Stabilisation and Motion Control of Unstable Objects by Alexander M. Formalskii
Author:Alexander M. Formalskii
Language: deu
Format: epub
Publisher: de Gruyter
Published: 2015-07-15T00:00:00+00:00
The allowable control functions are piecewise continuous functions α(t) belonging to segment (8.15). The set of admissible control functions will be denoted as U.
8.3 Optimalcontrolthat swings the pendulum
Let the initial state of the system be given
If angle α(0)= 0, then both links at t = 0 are stretched along the same line, and Ï(0)=p(0), as it follows from relations (8.9), (8.10). Then under initial conditions (8.16) âÏ < Ï(0)< 0. If âÏ < p(0)< 0 then there exists such value of α(0) (for example, α(0)= 0), that the moment of the gravity force is positive, i.e. f[p(0), α(0)] > 0 (see (8.13)). From differential equation (8.13) it follows that at such value of α(0)derivative K is positive. So the angular momentum K that was equal to zero at t =0, at the beginning of motion (when t > 0) will start to increase, and it will become positive. Let initial condition p(0)and the values of αmin, αmax be such that with each control function α(t) â U the angular momentum K at some time instant t1 >0 becomes zero: K(t1 )=0. Each control function α(t) â U has a different corresponding time t1.From equation (8.11) it follows that on entire time interval (0, t1 )the value of angle p increases strictly monotonically, because on this interval the angular momentum K >0.
The problem of optimalpendulumswinging isstatedasfollows. Itisrequiredto find such law of varying the control parameter α that angle p reaches its maximum possible value at some time t1 > 0, when the angular momentum K becomes zero (again) for the first time since the beginning of motion: K(t1 )=0. This can be symbolically written as:
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