Feedback linearization and equivalent-disturbance compensation control strategy for piezoelectric stage

Piezoelectric stages use piezoelectric actuators and flexure hinges as driving and amplifying mechanisms,respectively.These systems have high positioning accuracy and high-frequency responses,and they are widely used in various precision/ultra-precision positioning fields.However,the main challenge with these devices is the inherent hysteresis nonlinearity of piezoelectric actuators,which seriously affects the tracking accuracy of a piezoelectric stage.Inspired by this challenge,in this work,we developed a Hammerstein model to describe the hysteresis nonlinearity of a piezoelectric stage.In particular,in our proposed scheme,a feedback-linearization algorithm is used to eliminate the static hysteresis nonlinearity.In addition,a composite controller based on equivalent-disturbance compensation was designed to counteract model uncertainties and external disturbances.An analysis of the stability of a closed-loop system based on this feedback-linearization algorithm and composite controller was performed,and this was followed by extensive comparative experiments using a piezoelectric stage developed in the laboratory.The experimental results confirmed that the feedback-linearization algorithm and the composite controller offer improved linearization and trajectory-tracking performance.

An efficient approach for the equivalent linearization of frame structures with plastic hinges under nonstationary seismic excitations

An efficient approach is proposed for the equivalent linearization of frame structures with plastic hinges under nonstationary seismic excitations.The concentrated plastic hinges,described by the Bouc-Wen model,are assumed to occur at the two ends of a linear-elastic beam element.The auxiliary differential equations governing the plastic rotational displacements and their corresponding hysteretic displacements are replaced with linearized differential equations.Then,the two sets of equations of motion for the original nonlinear system can be reduced to an expanded-order equivalent linearized equation of motion for equivalent linear systems.To solve the equation of motion for equivalent linear systems,the nonstationary random vibration analysis is carried out based on the explicit time-domain method with high efficiency.Finally,the proposed treatment method for initial values of equivalent parameters is investigated in conjunction with parallel computing technology,which provides a new way of obtaining the equivalent linear systems at different time instants.Based on the explicit time-domain method,the key responses of interest of the converged equivalent linear system can be calculated through dimension reduction analysis with high efficiency.Numerical examples indicate that the proposed approach has high computational efficiency,and shows good applicability to weak nonlinear and medium-intensity nonlinear systems.

Linearization of Intrinsically Nonlinear Oscillators

We show that an intrinsically nonlinear oscillator can always be transformed into a linear or harmonic oscillator by addition of a constant force, which shifts the equilibrium position of the oscillator.

Equilibrium Control Theory and Direct Method of Feedback Linearization for Nonlinear Time Varying Systems

Aim To present a simple and effective method for the design of nonlinear and time varying control system. Methods A new concept of dynamic equilibrium of a system and its stability were presented first. It was pointed out that what is controlled directly by the input of a control system is the system's dynamic equilibrium rather than the states. Based on it, a new feedback linearization method for nonlinear system based on the Lyapunov direct method was given. Simulation studies were also carried out. Results The example and simulation show that by use of the method, the controller design becomes very simple and the control effect is quite satisfying. Conclusion The new method unifies the stabilizing problem(regulating problem) with the tracking problem. It is a very simple and effective method for the design of nonlinear and time varying control system.

MHD flow of upper-convected Maxwell fluid over porous stretching sheet using successive Taylor series linearization method

This paper investigates the magnetohydrodynamic （MHD） boundary layer flow of an incompressible upper-convected Maxwell （UCM） fluid over a porous stretching surface. Similarity transformations are used to reduce the governing partial differential equations into a kind of nonlinear ordinary differential equations. The nonlinear prob- lem is solved by using the successive Taylor series linearization method （STSLM）. The computations for velocity components are carried out for the emerging parameters. The numerical values of the skin friction coefficient are presented and analyzed for various parameters of interest in the problem.

Global Topological Linearization with Unbounded Nonlinear Term

In 1960s, Hartman and Grobman pointed out that if all eigenvalues of a matrix A have no zero real part and f(x) is small Lipchitzian, then x′=Ax+f(x) can be locally linearized on a neighborhood of the origin. Later, the above result was generalized to global under the condition that f(x) is a bounded function. In this paper, we delete the condition that f(x) is a bounded function, and prove that if f(x) has suitable structure, then x′=Ax+f(x) can be linearized.

Impact angle control over composite guidance law based on feedback linearization and finite time control

The impact angle control over guidance(IACG) law against stationary targets is proposed by using feedback linearization control(FLC) and finite time control(FTC). First, this paper transforms the kinematics equation of guidance systems into the feedbackable linearization model, in which the guidance law is obtained without considering the impact angle via FLC. For the purpose of the line of sight(LOS) angle and its rate converging to the desired values, the second-order LOS angle is considered as a double-integral system. Then, this paper utilizes FTC to design a controller which can guarantee the states of the double-integral system converging to the desired values. Numerical simulation illustrates the performance of the IACG, in contrast to the existing guidance law.

