S4-3.1 Control Fundamentals & Modelling

What this is and why it exists

Control theory begins with one distinction — act blindly, or measure and correct — and one manoeuvre: turn any physical system into the same mathematical object, the transfer function, so that one theory serves motors, ovens, aircraft and chemical plants alike. This unit gives you the distinction, two routes to the transfer function of a connected system, and the modelling trick that makes springs and masses answer to circuit mathematics.

The vocabulary

• Open-loop control — the output is never measured; the controller acts on the command alone, like a toaster timer.
• Closed-loop control — the output is measured, compared with the command, and the error drives the correction.
• Transfer function — output over input in the Laplace domain, with zero initial conditions; the system as one ratio of polynomials.
• Block diagram reduction — rules for collapsing series, parallel and feedback connections into a single block.
• Signal flow graph — the same system drawn as nodes and directed gains, ready for Mason's formula.
• Mason's gain formula — the overall gain read directly from a flow graph's forward paths and loops, with no step-by-step collapsing.
• Force-voltage analogy — mass, damper and spring map onto inductor, resistor and capacitor; mechanical equations become circuit equations.

The mental model

A toaster and a thermostat. The toaster is open-loop: the timer runs its course whether the bread burns or not — cheap, fast to design, and helpless against disturbance. The thermostat is closed-loop: it measures, compares and corrects, so a cold morning changes nothing about the room's final temperature. The price of feedback is complexity and a new possible failure — a loop that corrects too eagerly can overshoot and oscillate. Deciding whether a loop does that is most of this course.

Finding the loop's transfer function has two roads. Block reduction is local surgery: combine series blocks by multiplying, parallel by adding, collapse each feedback loop with the standard closed-loop formula, moving pickoff points where needed — transparent, but tedious when loops interlock. Mason's formula is the aerial view: list every forward path from input to output, list every loop, account for which loops touch, and the answer assembles in one expression. On heavily entangled diagrams Mason wins; on shallow ones reduction is quicker. Learn both and choose per problem.

The force-voltage analogy is the reason this is one subject rather than five. Newton's equation for a mass-damper-spring and Kirchhoff's equation for an inductor-resistor-capacitor loop are the same differential equation with the symbols renamed: force as voltage, velocity as current, mass as inductance, friction as resistance, springiness as capacitance. Model the mechanical system once, convert, and every circuit tool applies. The same unification is why control ideas reappear wherever dynamics do — robotics included.

What you should now be able to explain or do

Classify a described system as open- or closed-loop and defend the classification. Reduce a multi-loop block diagram to one transfer function. Apply Mason's gain formula to a flow graph with touching and non-touching loops. Convert a mass-damper-spring system to its electrical analog and back.

Check yourself