Time Optimal Swing-Up Control of Single Pendulum1

2000 ◽  
Vol 123 (3) ◽  
pp. 518-527 ◽  
Author(s):  
Yongcai Xu ◽  
Masami Iwase ◽  
Katsuhisa Furuta

Swing-up of a rotating type pendulum from the pendant to the inverted state is known to be one of most difficult control problems, since the system is nonlinear, underactuated, and has uncontrollable states. This paper studies a time optimal swing-up control of the pendulum using bounded input. Time optimal control of a nonlinear system can be formulated by Pontryagin’s Maximum Principle, which is, however, hard to compute practically. In this paper, a new computational approach is presented to attain a numerical solution of the time optimal swing-up problem. Time optimal control problem is described as minimization of the achievable time to attain the terminal state under the bounded input amplitude, although algorithms to solve this problem are known to be complicated. Therefore, in this paper, it is shown how the optimal time swing-up control is formulated as an auxiliary problem in that the minimal input amplitude is searched so that the terminal state satisfies a specification at a given time. Through the proposed approach, time optimal control can be solved by nonlinear optimization. Its approach is evaluated by numerical simulations of a simplified pendulum model, is checked satisfying the necessary condition of Maximum Principle, and is experimentally verified using the rotating type pendulum.

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 311
Author(s):  
Dongsheng Luo ◽  
Wei Wei ◽  
Hongyong Deng ◽  
Yumei Liao

In this paper, we consider the time-optimal control problem about a kind of Petrowsky system and its bang-bang property. To solve this problem, we first construct another control problem, whose null controllability is equivalent to the controllability of the time-optimal control problem of the Petrowsky system, and give the necessary condition for the null controllability. Then we show the existence of time-optimal control of the Petrowsky system through minimum sequences, for the null controllability of the constructed control problem is equivalent to the controllability of the time-optimal control of the Petrowsky system. At last, with the null controllability, we obtain the bang-bang property of the time-optimal control of the Petrowsky system by contradiction, moreover, we know the time-optimal control acts on one subset of the boundary of the vibration system.


Author(s):  
Natalya A. Il’ina

The task of organization a closed time-optimal control system of linear object with distributed parameters of parabolic type is considered. The object has two lumped internal controls for the power of heat sources excited in the electromagnetic field of an inductor. The proposed method for the synthesis of optimal controllers uses an alternance method for calculating the optimal program controls for each of the control actions. An example of the construction of a quasi-optimal time control system for the process of periodic induction heating of a metal workpiece with constant values of the feedback coefficients calculated for the most characteristic initial spatial distribution is given.


Micromachines ◽  
2020 ◽  
Vol 11 (9) ◽  
pp. 834
Author(s):  
Ilya Dikariev ◽  
Fabiola Angulo ◽  
David Angulo-Garcia

In this paper, we study the time optimal control problem in a DC-DC buck converter in the underdamped oscillatory regime. In particular, we derive analytic expressions for the admissible regions in the state space, satisfying the condition that every point within the region is reachable in optimal time with a single switching action. We then make use of the general result to establish the minimum and maximum variation allowed to the load in two predefined design set-ups that fulfills the time optimal single switching criteria. Finally, we make use of numerical simulations to show the performance of the proposed control under changes in the reference voltage and load resistance.


Author(s):  
Ivan Matychyn ◽  
Viktoriia Onyshchenko

AbstractThe problem of time-optimal control of linear systems with fractional dynamics is treated in the paper from the convex-analytic standpoint. A linear system of fractional differential equations involving Riemann- Liouville derivatives is considered. A method to construct a control function that brings trajectory of the system to the terminal state in the shortest time is proposed in terms of attainability sets and their support functions


2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Shaolin Ji ◽  
Qingmeng Wei ◽  
Xiumin Zhang

We study the optimal control problem of a controlled time-symmetric forward-backward doubly stochastic differential equation with initial-terminal state constraints. Applying the terminal perturbation method and Ekeland’s variation principle, a necessary condition of the stochastic optimal control, that is, stochastic maximum principle, is derived. Applications to backward doubly stochastic linear-quadratic control models are investigated.


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