By Karl Astrom
This monograph describes the response Wheel Pendulum, the latest inverted-pendulum-like gadget for keep an eye on schooling and examine. We talk about the background and historical past of the response wheel pendulum and different related experimental units. We improve mathematical versions of the response wheel pendulum intensive, together with linear and nonlinear types, and versions of the sensors and actuators which are used for suggestions regulate. We deal with numerous elements of the keep an eye on challenge, from linear regulate of themotor, to stabilization of the pendulum approximately an equilibrium configuration utilizing linear regulate, to the nonlinear keep an eye on challenge of swingup keep an eye on. We additionally talk about hybrid and switching keep an eye on, that's worthwhile for switching among the swingup and stability controllers. We additionally talk about very important sensible matters similar to friction modeling and friction repayment, quantization of sensor signs, and saturation. This monograph can be utilized as a complement for classes in suggestions keep an eye on on the undergraduate point, classes in mechatronics, or classes in linear and nonlinear nation house regulate on the graduate point. it could possibly even be used as a laboratory handbook and as a reference for learn in nonlinear regulate.
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Example text
Notice that there is a difference between the outputs. , an offset from the filtered velocity. The reason for this difference is that the friction force was neglected in the design of the model-based observer. To understand this we will investigate the consequences of neglecting the friction force. In the experiment the input signal is constant u = u 0 . Assume that the model has a constant friction force F0 but that the friction force is neglected in the observer. 4: Observer estimate (bottom) and first-order filter (top).
0302. 1405. Experiment 13 (Stabilization and Control of Wheel Velocity). 9). Find suitable values of the parameters k pp , k d p , and k dr . Investigate the behavior of controller with different closed loop poles. Explore the system by tapping the pendulum with a ruler or a metal rod. Investigate the behavior of the pendulum and the wheel. Explain your findings theoretically. 3 show results from experiments with these controllers. In the experiments a disturbance has been introduced by hitting the pendulum with a pencil.
The first question we have to answer is if it is possible to do this with only one control variable. For that purpose we will investigate the controllability of the system. Introduce the state variables x1 = θ, ˙ x2 = θ, x3 = θr , x4 = θ˙r Neglecting the friction force, Eq. 8) −ab p 0 0 0 This matrix is full rank and the system is thus controllable. This means that the dynamics of the closed loop system can be shaped arbitrarily using only one control variable. 2 CONTROL OF THE PENDULUM AND THE WHEEL VELOCITY Since the pendulum is influenced by the acceleration of the wheel, it may happen that the wheel velocity saturates after a while.
