Robust stability of nonlinear plants with a non-symmetric Prandtl-Ishlinskii hysteresis model

In this paper, robust stability of nonlinear plants represented by non-symmetric Prandtl-Ishlinskii (PI) hysteresis model is studied. In general, PI hysteresis model is the weighted superposition of play or stop hysteresis operators, and the slopes of the operators are considered to be the same. In order to make a hysteresis model, a modified form of non-symmetric play hysteresis operator with unknown slopes is given. The hysteresis model is described by a generalized Lipschitz operator term and a bounded parasitic term. Since the generalized Lipschitz operator is unknown, a new condition using robust right coprime factorization is proposed to guarantee robust stability of the controlled plant with the hysteresis nonlinearity. As a result, based on the proposed robust condition, a stabilized plant is obtained. A numerical example is presented to validate the effectiveness of the proposed method.     

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.     

Operator-based nonlinear feedback control design using robust right coprime factorization

In this note, robust stabilization and tracking performance of operator based nonlinear feedback control systems are studied by using robust right coprime factorization. Specifically, a new condition of robust right coprime factorization of nonlinear systems with unknown bounded perturbations is derived. Using the new condition, a broader class of nonlinear plants can be controlled robustly. When the spaces of the nonlinear plant output and the reference input are different, a space change filter is designed, and in this case this note considers tracking controller design using the exponential iteration theorem.     

Operator-Based Robust Nonlinear Control for SISO and MIMO Nonlinear Systems With PI Hysteresis

In this paper, operator based robust nonlinear control for single-input single-output (SISO) and multi-input multi-output (MIMO) nonlinear uncertain systems preceded by generalized Prandtl-Ishlinskii (PI) hysteresis is considered respectively. In detail, by using operator based robust right coprime factorization approach, the control system design structures including feedforward and feedback controllers for both SISO and MIMO nonlinear uncertain systems are given, respectively. In which, the controller design includes the information of PI hysteresis and its inverse, and some sufficient conditions for the controllers in both SISO and MIMO systems should be satisfied are also derived respectively. Based on the proposed conditions, influence from hysteresis is rejected, the systems are robustly stable and output tracking performance can be realized. Finally, the effectiveness of the proposed method is confirmed by numerical simulations.      

Operator-based robust nonlinear control system design for MIMO nonlinear plants with unknown coupling effects

In this article, operator-based robust nonlinear control system design for multi-input multi-output (MIMO) plants with unknown coupling effects is considered. That is, by using operator-based robust nonlinear control design, coupling effects existing in the MIMO nonlinear plants can be decoupled based on a feedback design and robust right coprime factorisation approach, the coupling effects caused by controllers and plant outputs can be stabilised by using definition of Lipschitz operator and contraction mapping theorem, and output tracking performance can be realised by a tracking design scheme. Finally, a simulation example about temperature control process of 3-input/3-output aluminum plate is given to support the theoretical analysis.     

Stable robust feedback control system design for unstable plants with input constraints using robust right coprime factorization

A stable robust control system design problem for unstable plants with input constraints is considered using robust right coprime factorization of nonlinear operator. For obtaining strong stability of the closed‐loop system of unstable plants with input constraints, a design scheme of robust nonhyphen‐linear control system is given based on robust right coprime factorization. Some conditions for the robustness and system output tracking of the unstable plant with input constraints are derived. Numerical examples are given to demonstrate the validity of the theoretical results. Copyright © 2007 John Wiley & Sons, Ltd.     

Operator-based robust control for nonlinear uncertain systems with unknown backlash-like hysteresis

In this paper, operator based robust control for nonlinear uncertain system with unknown backlash-like hysteresis is considered. In detail, a continuous backlash-like hysteresis operator is proved to be corresponding to a one-to-one operator, that is, it is suitable to be used in operator theoretic based control theory. Moreover, an internal model control (IMC) structure with one parallel compensating operator is proposed for nonlinear uncertain system with unknown backlash-like hysteresis. Based on the proposed control scheme, the designed system is robustly stable and the desired output tracking performance can be realized simultaneously. Finally, a simulation example about nonlinear plant preceded by backlash is given to show the design procedure of the proposed method.     

Robust consensus tracking for a class of heterogeneous second‐order nonlinear multi‐agent systems

This paper deals with the robust consensus tracking problem for a class of heterogeneous second‐order nonlinear multi‐agent systems with bounded external disturbances. First, a distributed adaptive control law is proposed based on the relative position and velocity information. It is shown that for any connected undirected communication graph, the proposed control law solves the robust consensus tracking problem. Then, by introducing a novel distributed observer and employing backstepping design techniques, a distributed adaptive control law is constructed based only on the relative position information. Compared with the existing results, the proposed adaptive consensus protocols are in a distributed fashion, and the nonlinear functions are not required to satisfy any globally Lipschitz or Lipschitz‐like condition. Numerical examples are given to verify our proposed protocols. Copyright © 2014 John Wiley & Sons, Ltd.     

