Count integrators, then check the error
Compare type 0/1/2 at K=2 with a step, then a ramp. Why can a loop track a fixed target but lag behind a moving one?
First, picture it
System type counts uncancelled poles at the origin in the loop transfer function L, not the total number of states. The examples contain zero, one and two integrators respectively. The type-2 loop includes a zero at −1: its damping is deliberate. Simply using K/s² would give undamped closed-loop poles and would not justify the usual steady-error table.
What the model says
For unity negative feedback, E=R/(1+L). A finite final value can be read from lim(sE) only when all poles of sE lie strictly in the left half-plane. The supported K>0 examples have denominators s+1+K, s+K and s²+Ks+K, which are stable. Initial states are zero, sensors exact, and actuators unlimited. Saturation, disturbances and model mismatch are outside this bench.
L₀=K/(s+1); L₁=K/s; L₂=K(s+1)/s² · E=R/(1+L)The graph records only 30 seconds. For K=0.2, type-2 transients decay slowly and the final plotted error can differ from the theoretical limit. Compare the numerical last sample with the separately labeled asymptotic prediction. Raising type is not a universal upgrade: additional phase lag, sensor noise and actuator effort still need design checks.
Type is not order
System type counts uncancelled poles at the origin in the loop transfer function L, not the total number of states. The examples contain zero, one and two integrators respectively. The type-2 loop includes a zero at −1: its damping is deliberate. Simply using K/s² would give undamped closed-loop poles and would not justify the usual steady-error table.
Ask stability before asking the final value
For unity negative feedback, E=R/(1+L). A finite final value can be read from lim(sE) only when all poles of sE lie strictly in the left half-plane. The supported K>0 examples have denominators s+1+K, s+K and s²+Ks+K, which are stable. Initial states are zero, sensors exact, and actuators unlimited. Saturation, disturbances and model mismatch are outside this bench.
Step and ramp ask different questions
A unit step asks the output to hold 1. Its errors are 1/(1+K), 0 and 0 for type 0/1/2. A unit ramp asks the output to keep moving with slope 1. Type 0 has no finite limiting error, type 1 has error 1/K, and type 2 tends to zero. These results follow by substituting R=1/s or 1/s² into E. An integrator stores the correction needed to remove a particular persistent error.
A last sample is not a limit
The graph records only 30 seconds. For K=0.2, type-2 transients decay slowly and the final plotted error can differ from the theoretical limit. Compare the numerical last sample with the separately labeled asymptotic prediction. Raising type is not a universal upgrade: additional phase lag, sensor noise and actuator effort still need design checks.
Make it concrete
Compare type 0/1/2 at K=2 with a step, then a ramp. Why can a loop track a fixed target but lag behind a moving one?
Open the concept benchThe graph records only 30 seconds. For K=0.2, type-2 transients decay slowly and the final plotted error can differ from the theoretical limit. Compare the numerical last sample with the separately labeled asymptotic prediction. Raising type is not a universal upgrade: additional phase lag, sensor noise and actuator effort still need design checks.
Check your understanding+
Compare type 0/1/2 at K=2 with a step, then a ramp. Why can a loop track a fixed target but lag behind a moving one?
A unit step asks the output to hold 1. Its errors are 1/(1+K), 0 and 0 for type 0/1/2. A unit ramp asks the output to keep moving with slope 1. Type 0 has no finite limiting error, type 1 has error 1/K, and type 2 tends to zero. These results follow by substituting R=1/s or 1/s² into E. An integrator stores the correction needed to remove a particular persistent error.