Mathematics for Dynamic Modeling by Edward J. Beltrami

By Edward J. Beltrami

This re-creation of arithmetic for Dynamic covers instruments corresponding to linearization, suggestions suggestions, using Liapunov features, and optimum keep an eye on. every one bankruptcy contains routines, a lot of which extend at the fabric within the textual content

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This device, the controller, uses the infor­ mation to move the pivot either right or left. The hand movement induces a restoring force v, which depends on Θ and 0, and it is called a feedback control. As the sensor data changes, so does v. Generally speaking, a control is externally imposed on a given system in an attempt to modify any undesirable behavior. In our case, we hope to keep the bob upright. Can it be done? Certainly if the unstable position is stabilized by some appropriate feedback control v, the answer is yes, but the problem is to show that the modified pendulum does indeed possess a stable equilibrium.

The orbits are shown for energy levels 0 < Ex < E2 < E3. The minimum potential energy occurs at E = 0. (decreases), the direction of the orbits is clockwise as shown by the arrows. Such motion reflects the back and forth swings of the pendulum about its stable equilibrium. Because U(p) = E2 at p = ±mn, the relation U(p) < E2 together with the knowledge that total energy is E2 combine to show that p = 0 at these points. 1, such points are the unstable equilibria of the pendulum in the 0, Θ plane.

Cos χΛ -\— sin χΛ. 23) At this point we make a modeling assumption, which is that the control v depends linearly on the position angle Θ and its velocity Θ. We do this for mathematical convenience in the hope that it is not too restrictive a condition. 24) for suitable constants cl9 c 2 , which remain to be determined. 23 gives x2 = —rx2 Λ cxxx cos χλ Λ c2x2 g m — cos xx + — sin xx. 25) An equilibrium occurs at xx = x2 = 0, which is the straight up position at which 0 = 0 = 0. Our goal is to choose v at each instant so that the equilibrium x = 0 becomes asymptotically stable.

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