Inverted pendulum倒立摆的matlab建模概要1.docxVIP

ECE451 Controll Engineering Inverted pendulum09/29/2013Introduction:Inverted pendulum is a typical fast, multi-varaibles, nonlinear, unstable system, it has significant meaning. We choose the PID controller to fot the inverted pendulum. Assume the input is a step signal , the gravitational accelerationg=9.8m/s^2 and linearize the nonlinear model around the operating point.1.Mathematic ModlingMmass of the car0.5 kgmmass of the pendulum0.2 kgbcoefficient of friction for cart0.1 N/m/secllength to pendulum center of mass0.3 mImass moment of inertia of the pendulum 0.006 kg.m^2Fforce applied to the cartxcoordinate of cart positionθpendulum angle from vertical (down)N and F are the force from horizontal and vertical direction.)Force analysisConsider the horizontal direction cart force, we get the equation:Consider the horizontal direction pendulum force, we get the equation:To get rid of P and N, we get this equation:Merge these two equations, about to PAnd N, to obtain a second motion equation:u to represent the controlled object with the input force F, linearized two motion equationsApply Laplace transform to the equation aboveThe transfer function of angle and positionLet v = put the equation above into the second equationWe get the transfer functionState space equation:Solve the algebraic equation, obtain solution as follows:Finally we get the system state space equations.2. PID Controller DesignWe now design a PID controller for the inverted pendulum system.KD(s) is the transfer function of the controller.G(s) is the transfer function of the controlled car.Considering that the input r(s) =0, the block diagram can be transformed as:The output of the system is num—the numerator of the objectden —the denominator of the objectnumPID – the numerator of the PID controller transfer functiondenPID – the denominator of the PID controller transfer functionThe object transfer function is ,in which.PID Controller Transfer Function is Now, we add the car’s position as another ou