Bilinear Approximation Method Research Articles

Bilinear Systems With Initial Gaps Involving Inelastic Collision: Forced Response Experiments and Simulations

Abstract In this paper, the forced response of a two degrees-of-freedom (DOF) bilinear oscillator with initial gaps involving inelastic collision is discussed. In particular, a focus is placed upon the experimental verification of the generalized bilinear amplitude approximation (BAA) method, which can be used for the accurate estimation of forced responses for bilinear systems with initial gaps. Both experimental and numerical investigations on the system have been carried out. An experimental setup that is capable of representing the dynamics of a 2DOF oscillator has been developed, and forced response tests have been conducted under swept-sine base excitation for different initial gap sizes. The steady-state response of the system under base excitation was computed by both traditional time integration and BAA. It is shown that the results of experiments and numerical predictions are in good agreement especially at resonance. However, slight differences in the responses obtained from both numerical methods are observed. It was found that the time duration where the DOFs are in contact with each other predicted by BAA is longer than that predicted by time integration. Spectral analyses have also been conducted on both experimental and numerical results. It was observed that in a frequency range where intermittent contact between the masses occurs, super-harmonic components of the excitation frequency are present in the spectra. Moreover, as the initial gap size increases, the frequency band where the super-harmonic components are observed decreases.

Digital and analog integer delayed modeling and control for multivariable systems with multiple time delays in states, inputs and outputs

This paper presents a new approximated integer delayed modeling and control technique for Continuous-time Fractional Delay Differential Equations (CFDDEs). Using the impulse response sequences of the CFDDE and the balanced model-reduction method, a delay-free discrete-time state-space model is constructed. Then, an equivalent Discrete-time Integer Delay Difference Equation (DIDDE) is obtained by transforming the obtained discrete-time state-space model into a controller-type block companion form. Furthermore, based on the obtained DIDDE, an equivalent Continuous-time Integer Delay Differential Equation (CIDDE) is determined by means of the newly developed Chebyshev's bilinear approximation method. For digital control of the CFDDE, an optimal Discrete-time Integer Delayed Control Law (DIDCL) is designed using the conventional discrete-time LQR approach together with the obtained delay-free discrete-time state-space model. On the other hand, for continuous-time control of the CFDDE, a Continuous-time Integer Delayed Control Law (CIDCL) is determined from the designed DIDCL by means of the inverse Chebyshev's bilinear approximation method. Finally, digital and analog integer delayed observers are constructed for the implementations of the developed DIDCL and CIDCL, respectively. An illustrative example is given to demonstrate the effectiveness of the proposed method.

Observer-based sampled-data control for nonlinear systems: Robust intelligent digital redesign approach

This paper suggests a solution for a robust digital implementation of the output-feedback control system by utilizing intelligent digital redesign (IDR). The general IDR method involves designing a digital fuzzy controller which is comparable to a pre-designed analog one via state-matching. However, under containing the parametric uncertainties, the new problem about the structural property of the uncertainties is inevitably occurred in the procedure of discretization. For solving that, we use the bilinear and inverse-bilinear approximation method and the concerned IDR is viewed as a convex minimization problem of the norm distances between linear operators to be matched. Also, we consider the state estimation error in the plant dynamics so that we attempt to improve the control performance. The robust stability property is preserved by the proposed IDR method and its condition is represented in terms of linear matrix inequalities (LMIs). The simulation results for permanent magnet synchronous motor (PMSM) system are demonstrated to visualize the feasibility of the proposed method.

관측기 기반 시스템에 대한 강인 디지털 재설계

본 논문은 관측기 기반 시스템에 대한 강인 디지털 재설계 방안을 제안한다. 디지털 재설계란, 기존의 안정화 된 연속시간 플랜트와 이산 시간에서 설계된 디지털 제어기와의 상태 접합 및 안정도 분석을 통해 전체 시스템을 재구성하는 것을 말한다. 그리고 전 역적 접근을 위한 방안으로서 문제를 볼록 최적화 관점으로 변환 후, 에러가 가질 수 있는 놈의 영역을 최소화하여 상태 접합을 이루고자 하였다. 본 논문에서는 관측기 기반 시스템에 대한 디지털 재설계를 목표로 하되, 추가적인 파라미터 불확실성을 고려한 강인 디지털 재설계를 구성하게 된다. 파라미터 불확실성은 이산화 과정에서 구조적 형태가 변화하기 때문에, 이를 고려하여 주어진 식을 선형 행렬 부등식 형태로 나타내게 된다. 이 조건들을 통해 디지털 재 설계의 상태 접합 및 안정도가 유도 가능하다는 것을 본 논문에서 증명하게 된다. In this paper, we presents robust digital redesign (DR) method for observer-based linear time-invariant (LTI) system. The term of DR involves converting an analog controller into an equivalent digital one by considering two condition: state-matching and stability. The design problems viewed as a convex optimization problem that we minimize the error of the norm bounds between interpolated linear operators to be matched. Also, by using the bilinear and inverse bilinear approximation method, we analyzed the uncertain parts of given observer-based system more precisely, When a sampling period is sufficiently small, the conversion of a analog structured uncertain system to an equivalent discrete-time system have proper reason. Sufficiently conditions for the state-matching of the digitally controlled system are formulated in terms of linear matrix inequalities (LMIs).

