Inverted pendulum

Design a fuzzy logie eontroller for the inverted pendulum system shown in Figure 10.13 so that the pendulum remains in the vertieal position. 

The inverted pendulum problem is a elassie example of produeing a stable elosed-loop eontrol system from an unstable plant. 

Figure 10.15 shows the time response of the inverted pendulum state variables from an initial eondition of 0 = 0.1 radians. On eaeh graph, three eontrol strategies are shown, the 11 set rulebase of equation (10.52), the 22 set rulebase that ineludes X and X, and the state feedbaek method given by equation (10.51). 

Fig. 10.14 Input and output fuzzy windows for the inverted pendulum problem.

Fig. 10.15 Inverted pendulum state variable time response for three control strategies.

This is the inverted pendulum eontrol problem and, as a benehmark, is initially solved as a multivariable eontrol problem, using pole plaeement (Aekermann s formula) to ealeulate the feedbaek gain matrix in examplOS.m. 

Fig. A1.4 Inverted pendulum control system design using pole placement...

Fig. A1.8 Simulink implementation of inverted pendulum fuzzy logic control problem.

There is also an openloop unstable mechanical system the inverted pendulum. This is the problem of balancing a broom on the palm of your hand. You must keep moving your hand to keep the broom balanced. If you put your brain on manual and hold your hand still, the broom will topple over. So the process is openloop unstable. 

There are several advantages to the cylindrical representation. Now the periodic whirling motions look periodic—they are the closed orbits that encircle the cylinder for E>1. Also, it becomes obvious that the saddle points in Figure 6.7.3 are all the same physical state (an inverted pendulum at rest). The heteroclinic trajectories of Figure... 

The gravitational torque is like that of an inverted pendulum, since water is pumped in at the top of wheel (Figure 9.1.6). 

To check that the sign is correct, observe that when sin 6 > 0 the torque tends to increase co, just as in an inverted pendulum. Here g is the effective gravitational constant, given by g = gg sin a where gg is the usual gravitational constant and a is the tilt of the wheel from horizontal (Figure 9.1.1). 

Kuo AD, Donelan JM, Ruina A (2005) Energetic consequences of walking like an inverted pendulum step-to-step transitions. Exerc Sport Sci Rev 33 88-97... 

Parametrically Forced Inverted Pendulum The basic example for parametrically excited systems is the inverted pendulum with periodically vibrating suspension (see Fig. 17). The equation of motion for the system reads... 

The Strutt-frice diagram is shown in the left panel of Fig. 18. The case > 0 corresponds to the downward position of the pendulum, while the case < 0 corresponds to the upward position (inverted pendulum). The diagram in the right panel of Fig. 18 presents the stability domains transformed to the plane of the forcing frequency / = coHn and the forcing amplitude r for a pendulum of length / = 50 cm. The case in which the maximum acceleration of the pivot point is equal to the gravitational acceleration g... 

Hunt, A., Chen, Z., Tan, X. and Kruusmaa, M. (2010). Control of an inverted pendulum using an ionic polymer-metal composite actuator, in Proceedings of 2010 lEEE/ASME International Conference on Advanced Intelligent Mechatronics (Montreal, Canada), pp. 163-168. 

Bagheri Shouraki and Honda (1998) showed the ability of ALM for stable controlling of dynamic systems such as invert pendulum... 

In the mid-eighteenth century, however, the British Isles experienced a series of severe earthquakes, which created a tsunami that destroyed Lisbon, Portugal, killing tens of thousands of people. Scientists quickly developed an interest in cataloging and understanding seismic events. In the early nineteenth century, Scottish physicist and glaciologist James D. Forbes invented the inverted pendulum seismometer, which gauged not only the severity of an earthquake but also its duration. 

Many discussions have taken place about its exact internal mechanism, but a consensus has not been reached. Models with common or inverted pendulums and a model assuming that the oscillating element was the external jar suspended from the pillar have been proposed. Assuming the chronicles are correct, the most puzzling question is that only one ball was delivered and that it pointed in the direction of the epicenter. Thinking on simple mechanisms with pendulums, it looks like more than one ball can be delivered and the pointing direction may be toward or opposite to the epicenter. Anyway, the use of more complex designs, already known at that time, has shown the feasibility of such a mechanism (Fig. 7b-e). 

Historical Seismometer, Fig. 14 Agamennone double-action electric seismoscope. Two inverted pendulums, with different free period, are set aside. The close proximity of the two upper extremities of the pendulums had the effect, in the case of a tremor, of making them come into contact with one another, completing an electric circuit (From Ferrari 1992)... 

Historical Seismometer, Fig. 20 Different pendulum settings used for earthquake recording. It is possible to establish a seismograph classification according to the mechanical constitution of the transducer (i) conical or bifilar pendulum, (2) rigid pendulum, (5) Zdllner pendulum, (4) vertical pendulum, (5) torsion pendulum, (6) inverted pendulum, (7) fiexure pendulum, (S) vertical oscillator, (9) vertical rotational oscillator (From Batiio 2004)... 

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