Formulation of the Time-Optimal Problem and Maximum Principle

1978 ◽  
pp. 1-19
Author(s):  
R. V. Gamkrelidze
Keyword(s):  
Maximum Principle ◽  
Time Optimal

Strong Lagrange duality and the maximum principle for nonlinear discrete time optimal control problems

2019 ◽  
Vol 25 ◽  
pp. 20
Author(s):  
Eero V. Tamminen
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Discrete Time ◽  
Optimal Control Problems ◽  
Discrete Maximum Principle ◽  
Control Problems ◽  
Convexity Condition ◽  
Lagrange Duality ◽  
The Maximum Principle ◽  
Time Optimal

We examine discrete-time optimal control problems with general, possibly non-linear or non-smooth dynamic equations, and state-control inequality and equality constraints. A new generalized convexity condition for the dynamics and constraints is defined, and it is proved that this property, together with a constraint qualification constitute sufficient conditions for the strong Lagrange duality result and saddle-point optimality conditions for the problem. The discrete maximum principle of Pontryagin is obtained in a straightforward manner from the strong Lagrange duality theorem, first in a new form in which the Lagrangian is minimized both with respect to the state and to the control variables. Assuming differentiability, the maximum principle is obtained in the usual form. It is shown that dynamic systems satisfying a global controllability condition with convex costs, have the required convexity property. This controllability condition is a natural extension of the customary directional convexity condition applied in the derivation of the discrete maximum principle for local optima in the literature.

Time Optimal Swing-Up Control of Single Pendulum1

2000 ◽  
Vol 123 (3) ◽  
pp. 518-527 ◽  
Author(s):  
Yongcai Xu ◽  
Masami Iwase ◽  
Katsuhisa Furuta
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Auxiliary Problem ◽  
Terminal State ◽  
Time Optimal Control ◽  
Necessary Condition ◽  
Optimal Time ◽  
Input Amplitude ◽  
Time Optimal ◽  
Bounded Input

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.

Sign in / Sign up

Export Citation Format

Share Document