SC · Laplace

Block Diagrams and Feedback

Draw systems as boxes and connect them, and their transfer functions combine algebraically. In series they multiply, in parallel they add, and feedback, which routes the output back toward the input, closes a loop and creates a new transfer function. In this lesson you raise the feedback gain, move an unstable plant's pole into the left half-plane, and see the system become stable.

Stabilize the unstable with feedback

The upper diagram is a negative feedback loop. The input X enters the summing junction (+), and the output Y, after passing through the forward-path plant H(s) = 1/(s − 1), goes back through a feedback path of gain k and is subtracted at the summing junction (−). This plant has its pole at +1 in the right half-plane, so on its own it diverges. The lower plot is the closed-loop impulse response e^((1 − k)t), and k starts at 0.4. Raise the gain k to push the closed-loop pole s = 1 − k to the left of the imaginary axis, and you will see the growing response die away to 0.

negative feedback loop

closed-loop response

feedback gain k0.40

Closed loop and pole

T(s) = 1 / (s − 0.60)   pole s = 0.60

The feedback gain is still small, so the closed-loop pole s = 1 − k is in the right half-plane. The response grows exponentially and runs off the top of the plot. Even with feedback in place, as long as the gain is below 1 the unstable plant is not yet stabilized.

Unstable

The algebra of wiring boxes

A block diagram draws each system as a box labeled with its transfer function and connects the boxes with arrows in the direction the signal flows. Connect two systems in series and the first one's output becomes the second one's input, so the overall transfer function is the product of the two, H₁H₂. In parallel, where the same input is split between two systems and their outputs are added, the overall transfer function is the sum H₁ + H₂. However tangled the signal flow, applying these multiply and add rules step by step reduces it to a single transfer function.

Observeseries: H₁ H₂

Connecting in series multiplies the transfer functions.

Chooseparallel: ?

Connecting in parallel adds the transfer functions.

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

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

On your ownpole: 1 → ?

Feedback moves the poles to new places.

Feedback closes a loop

Feedback is a structure that sends the output back and subtracts it from the input. The signal entering the forward path H is the original input X minus the output after the feedback path G, GY, so Y = H(X − GY). Rearranging gives Y(1 + HG) = HX, that is, the closed-loop transfer function T = Y/X = H/(1 + HG). The 1 + HG in the denominator is the part feedback newly creates. With negative feedback, the signal entering the forward path shrinks when the output overshoots and grows when it falls short, so the output adjusts itself to follow the target. In the first screen's diagram, the feedback path G is the constant k.

Feedback moves the poles

The closed-loop poles are the roots of the denominator, 1 + HG = 0. This equation differs from the one that set the poles of the open-loop H, so closing a feedback loop moves the poles to new places. Apply negative feedback with gain k to the first screen's unstable plant H = 1/(s − 1) and the closed loop becomes 1/(s − 1 + k), moving the pole to s = 1 − k. So making k larger than 1 pushes the pole into the left half-plane and the system becomes stable. Placing poles with gain like this is the heart of control, and following the path the poles trace as the gain changes is the root locus, which comes later.

Back to the first screen

The plant H = 1/(s − 1) had its pole in the right half-plane, so on its own it diverged. Once the feedback loop was closed and the gain k raised, the closed-loop pole s = 1 − k crossed the imaginary axis into the left half, and the growing response died away to 0. Feedback did more than attach one more box; it changed the denominator to 1 + HG and moved the pole to a new place. Using a single gain to place poles and turn instability into stability is the heart of feedback control.

A block diagram draws systems as transfer-function boxes connected together. Series means product H₁H₂, and parallel means sum H₁ + H₂. Negative feedback closes a loop with a forward path H and a feedback path G, producing the closed loop T(s) = H/(1 + HG). The closed-loop poles are the roots of 1 + HG = 0, so they differ from the open-loop poles. That is why feedback moves poles, stabilizing unstable systems and adjusting the speed and gain of the response; this is the foundation of feedback control.

On to the next unit

If feedback can put poles wherever you want, then to finish a design you need to know exactly what response each pole location produces. The next unit looks at the responses of standard first-order and second-order systems. A first-order system is a simple decay set by a single time constant τ, while in a second-order system the damping ratio ζ and the natural frequency ωn set the transient response, so practical specifications such as overshoot and settling time can be read straight from the pole locations. That is where it becomes clear what response the poles you moved with feedback will produce.