Apps

Bryson-Denham Problem

Apps.BrysonDenhamProblem History

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Changed line 9 from:

The parameter u is the acceleration can be adjusted over the time horizon from a starting time of zero to a final time of one. The variable x is the position and v is the velocity.

to:

The parameter u (acceleration) is adjusted over the time horizon from a starting time of zero to a final time of one. The variable x is the position and v is the velocity.

Added lines 24-36:

Solution

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Attach:bryson_denham_solution.png Δ

Changed line 9 from:

The parameter u is the acceleration can be adjusted over the time horizon from a starting time of zero to a time of one. The variable x is the position and v is the velocity.

to:

The parameter u is the acceleration can be adjusted over the time horizon from a starting time of zero to a final time of one. The variable x is the position and v is the velocity.

Changed lines 15-23 from:

$$\quad \frac{dx(t)}{dt} = v $$

$$\quad \frac{dv(t)}{dt} = u$$

$$\quad x(0) \; = \; x(1) \; = \; 0$$

$$\quad v(0) \; = \; -v(1) \; = \; 1$$

$$\quad x(t) \le \ell, \ell=\frac{1}{9}$$

to:

$$\frac{dx(t)}{dt} = v(t) $$

$$\frac{dv(t)}{dt} = u(t) $$

$$x(0) \; = \; x(1) \; = \; 0$$

$$v(0) \; = \; -v(1) \; = \; 1$$

$$x(t) \le \ell, \; \ell=\frac{1}{9}$$

Added lines 1-42:

(:title Bryson-Denham Problem:) (:keywords Benchmark, Python, nonlinear control, dynamic programming, optimal control:) (:description Minimize the integral of the control input while meeting certain path and final time constraints.:)

The Bryson-Denham optimal control problem is a benchmark test problem for optimal control algorithms.

Problem Statement

The parameter u is the acceleration can be adjusted over the time horizon from a starting time of zero to a time of one. The variable x is the position and v is the velocity.

$$\min J = \frac{1}{2} \; \int_0^1 u^2(t) dt$$

$$\mathrm{subject\;to}$$

$$\quad \frac{dx(t)}{dt} = v $$

$$\quad \frac{dv(t)}{dt} = u$$

$$\quad x(0) \; = \; x(1) \; = \; 0$$

$$\quad v(0) \; = \; -v(1) \; = \; 1$$

$$\quad x(t) \le \ell, \ell=\frac{1}{9}$$


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