Tracking control for air-breathing hypersonic cruise vehicle based on tangent linearization approach

This paper is focused on developing a tracking controller for a hypersonic cruise vehicle using tangent linearization approach.The design of flight control systems for air-breathing hypersonic vehicles is a highly challenging task due to the unique characteristics of the vehicle dynamics.Motivated by recent results on tangent linearization control,the tracking control problem for the hypersonic cruise vehicle is reduced to that of a feedback stabilizing controller design for a linear time-varying system which can be accomplished by a standard design method of frozen-time control.Through a proper model transformation,it can be proven that the tracking error of the designed closed-loop system decays exponentially.Simulation studies are conducted for trimmed cruise conditions of 110000 ft and Mach 15 where the responses of the vehicle to step changes in altitude and velocity are evaluated.The effectiveness of the controller is demonstrated by simulation results.

Trajectory linearization control of an aerospace vehicle based on RBF neural network

An enhanced trajectory linearization control （TLC） structure based on radial basis function neural network （RBFNN） and its application on an aerospace vehicle （ASV） flight control system are presensted. The influence of unknown disturbances and uncertainties is reduced by RBFNN thanks to its approaching ability, and a robustifying itera is used to overcome the approximate error of RBFNN. The parameters adaptive adjusting laws are designed on the Lyapunov theory. The uniform ultimate boundedness of all signals of the composite closed-loop system is proved based on Lyapunov theory. Finally, the flight control system of an ASV is designed based on the proposed method. Simulation results demonstrate the effectiveness and robustness of the designed approach.

Feedback linearization of the nonlinear model of a small-scale helicopter

In order to design a nonlinear controller for small-scale autonomous helicopters, the dynamic characteristics of a model helicopter are investigated, and an integrated nonlinear model of a small-scale helicopter for hovering control is presented. It is proved that the nonlinear system of the small-scale helicopter can be transformed to a linear system using the dynamic feedback linearization technique. Finally, simulations are carried out to validate the nonlinear controller.

Linearization of T-S fuzzy systems and robust H_∞ control

Takagi-Sugeno(T-S) fuzzy model is difficult to be linearized because of membership functions included.So,novel T-S fuzzy state transformation and T-S fuzzy feedback are proposed for the linearization of T-S fuzzy system.The novel T-S fuzzy state transformation is the fuzzy combination of local linear transformation which transforms local linear models in the T-S fuzzy model into the local linear controllable canonical models.The fuzzy combination of local linear controllable canonical model gives controllable canonical T-S fuzzy model and then nonlinear feedback is obtained easily.After the linearization of T-S fuzzy model,a robust H∞ controller with the robustness of sliding model control(SMC) is designed.As a result,controlled T-S fuzzy system shows the performance of H∞ control and the robustness of SMC.

Linearization Learning Method of BP Neural Networks

Feedforward multi layer neural networks have very strong mapping capability that is based on the non linearity of the activation function, however, the non linearity of the activation function can cause the multiple local minima on the learning error surfaces, which affect the learning rate and solving optimal weights. This paper proposes a learning method linearizing non linearity of the activation function and discusses its merits and demerits theoretically.

Research of robust adaptive trajectory linearization control based on T-S fuzzy system

A robust adaptive trajectory linearization control （RATLC） algorithm for a class of nonlinear systems with uncertainty and disturbance based on the T-S fuzzy system is presented. The unknown disturbance and uncertainty are estimated by the T-S fuzzy system, and a robust adaptive control law is designed by the Lyapunov theory. Irrespective of whether the dimensions of the system and the rules of the fuzzy system are large or small, there is only one parameter adjusting on line. Uniformly ultimately boundedness of all signals of the composite closed-loop system are proved by theory analysis. Finally, a numerical example is studied based on the proposed method. The simulation results demonstrate the effectiveness and robustness of the control scheme.

Sliding Mode Control of Hydraulic Pressure in Electro-Hydraulic Brake System Based on the Linearization of Higher-Order Model

The possibility to enhance the stability and robustness of electrohydraulic brake(EHB)systems is considered a subject of great importance in the automotive field.In such a context,the present study focuses on an actuator with a four-way sliding valve and a hydraulic cylinder.A 4-order nonlinear mathematical model is introduced accordingly.Through the linearization of the feedback law of the high order EHB model,a sliding mode control method is proposed for the hydraulic pressure.The hydraulic pressure tracking controls are simulated and analyzed by MATLAB/Simulink soft considering separately different conditions,i.e.,a sine wave,a square wave and a square wave with superimposed sine disturbance.The results show that the proposed strategy can track the target within 0.25 s,and the mean observed error is less than 1.2 bar.Moreover,with such a strategy,faster response and less overshoot are possible,which should be regarded as significant advantages.