불확실성을 갖는 비선형 시스템의 강인한 지능형 디지털 재설계

본 논문은 불확실성을 포함한 비선형 시스템에 대한 제어를 위해 강인 지능형 디지털 재설계의 전 역적 접근 방안에 대해 제안하고자 한다. 이산화를 통한 제어기 설계에 있어서 불확실성이 포함된 실시간 비선형 시스템에 대해 보다 효율적이고 안정적인 접근을 위해 T-S 퍼지 모델이 사용되었다. 그리고 전역적 접근을 위한 방안으로서 문제를 볼록 최적화 관점으로 변환 후, 오차가 가질 수 있는 놈의 영역을 최소화 하여 상태 접합을 이루고자 하였다. 또한 쌍선형과 역 쌍선형 기법을 사용함으로써 불확실성이 포함된 비선형 시스템을 보다. 더 정확하게 분석하였다. 샘플링 기간이 충분히 작다면, 불확실 비선형 시스템의 실시간 시스템으로의 전환이 충분한 이유를 가지게 된다. 전 역적 접근을 통한 디지털로 제어된 시스템은 선형 행렬 부등식 형태로 바꾸어 시스템의 안정성을 보장하고자 하였다. 마지막으로 T-S 퍼지 모델로 분석된 혼돈 Lorenz system에 적용함으로써 제안된 방법의 안정성과 효율성을 확인한다. This paper presents intelligent digital redesign method for hybrid state space fuzzy-model-based controllers. For effectiveness and stabilization of continuous-time uncertain nonlinear systems under discrete-time controller, Takagi-Sugeno(TS) fuzzy model is used to represent the complex system. And global approach design problems viewed as a convex optimization problem that we minimize the error of the norm bounds between nonlinearly interpolated linear operators to be matched. Also, by using the bilinear and inverse bilinear approximation method, we analyzed nonlinear system's uncertain parts more precisely. When a sampling period is sufficiently small, the conversion of a continuous-time structured uncertain nonlinear system to an equivalent discrete-time system have proper reason. Sufficiently conditions for the global state-matching of the digitally controlled system are formulated in terms of linear matrix inequalities (LMIs). Finally, a TS fuzzy model for the chaotic Lorentz system is used as an . example to guarantee the stability and effectiveness of the proposed method.

Digital modelling and robust digital redesign of sampled-data uncertain systems via the interval tuning bilinear approximation method

Based on the bilinear approximation method, we present an interval tuning bilinear approximation method to do the digital model conversion and digital redesign of sampled-data uncertain systems. The tuning bilinear method together with interval arithmetic operation is employed to obtain the approximate discrete-time model and the digital robust control law from the continuous-time uncertain state equation and the given continuous-time robust control law, respectively. The proposed digitally tuning redesigned control law with one parameter can be adjusted so that the resulting dynamic states of the digitally controlled sampled-data uncertain system are able to closely match those of the original continuous-time well-designed uncertain system for a relatively longer sampling period.

Improved Bilinear Routh Approximation Method for Discrete-Time Systems

Hwang and Shieh proposed a bilinear Routh approximation method for reducing the order of discrete-time systems. A reduced model derived by the method is not only stable whenever an original model is stable, but also fits the first few time-moments of the original one. This paper addresses the possibility of improving the method by letting the impulse response energy of the original model also be conserved in the reduced model without destroying the stability preserving and time-moments matching properties.

Digital Redesign of State-Delayed Systems Counting the Computer-Control Delay via Law of Mean.

This paper presents a new approach for digital redesign of the continuous-time state-delayed system with the state-feedback controller. The bilinear approximation method together with the linear interpolation is utilized to approximate some integration functions, and the concept of the Law of Mean from the integral calculus is applied to establish an equivalent state-feedback digital controller from the available statefeedback analog controller. In addition, for practical implementation of the developed digital controller by a microcomputer, the computation time delay of the digitally redesigned control law is considered and equivalently modeled at the input, and then a digital predictor controller is proposed for the sampled-data state-delayed system with the control delay.

Model Conversions of Uncertain Linear System via the Interval Tuning Bilinear Approximation Method.

