3.2 Continuous, Discrete, and Functional Variables

Three kinds of decisions¶

Continuous variables take real values within bounds. Spring stiffness, link length, controller gain, sensor sampling period, and motor rating are commonly modeled this way during preliminary design.

Discrete variables take values from a countable set. Examples include the number of battery modules, a catalog motor, gear tooth count, sensor location index, or a binary decision that determines whether a component exists.

Functional variables are entire functions, usually trajectories over time or space. Open-loop control $u(t)$ is the most important CCD example. A distributed blade-thickness profile or a gain schedule $K(\rho)$ can also be functional.

A mixed CCD decision set may therefore be written

where $p_c$ is continuous, $p_d$ is integer-valued, $a$ contains architecture binaries, and $u(\cdot)$ is time dependent.

Functional decisions become finite vectors¶

A computer cannot optimize an uncountable number of values directly. Numerical optimal-control methods parameterize or discretize a trajectory:

or treat values $u_k=u(t_k)$ at mesh points as decision variables. The coefficients $\alpha_j$ or samples $u_k$ form a finite vector. More freedom can improve the discrete solution but increases problem size and may expose unrealistic high-frequency behavior.

Modeling choices can change the answer¶

Treating a catalog actuator rating as continuous may be useful for discovering an ideal scale, but the result must later be mapped to an available component and revalidated. Relaxing a binary architecture variable to the interval $[0,1]$ can aid computation, but fractional hardware has no direct physical meaning unless the relaxation is interpreted carefully.

Activity 3.2: choose variable types¶