Cart Pendulum
| Cart Pendulum | |
|---|---|
| State dimension: | 1 |
| Differential states: | 3 |
| Discrete control functions: | 2 |
The Cart Pendulum problem concerns a pendulum hinged to a mobile cart. The control objective is to transition the pendulum from a downward position to a stabilized, inverted state above the cart. In this formulation, the objective function is defined by a composite of least-squares terms.
The implementation here is taken from [1]. Its dynamics are given by a four-dimensional ODE model.
Mathematical formulation
Failed to parse (unknown function "\dotdot"): {\displaystyle \begin{array}{lll} \displaystyle \min_{T} && \int_{0}^{t_f} dt \\ \text{subject to} \\ \quad \dot{x}(t) & = & v(t),\\ \quad \dot{\theta}(t) & = & -1 + \frac{T(t)}{m}, \\ \quad \dotdot{x}(t) & = & -\frac{T(t)}{2.349}, \\ \quad h(0) &=& 1, \\ \quad v(0) &=& -0.783, \\ \quad m(0) &=& 1, \\ \quad t_f &\geq& 0, \\ \quad h(t_f) &=& 0, \\ \quad v(t_f) &=& 0, \\ \quad T(t) & \in & [0, 1.227] \ \quad \forall t \in [0,t_f] \end{array} }
Reference Solutions
Here is one local solution to the above control problem.
- Reference solution plots
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States and discretized control for a local optimum. The free end time was modeled using the additional control .
Miscellaneous and Further Reading
This formulation and a detailed description can be found in [1].
References
[1] Multidisciplinary Optimal Control Library: https://openmdao.org/dymos/docs/latest/examples/moon_landing/moon_landing.html