Non‐parametric linear time‐invariant extensions of non‐invertible and backlash plant

A rigorous validation for the use of a set of linear time‐invariant models as a surrogate in the design of controllers for uncertain nonlinear systems, which are invertible as one‐to‐one operators, such as used in the nonlinear quantitative feedback theory (NLQFT) design methodology has been given by Baños and Bailey. This paper presents a similar validation but weakens the requirement on the invertibility of the nonlinear plant by application of Kakutani's fixed‐point theorem and an incremental gain constraint on the plant within its operational envelope. The set of linear time‐invariant models to be used for design is shown to be an extension (termed here the linear time‐invariant extension—LTIE) of the nonlinear plant restricted to the desired output operating space. A new non‐parametric approach to the modelling of the LTIE is proposed which is based on Fourier transforms of the plant I/O data and which accordingly may be based solely on experimental testing without the need for an explicit parametric plant model. This new approach thus extends the application of robust linear controller design methods (including those of NLQFT) to nonlinear plants with set‐valued (multi‐valued) inverses such as those containing backlash and also to plants for which explicit parametric models are difficult to obtain. The method is illustrated by application of the non‐parametric approach to an NLQFT tracking controller design for a mechanical backlashed servomechanism problem. Copyright © 2013 John Wiley & Sons, Ltd.     

Operator-Based Robust Nonlinear Free Vibration Control of a Flexible Plate With Unknown Input Nonlinearity

In this paper,a robust nonlinear free vibration control design using an operator based robust right coprime factorization approach is considered for a flexible plate with unknown input nonlinearity.With considering the effect of unknown input nonlinearity from the piezoelectric actuator,operator based controllers are designed to guarantee the robust stability of the nonlinear free vibration control system.Simultaneously,for ensuring the desired tracking performance and reducing the effect of unknown input nonlinearity,operator based tracking compensator and estimation structure are given,respectively.Finally,both simulation and experimental results are shown to verify the effectiveness of the proposed control scheme.     

ROBUST ADAPTIVE CONTROL FOR STRICT‐FEEDBACK NONLINEAR SYSTEMS

In this paper, we propose a robust adaptive tracking control based on the backstepping strategy for strict‐feedback nonlinear systems with nonparametric uncertain nonlinearities. It is shown that one can design a stable adaptive control system provided that the uncertain nonlinearities can be decomposed by unknown bounded nonlinear functions and known nonlinear functions. The proposed method can deal with uncertain nonlinearities that appear at the control input term too. It is also shown that suitable choice of design parameters guarantees the convergence of tracking error to any desired bound.     

Robust Control for Uncertain Nonlinear Feedback Systems 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.      

Synchronization and tracking control for multi‐motor driving servo systems with backlash and friction

This paper presents a novel switching controller incorporated with backlash and friction compensations, which is utilized to achieve speed synchronization among multi‐motor and load position tracking. The proposed controller consists of two parts: synchronization and tracking control in contact mode and robust control in backlash mode, where a function characterizing whether backlash occurs is used for switching between two modes. Using the proposed switching controller, several control objectives are achieved. Firstly, the coupling problem of speed synchronization and load tracking in contact mode is addressed by introducing a switching plane. Secondly, based on the switching plane, an improved prescribed performance function is introduced to attain load tracking with prescribed performances, and L_∞ performance of speed synchronization is guaranteed by initialization method, maintaining the transient performance of synchronization behavior. Thirdly, the lumped uncertain nonlinearity including friction and other uncertain functions is compensated by Chebyshev neural network in contact mode. Furthermore, a robust control is adopted in backlash mode to make system traverse backlash at an exponential rate and simultaneously eliminate low‐speed crawling phenomenon of LuGre friction. Finally, comparative simulations on four‐motor driving servo system are provided to verify the effectiveness and reliability. Copyright © 2015 John Wiley & Sons, Ltd.     

Perturbed nonlinear systems control using extended robust right coprime factorisation

In this paper, extended operator-based robust right coprime factorisation is investigated for dealing with a class of nonlinear systems with unknown bounded perturbations. First, a new kind of operator is introduced, by which operator-based right coprime factorisation approach is extended to consider the perturbed nonlinear systems. By regulating the exponent of the proposed operator, a broader class of nonlinear systems can be handled using the extended right coprime factorisation approach. Second, for guaranteeing robust stability of the perturbed nonlinear systems, feasible design schemes based on some sufficient conditions are discussed, which can reduce complicated calculation in control processing using a different unimodular operator. Finally, a simulation example is involved to illustrate the proposed design scheme for confirming the effectiveness of the proposed method.     

