From Formulation to Numerical Optimization

Finite-dimensional transcription¶

A computer cannot optimize arbitrary functions directly. A continuous-time formulation is converted into a nonlinear program by:

parameterizing state and control trajectories;
sampling trajectories at finite points;
enforcing dynamics through simulation or defect equations; and
approximating integral objectives numerically.

For example, quadrature gives

\int_{t_0}^{t_f}L(t)\,dt\approx\sum_{i=1}^{N}w_iL(t_i).

(1)

Later chapters develop these transcription choices. The important point here is that reliable numerical methods begin with a sound mathematical formulation.

Common formulation mistakes¶

Leaving out essential physics. The result may not transfer to the real system.
Confusing variables and parameters. A quantity is a decision variable only if the optimizer may change it.
Using an objective that misses the engineering goal. Minimizing control effort alone, for example, may destroy tracking.
Treating path constraints as endpoint constraints. This permits dangerous intermediate violations.
Over-constraining the problem. Excessively tight limits may eliminate all feasible designs.
Forgetting variable bounds. The optimizer may enter unrealistic model regions.
Mixing open- and closed-loop representations. A feedback-parameterized controller must be reflected consistently in the dynamics and decisions.