The ball and plate system is an inherently nonlinear under actuated benchmark system used for validating the performance of various control schemes. A mathematical model depicting the dynamics close to that of the sys...The ball and plate system is an inherently nonlinear under actuated benchmark system used for validating the performance of various control schemes. A mathematical model depicting the dynamics close to that of the system is very much required for such a test bed. In this correspondence, the complete nonlinear model, a simplified nonlinear model, and a linearized model of the ball and plate system are developed. The system comprises a ball and plate mechanism and a rotary servo unit. The ball and plate mechanism is modelled using the Euler–Lagrange method, whereas the rotary servo subsystem is modelled from the first principles. The nonlinear model of the combined system is developed by including the dynamics of the servo motor with gear and rolling resistance between the ball and the plate. The simplified nonlinear model of the system is obtained with suitable assumptions. The model is linearized around the operating point using the Jacobian. The validity of the developed models is investigated through correlation function analysis. The open-loop response of the three models, viz., nonlinear, simplified nonlinear, and linearized models, is analyzed in the MATLAB/Simulink platform. Since the open-loop system is unstable, the experimental validation of the model is performed with a double-loop PSO (particle swarm optimization) PID control scheme.展开更多
The electrostrictive properties above Te max represented by the field-related and polarization-related M and Q coefficients have been measured for PbZr_(1-x)Sn_(x)O_(3)single crystals with ani-ferroelectric phase tran...The electrostrictive properties above Te max represented by the field-related and polarization-related M and Q coefficients have been measured for PbZr_(1-x)Sn_(x)O_(3)single crystals with ani-ferroelectric phase transitions.The investigations presented in this paper have proved that the M_(11)(T)and Q_(11)(T)runs bring direct information of the pre-transitional effects connected with the co-existence of local polar objects with nonpolar matrix.Due to this co-existence,nonlinear Q_(11)(T)dependence in the neighborhood of T_(C)has been detected.Observed even in a wide tem-perature above T_(C),this co-existence is a natural feature of the material with ABO_(3)perovskite structure that undergoes structural transition to po lar phase.展开更多
文摘The ball and plate system is an inherently nonlinear under actuated benchmark system used for validating the performance of various control schemes. A mathematical model depicting the dynamics close to that of the system is very much required for such a test bed. In this correspondence, the complete nonlinear model, a simplified nonlinear model, and a linearized model of the ball and plate system are developed. The system comprises a ball and plate mechanism and a rotary servo unit. The ball and plate mechanism is modelled using the Euler–Lagrange method, whereas the rotary servo subsystem is modelled from the first principles. The nonlinear model of the combined system is developed by including the dynamics of the servo motor with gear and rolling resistance between the ball and the plate. The simplified nonlinear model of the system is obtained with suitable assumptions. The model is linearized around the operating point using the Jacobian. The validity of the developed models is investigated through correlation function analysis. The open-loop response of the three models, viz., nonlinear, simplified nonlinear, and linearized models, is analyzed in the MATLAB/Simulink platform. Since the open-loop system is unstable, the experimental validation of the model is performed with a double-loop PSO (particle swarm optimization) PID control scheme.
文摘The electrostrictive properties above Te max represented by the field-related and polarization-related M and Q coefficients have been measured for PbZr_(1-x)Sn_(x)O_(3)single crystals with ani-ferroelectric phase transitions.The investigations presented in this paper have proved that the M_(11)(T)and Q_(11)(T)runs bring direct information of the pre-transitional effects connected with the co-existence of local polar objects with nonpolar matrix.Due to this co-existence,nonlinear Q_(11)(T)dependence in the neighborhood of T_(C)has been detected.Observed even in a wide tem-perature above T_(C),this co-existence is a natural feature of the material with ABO_(3)perovskite structure that undergoes structural transition to po lar phase.
