SC · Laplace

Block Diagrams and Feedback

Draw systems as boxes and wire them, and their transfer functions combine by algebra: series multiplies, parallel adds, and feedback — output fed back to the input — closes a loop. Raise the feedback gain and see it move an unstable plant’s pole into the left half-plane, stabilizing it.

Stabilize the unstable with feedback

The top is a negative feedback loop. The forward-path plant H(s) = 1/(s−1) has its pole in the right half (+1), so left alone it diverges. Below is the closed-loop response. Raise the feedback gain k to push the closed-loop pole s = 1−k left of the imaginary axis, turning the diverging response into one that decays to 0.

negative feedback loop

closed-loop response

feedback gain k0.40

Closed loop and pole

T(s) = 1 / (s − 0.60)   극 s = 0.60

The feedback gain is too weak, so the closed-loop pole s = 1−k is still in the right half. The response grows and diverges. The unstable plant is not yet tamed.

Unstable

The algebra of wiring boxes

A block diagram draws systems as boxes labeled with transfer functions and wires the signal flow with arrows. Wire two systems in series and the first output becomes the second input, so the overall transfer function is the product H₁H₂. Split the same input into two systems and add the outputs — parallel — and it is the sum H₁+H₂. Even a tangled signal flow can be reduced step by step into a single transfer function by these multiply-and-add rules.

Observeseries: H₁ H₂

Series multiplies the transfer functions.

Chooseparallel: ?

Parallel adds the transfer functions.

Fill inclosed loop = H / (1 + H ?)

The closed loop is H/(1+HG).

On your ownpole: 1 → ?

Feedback moves the pole to a new place.

Feedback closes a loop

Feedback feeds the output back and subtracts it at the input. The input to the forward path H is the original input X minus the output sent through the feedback path G, so Y = H(X − GY). Solving, Y(1 + HG) = HX, giving the closed-loop transfer function T = Y/X = H/(1 + HG). The 1 + HG in the denominator is the new expression feedback creates. Being negative feedback, it cuts the input when the output is too high and raises it when too low, so the system follows its target on its own.

Feedback moves the poles

The closed-loop poles are the roots of the denominator 1 + HG = 0. This is a different expression from the one that made the open-loop poles of H, so feedback moves the poles to new places. Apply gain-k negative feedback to the unstable plant H = 1/(s−1) you just saw, and the closed loop becomes 1/(s − 1 + k), moving the pole to s = 1 − k. Raise k past 1 and the pole crosses into the left half, stabilizing the system. Placing poles with gain like this is the heart of control, and tracing the path the poles draw as the gain varies is the root locus to come.

Back to the first screen

The plant H = 1/(s−1) had its pole in the right half and diverged on its own. Closing the feedback loop and raising the gain k moved the closed-loop pole s = 1−k across the imaginary axis into the left half, and the growing response decayed to 0. Feedback did not merely wire boxes together — it changed the denominator to 1+HG and moved the pole to a new place. Placing a system’s poles with a single gain like this, turning instability into stability, is the heart of feedback control.

A block diagram wires systems as transfer-function boxes. Series multiplies H₁H₂, parallel adds H₁+H₂. Negative feedback closes a loop with forward path H and feedback path G, giving the closed loop T(s) = H/(1+HG). The closed-loop poles are the roots of 1+HG = 0, different from the open loop. So feedback moves the poles to stabilize a system and tune its speed and gain. This is the foundation of feedback control.

On to the next unit

If feedback can place the poles where you want, then to finish a design you must know exactly the response shape those pole positions produce. The next unit, first- and second-order responses, looks at the standard forms. A first-order system is simple decay set by a single time constant τ; a second-order system has its transient set by a damping ratio ζ and natural frequency ωn, so practical specs like overshoot and settling time are read straight from the pole positions. What response a feedback-placed pole produces becomes clear there.