Lecture 28 - Block Diagram Algebra

With an introduction to block diagrams, we can take the next step to compute the closed-loop dynamic response. Lecture 28 covers the rearrangement of the block diagram to obtain the response of the system (Y(s)) with a change in the closed-loop setpoint (Ysp(s)). This closed loop transfer function (Gcl(s)=Y(s)/Ysp(s)) can help design controller parameters, determine oscillatory closed-loop behavior, and determine the degree of steady-state offset.

• Lecture 28 Notes
• Offset Worksheet
• Block Diagram Algebra Worksheet

Please use the following videos to check your work for 11.7 and 11.10 after you have completed the problem.

Solution to 11.7

Solution to 11.10 with MATLAB

Relate each problem in the context of the overall course objectives.

Discussion Questions

Test the lecture's ideas before or after class - commit to an answer first, then check it with a classmate or an AI tutor.

1. The setpoint transfer function is G_cG_vG_p/(1+G_cG_vG_pG_m) and the disturbance transfer function is G_d/(1+G_cG_vG_pG_m). The numerators differ; the denominators are identical. What does the shared denominator control, and why must it be the same for both?
2. Set s = 0 in the closed-loop setpoint transfer function for a P-only controller. Show that the result is K_cK_p/(1+K_cK_p), not 1 - and connect that to the offset you saw in Lecture 5.
3. The sensor G_m and valve G_v are "fast, so we ignore them." At what point in a block-diagram reduction does that shortcut change the predicted stability limit, and how would you check whether it matters for your loop?
4. In problem 11.7 you solve around the loop for an internal variable. Why does every internal signal in the loop share the same characteristic equation, no matter which input-output pair you pick?

App: The Feedforward Cascade Control Studio draws the loop as a live P&ID with the signal path lit - watch a disturbance propagate around the physical loop that your block diagram abstracts, then trace the same path through your algebra.