Consider the cart and pendulum system describing the evolution of the cart velocity, the pendulum angle and the pendulum

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Consider The Cart And Pendulum System Describing The Evolution Of The Cart Velocity The Pendulum Angle And The Pendulum 1 (557.64 KiB) Viewed 44 times

Consider the cart and pendulum system describing the evolution of the cart velocity, the pendulum angle and the pendulum angular velocity. The linearized equations around the unstable equilibrium (angle 180deg, zero input, zero velocities) for the deviations of the state variables from the linearization point, are: -C₂v+me+F dt IZ W -cw+mgle + ml- (m+M)- de dt av me² dw de aw dv The various constants are (all values in SI): friction coefficients cucc-0.1, pendulum mass m=0.2, pendulum length = 0.2, cart total mass M=0.4, gravitational constant g=9.81. The Force term is applied by the cart wheels driven by a DC motor with a model from the applied voltage (say, - 5:5Volts) to the force. di ki V = Ri + L+Vemf Vemf = kv. F: Where the back-emf is taken as proportional to the motor angular velocity, which is also proportional to the cart velocity, while the motor current generates the torque that is converted to force by the wheels of radius r. The constants are motor resistance R=5, inductance L=0.1, emf/torque-constant k=0.3, and wheel radius r=0.1. We want to design a controller for this system to stabilize the inverted pendulum and be able to follow commands for the cart speed. We want to use a sequential approach where we first stabilize the angle with an "inner-loop controller". Then form the inner closed loop and design an "outer loop controller" for the cart velocity. There are two difficulties associated with this problem. One is that the angle subsystem has a RHP pole and a zero at the origin. Its stabilization requires a controller with a RHP pole. We can solve this as a modified PID problem where instead of the integrator we use a RHP pole determined iteratively. The other difficulty is technical, namely, how to create the various systems and loops without leaving stray pole- zero cancellations (possibly in the RHP) and without resorting to tedious hand calculations. (One approach for this is to implement the model in Simulink and use the "ljomod" command, and the other -taken here- is to work with the state space model using the "feedback" command.) 1. Form the state-space description of the system with one input (Voltage) and three outputs, (velocity, angle, Voltage). It is convenient to keep the voltage as an output to make it easy to simulate with Matlab.commands We want to implement the controller in DT with a sampling rate 100Hz. For this system, it is more convenient to follow the w-plane approach, find the ZOH equivalent of the plant now and convert to the w-plane and do not have to do corrections during the subsequent iterations.