Neural Network Based Feedback Linearization Control of an Unmanned Aerial Vehicle

This paper presents a flight control design for an unmanned aerial vehicle （UAV） using a nonlinear autoregressive moving average （NARMA-L2） neural network based feedback linearization and output redefinition technique. The UAV investigated is non- minimum phase. The output redefinition technique is used in such a way that the resulting system to be inverted is a minimum phase system. The NARMA-L2 neural network is trained off-line for forward dynamics of the UAV model with redefined output and is then inverted to force the real output to approximately track a command input. Simulation results show that the proposed approaches have good performance.

A New Nonlinear Companding Algorithm Based on Tangent Linearization Processing for PAPR Reduction in OFDM Systems

In this paper, a novel nonlinear companding transform(NCT) is proposed to reduce the Peak-to-Average Power Ratio(PAPR) of orthogonal frequency division multiplexing(OFDM) signals. The companding function is designed based on continuously differentiable reshaping of the probability density function(PDF) of signal amplitudes. The original PDF is cut off for PAPR reduction, and lower and medium segments of original PDF are scaled and linearized respectively, for maintaining power and cumulative distribution constraints. The linearized segment is set to be the tangent of the scaled version at the inflexion point, so as to reduce the out-ofband(OOB) radiation as much as possible. Parameters of the proposed scheme are solved under joint constraints of constant power and unity cumulative distribution. A new receiving method is also proposed to improve the bit error rate(BER) performance of OFDM systems. Simulation results indicate the proposed scheme can achieve better OOB radiation and BER performance at same PAPR levels, compared with existing similar companding algorithms.

Random vibration of nonlinear structures with stiffness and strength deterioration by modified tail equivalent linearization method

In this research the tail equivalent linearization method (TELM) has been extended to study structures with degrading materials. The responses of such structures to excitations are non-stationary, even if the excitations are stationary. Non-stationary behavior of the system cannot be considered by conventional TELM. Applying the conventional TELM, the only distinction in the design point excitation for two stationary excitations with diff erent durations is in the addition of a zero value part at the beginning of the design point of the longer excitation. This means that the failure probability is the same for the non-stationary systems under excitations with diff erent durations. Therefore, this solution cannot be correct. In this study, in using TELM for systems with degrading materials, hysteretic energy is replaced by average hysteretic energy, calculated by averaging the obtained hysteretic energy of the structure subjected to a few random sample load realization. In this way, the degradation parameters under design point coincide with those under sample load realizations. Since the average of the hysteretic energy is converges very fast, the modifi ed TELM only requires about tens to hundreds solutions of the response in addition to the ordinary calculations of conventional TELM.

Statistical Linearization on 2 DOFs Hydropneumatic Suspension with Asymmetric Non-linear Stiffness

Most references on hydropneumatic suspension analysis regard it as harden Duffing spring and take the white noise as the system input, which is quite different from real physical model. It will introduce considerable errors to the analytical result compared with the numerical simulation which makes it impossible to give a good depiction of the hydropneumatic suspension dynamics. In this paper, the dynamic response of the hydropneumatic suspension is worked out using statistical linearization based on 2 DOFs nonlinear suspension model. The damping of the suspension and the tire stiffness are both regarded as linear components and the real road roughness spectrum is used to work out the system input. The explicit analytical equivalent stiffness, dynamic mean value offset from statistic equilibrium position and the sprung acceleration varied with parameters of hydropneumatic spring, road roughness and vehicle velocity are worked out by substituting the nonlinear stiffness of hydropneumatic spring with its first three terms Tyler series at the static equilibrium position using James formula. The comparison of the numerical simulation and analytical result both on statistical parameters and distribution shows the validity of the analysis. The explicit form provides a concise and valid method on hydropneumatic suspension design and optimization.

THIRD ORDER SHEAR DEFORMATION MODEL FOR LAMINATED SHELLS WITH FINITE ROTATIONS:FORMULATION AND CONSISTENT LINEARIZATION

The paper presents an approach for the formulation of general laminated shells based on a third order shear deformation theory. These shells undergo finite (unlimited in size) rotations and large overall motions but with small strains. A singularity-free parametrization of the rotation field is adopted. The constitutive equations, derived with respect to laminate curvilinear coordinates, are applicable to shell elements with an arbitrary number of orthotropic layers and where the material principal axes can vary from layer to layer. A careful consideration of the consistent linearization procedure pertinent to the proposed parametrization of finite rotations leads to symmetric tangent stiffness matrices. The matrix formulation adopted here makes it possible to implement the present formulation within the framework of the finite element method as a straightforward task.

CONTROLLING CHAOTIC OSCILLATIONS OF VISCO-ELASTIC PLATESBY THE LINEARIZATION VIA OUTPUTFEEDBACK

Controlling chaotic oscillations of viscoelastic plates are investigated in this paper. Based on the exact linearization method in nonlinear system control theory, a nonlinear feedback control law is presented for a class of non_affine control systems. The mathematical model describing motion of nonlinear viscoelastic plates is established, and it is simplified by the Galerkin method. The phase space portrait and the power spectrum are employed to demonstrate chaos in the system. The deflection is treated as an output, and is controlled to given periodic goals.