GENERALIZED QUADRATIC STABILIZATION FOR DISCRETE‐TIME SINGULAR SYSTEMS WITH TIME‐DELAY AND NONLINEAR PERTURBATION

This paper discusses a generalized quadratic stabilization problem for a class of discrete‐time singular systems with time‐delay and nonlinear perturbation (DSSDP), which the satisfies Lipschitz condition. By means of the S‐procedure approach, necessary and sufficient conditions are presented via a matrix inequality such that the control system is generalized quadratically stabilizable. An explicit expression of the static state feedback controllers is obtained via some free choices of parameters. It is shown in this paper that generalized quadratic stability also implies exponential stability for linear discrete‐time singular systems or more generally, DSSDP. In addition, this new approach for discrete singular systems (DSS) is developed in order to cast the problem as a convex optimization involving linear matrix inequalities (LMIs), such that the controller can stabilize the overall system. This approach provides generalized quadratic stabilization for uncertain DSS and also extends the existing robust stabilization results for non‐singular discrete systems with perturbation. The approach is illustrated here by means of numerical examples.     

Adaptive Tracking Control for Nonlinear Systems with a Class of Input Nonlinearities

An adaptive tracking control approach is presented for nonlinear systems with a class of input nonlinearities. A generalized model has been developed for a class of non‐smooth nonlinearities that include dead‐zone, backlash and ‘backlash‐like’ hysteresis. By using the developed model and Nussbaum‐gain technique, the problem of input nonlinearity is solved perfectly. The proposed method is available even when the designer is uncertain about the type of input nonlinearities mentioned above, and the knowledge on the bounds of these nonlinearity parameters is not required. Furthermore, it is proved that all closed‐loop signals are bounded and the tracking error converges to a small residual set asymptotically. Two simulation examples are provided to demonstrate the effectiveness of the proposed method.     

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

In 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.     

Operator-based robust control design for a human arm-like manipulator with time-varying delay measurements

In this paper, an operator-based robust nonlinear control for a human multi-joint arm-like manipulator with time-varying delay measurements is proposed by using robust right coprime factorization approach, a delay compensation operator and a forward predictive operator. That is, first, considering the uncertainties of dynamic model consisting of measurement error and disturbances, an operator-based nonlinear feedback control scheme is designed to eliminate effect of uncertainties. Second, an operator controller based on real measured data from human multi-joint arm viscoelasticity is presented to obtain desired motion mechanism of human multi-joint arm viscoelastic properties, the unknown time-varying delay measurements are described by a delay operator, the delay compensation operator is designed to remove the effect of unknown time-varying delay measurements, and the forward predictive operator is designed to compensate the term related to effect of central nervous system (CNS) during human multi-joint arm movements. The BIBO stability can be guaranteed and the tracking performance can be realized by the designed operator controller, the delay compensation operator and the forward predictive operator. Finally, the effectiveness of the proposed design scheme is confirmed by the simulation results based on experimental data.     

Eso‐Based Adaptive Robust Control of Dual Motor Driving Servo System

Based on extended state observer (ESO), we propose an adaptive robust control (ARC) for a dual motor driving servo system, in which there exist nonlinearities affecting control performance. To apply ESO and estimate the lumped uncertainty online, backlash and friction are analyzed and the nonlinear model of the plant is derived. We achieve several control objectives. First, the bias torque is considered in order to eliminate the effect of backlash. Second, the speed feedback is used to maintain the speed synchronization of motors. Then, to achieve feedforward control, finite‐time ESO is designed to estimate the unknown nonlinearities online. Furthermore, the ESO‐based adaptive robust controller is designed to guarantee L_∞ of tracking error by an initialization method, maintaining the transient performance of tracking behavior. Finally, extensive experimental results on a practical test rig validate the effectiveness of our proposed method.     

Parameterizing robust manipulator controllers under approximate inverse dynamics: A double‐Youla approach

We consider the goal of ensuring robust stability when a given manipulator feedback control law is modified online, for example, to safely improve the performance by a learning module. To this end, the factorization approach is applied to both the plant and controller models to characterize robustly stabilizing controllers for rigid‐body manipulators under approximate inverse dynamics control. Outer‐loop controllers to stabilize the nonlinear uncertain loop that results from approximate inverse dynamics are often derived by lumping uncertainty in a single term and subsequent analysis of the error system. Here, by contrast, the well‐known norm bounds of these uncertain dynamics are first recast into a generalized plant configuration that preserves the characteristic uncertainty structure. Then, the overall loop uncertainty is expressed with respect to the nominal outer‐loop feedback controller by means of an uncertain dual‐Youla operator. Therefore, using the dual‐Youla parameterization, we provide a novel way to rigorously quantify permissible perturbations of robot manipulator feedforward/feedback controllers. The method proposed in this paper does not constitute another robust control law for rigid‐body manipulators, but rather a characterization of a set of robustly stabilizing controllers. The resulting double‐Youla parameterization for the control of robot manipulators is amenable to numerous advanced design methods. The result is thoroughly discussed by a planar elbow manipulator and exemplified with a six‐degree‐of‐freedom robot scenario with varying payload.