decoupling controller. The configuration of the two-parameter compensation is employed. The Bezout identity makes the feedforward controller easy to determine. A brief algorithm is presented. In addition, the proposed synthesis is illustrated with a numerical example. The robust bounds of the feedback controller can be assessed for both the additive uncertainty and the coprime factor uncertainties. The result shows that the compensated system is decoupled and is guaranteed to be internally stable within the specified robust bound although the pole number varies on RHP.

System Design for Nonlinear Plants Using Operator-Based Robust Right Coprime Factorization and Isomorphism

The nonlinear system design problem is investigated concerning with robust stability and output tracking problems for nonlinear perturbed feedback control systems using operator-based robust right coprime factorization and isomorphism. First, a compensator is designed to factorize a nonlinear perturbed plant using isomorphism, then the right factorization of the plants can be realized. Further, by guaranteeing an output tracking property, the existence of two stable controllers is proved so that a Bezout identity for the right factorization of the plant can be satisfied. Meanwhile, the robust stability can be guaranteed by the two controllers. Therefore, the right factorization is exactly to be a robust right coprime factorization. That is, by the proposed design scheme, the robust right coprime factorization is realized and the plant output can also track to the reference input. A numerical example is given to show the effectiveness of the proposed design scheme.