4.2 Control Trajectories and Controller Parameters

Two representations of control design¶

The trajectory $u(t)$ is the command delivered to the plant model: motor voltage, torque, valve position, thrust, pitch angle, or another manipulated input. In open-loop CCD, the entire function may be optimized:

This representation makes few assumptions about controller structure and reveals performance limits, but it does not by itself specify a causal feedback law.

The vector $c$ contains time-independent controller parameters. For proportional-derivative position control,

The implemented command depends on the reference $r$, measured or estimated states, saturation, and plant-dependent authority. More elaborate $c$ may contain estimator covariances, filter coefficients, MPC weights and horizons, gain-schedule parameters, or controller-order coefficients.

Feedback requires an information model¶

A general causal policy is

where $\mathcal I_t$ is information available by time $t$ and $a_c$ is the control architecture. The information set may include sensor histories, state estimates, references, preview, and model predictions. Writing $u=\pi(x)$ while assuming an unmeasured state is available hides a sensor and estimation requirement.

Do not mix command and delivered actuation¶

Motor torque, joint torque after gearing, and commanded torque are different quantities. Efficiency, bandwidth, saturation, rate limits, and delays map one to another. The formulation should name the optimized quantity and enforce the actuator dynamics that connect it to the plant.

Activity 4.2: identify the realizable controller¶