2.4 Feedback and Feedforward Control

Feedback reacts to measured consequences¶

Feedback computes a command from measured or estimated behavior,

u(t)=\kappa\!\left(r_c(t),\hat{x}(t),y(t),c\right),

where $r_c$ is a reference and $c$ contains controller parameters. Feedback can reject unmeasured disturbances, stabilize unstable plants, reduce sensitivity to parameter error, and enforce desired dynamics. It acts only after information about the disturbance or resulting error becomes available.

Feedforward anticipates known causes¶

Feedforward uses a reference or measured disturbance directly,

u(t)=u_{\mathrm{ff}}\!\left(r_c,d_m,p\right)+u_{\mathrm{fb}}\!\left(\hat{x},r_c,c\right).

It can cancel predictable effects without waiting for error, but its quality depends on model accuracy and information. Road preview, wind preview, commanded robot trajectories, and known payload changes can support feedforward action.

Feedback and feedforward paths with plant design variables shown explicitly.

Why the distinction matters in CCD¶

An open-loop optimal trajectory often assumes the entire disturbance is known. A realizable feedback controller may know only current measurements. A preview controller lies between these cases. Because information changes achievable performance, each assumption can favor a different plant.

For the suspension, road preview allows the actuator to prepare for an upcoming bump. Without preview, the controller reacts after body or wheel motion begins. More preview may justify softer passive elements or a smaller actuator; limited sensing may favor a more forgiving passive design.

Two-degree-of-freedom control¶

Separating feedforward and feedback clarifies design responsibilities. Feedforward shapes nominal reference tracking; feedback supplies robustness and disturbance rejection. CCD should evaluate both nominal performance and sensitivity to modeling error. An architecture with excellent feedforward tracking but weak feedback robustness may be undesirable.

Controller structure is a design choice¶

PID, state feedback, output feedback, gain scheduling, robust control, and MPC expose different parameters and information requirements. A fair CCD comparison must hold physical design freedom and requirements constant while acknowledging that controller classes have different implementation costs.

Activity 2.4: information audit¶

Activity 2.4: information audit

A reduced lateral-error model for an autonomous vehicle is

\dot e=-ae+bu+d, \qquad u=-k\tilde e-\frac{\hat d}{b},

where $a=0.8$ s $^{-1}$ , $b=1$ , $k=2.2$ s $^{-1}$ , and a constant crosswind produces $d=0.60$ m/s. The camera measurement $\tilde e$ has 80 ms delay and 0.03 m RMS noise. A wind sensor has 30 ms delay and 10% amplitude error. The planned path is known 25 m ahead while the vehicle travels at 20 m/s; future crosswind is not forecast.

Classify the planned path, delayed lateral-error measurement, current wind estimate, future wind, and estimated tire force as feedback, feedforward, preview, estimated, or unavailable information.
For a constant disturbance estimate, show that

e_{\mathrm{ss}}=\frac{d-\hat d}{a+bk}.

Calculate $e_{\mathrm{ss}}$ for feedback only, perfect wind feedforward, and a wind estimate that is 10% low. 3. Compute the available path-preview time. Is any crosswind-preview time available under the stated architecture? 4. At the nominal closed-loop bandwidth $\omega_b=a+bk$ , calculate the phase lag caused separately by the camera and wind-sensor delays using $\phi=\omega_b\tau(180/\pi)$ . 5. The actuator-noise command is approximated by $u_{n,\mathrm{rms}}=k\sigma_e$ . Calculate it and compare it with a 0.10-unit RMS actuator-noise budget. 6. Identify every signal used by the proposed law that is realizable in real time. Then state one additional sensor or communication channel required to use future wind as genuine preview rather than assumed knowledge.