This paper presents a new tuning bilinear approximation method together with the interval arithmetic operation to convert a continuous-time (descrete-time) uncertain system to an equivalent discrete-time (continuous-time) uncertain model.Based on the bilinear approximation method, we induct one parameter for tuning the error bound between the exact and approximate uncertain system matrices.The bounds of interval system matrices via the proposed interval tuning bilinear approximation method are narrower than those of the existing interval bilinear and the interval pade approximation methods(1).The resulting interval models (the enclosing interval models) are able to tightly enclose the exact uncertian models.Also the approximate discrete-time interval solution is able to tightly enclose the exact interval solution of the continuous-time uncertain state-space equation.The proposed method will be helpful for analysis and synthesis of uncertain systems.

Digital redesign of H8 controller via bilinear approximation method for state-delayed systems

Two issues are addressed: digitial modeling of the continuous-time state-delayed system; and digital redesign of the observer-based H8 controller for the continuous-time state-delayed system. The bilinear approximation method together with the linear interpolation is utilized to find an equivalent discrete-time model of the original continuous-time state-delayed system, so that the states of the obtained discrete-time model closely match those of the original continuous-time statedelayed system. In addition, the same idea is used to establish an equivalent digital controller from the analogue H8 controller for the continuous-time state-delayed system, so that the states of the digitally controlled sampled-data state-delayed system closely match those of the original analogously controlled continuoustime state-delayed system. A discrete-time observer is constructed from the original continuous-time observer via a modified lifting technique, such that the estimated states of the obtained discrete-time observer match those of the original continuous-time observer at the sampling instants. Using the newly digitally redesigned observer-based controllers, the resulting dynamic states of the overall digitally controlled sampled-data state-delayed systems are able to closely match those of the original observer-based analogously controlled continuous-time state-delayed systems. An illustrative example is given to show the digital modeling and digital redesign results for a state-delayed system.

Model conversion of uncertain linear systems with input time-delay via the interval bilinear approximation method

This paper presents a method for model conversion of a continuous-time (discrete-time) uncertain linear system with input time-delay to an equivalent discrete-time (continuous-time) uncertain linear time-delay model. The bilinear approximation method and interval arithmetic operations are proposed for converting continuous-time (discrete-time) uncertain input time-delay models to equivalent discrete-time (continuous-time) uncertain input time-delay models. Here, we also present a method to estimate tighter error bounds of discrete-time (continuous-time) structured uncertainties based on the given continuous-time (discrete-time) structures uncertainties. By applying the above proposed methods, the converted model tightly encloses the original system; consequently, the analysis and design can be carried out using the obtained models instead of the original systems. An example is given to demonstrate the effectiveness of the proposed method.

Digital redesign of cascaded analogue controllers for sampled-data interval systems

The paper presents a new method for the digital redesign of cascaded analogue controllers for continuous-time parametric interval systems. The bilinear and inverse-bilinear approximation method is developed to carry out discretisation of the predesigned cascaded analogue controller, taking into account intersample behaviour and implementation errors. A dual concept of the proposed digital redesign method is utilised to construct a new pseudodigital observer such that the estimated states of the digitally redesigned digital observer closely match those of the original continuous-time observer at the sampling instants. Using the newly digitally redesigned observer based controllers, the resulting dynamic states of the digitally controlled cascaded sampled-data interval systems are able to closely match those of the original analogously controlled cascaded continuous-time parametric interval systems.

Order Reduction of Discrete-Time System Via Bilinear Routh Approximation

In this paper, a method of combining the Routh approximation method with the bilinear transformation is presented for deriving stable reduced-order models of a strictly proper z-transfer function Gn(z). It is based on applying the bilinear transformation to the (z+1)Gn(z), and then deriving a new bilinear Routh γ − δ canonical expansion for Gn(z). The proposed bilinear Routh approximation method has all the advantages of the Routh approximation method [5] while without having initial-value problem caused by the bilinear transformation. A numerical example is included to illustrate the procedure.

Analysis on Shock Response of Nonlinear Systems by Bilinear Approximation Method

Analysis on Shock Response of Nonlinear Systems by Bilinear Approximation Method

Transient Response of a Nonlinear System by a Bilinear Approximation Method

Abstract A line-segment approximation for a nonlinear elastic force has been developed which makes it possible to obtain analytical solutions to a number of transient-load problems which formerly could be solved only by graphical or numerical methods. It is shown that for many problems involving even large nonlinearities, two line segments only are sufficient to give satisfactory accuracy. For some problems where the classical analytical approximate methods of Lindstedt or of Kryloff and Bogoliuboff can be applied, it is shown that the bilinear method leads to an improved accuracy.