Algebra

For the unity-feedback system shown, with forward path containing: - An integrator: 1/s - A plant: 3/(s+4) - An inner feedback loop with gain 2 - An outer feedback path with gain 6s (i) Obtain the open loop transfer function G(s) for the system and hence determine the system type. (ii) Determine the position error constant K_p, velocity error constant K_v, and acceleration error constant K_a for the system. (iii) Calculate the steady-state error that may be expected for a unit ramp input.

Step-by-step solution with explanation

Final Answer

(i) Equivalent open-loop TF: — System Type 0 (no free integrators at s=0). (ii) , , . (iii) Steady-state error for unit ramp = (the Type 0 system cannot track a ramp input).

Step-by-step solution

Simplify the inner feedback loop

The plant 3/(s+4) has an inner feedback loop with gain 2. We use the standard closed-loop formula: G_closed = G/(1+GH). Multiply numerator and denominator by (s+4) to simplify. This collapses the inner loop into one block.

Find the forward path transfer function

Cascade the integrator 1/s with the simplified inner-loop block 3/(s+10). Cascaded blocks simply multiply. This is the total forward path gain before the outer feedback is applied.

Find the open-loop transfer function G(s)

For a unity-feedback system the open-loop transfer function is G(s) = forward path / (1 + inner loops already absorbed). Here the outer feedback path H(s) = 6s is the feedback in the unity-feedback sense. The open-loop TF seen by the error signal is: L(s) = G_forward(s) = 3/[s(s+10)], but we must include the outer feedback gain 6s in the loop. So the open-loop transfer function (loop gain) is: G(s)H(s) = [3/(s(s+10))] × 6s = 18/(s+10). However, for error-constant analysis we need G(s) for the equivalent unity-feedback system.

Form the equivalent unity-feedback open-loop TF

When the outer feedback is H(s)=6s (not unity), we convert to an equivalent unity-feedback open-loop TF using G_eq = G_fwd / [1 + G_fwd(H-1)]. Expanding: s(s+10)+3(6s-1) = s²+10s+18s-3 = s²+28s-3. This lets us apply standard error-constant formulas correctly.

Determine system type from G_eq(s)

System type is the number of poles at s=0 in the open-loop TF. G_eq(s) has no s in the denominator as a standalone factor, so there are zero poles at the origin. The system is Type 0.

Compute the three error constants

For a Type 0 system, K_p is finite, while K_v and K_a are both zero. We evaluate each limit by substituting s→0. Note K_p = -1 (negative), which signals the system as described may be unstable or the feedback polarity convention differs — this is a red flag worth checking in the physical system.

Calculate steady-state error for unit ramp input

For a unit ramp input r(t)=t, the steady-state error formula is e_ss = 1/K_v. Since K_v = 0 (Type 0 system), the error grows without bound. A Type 0 system cannot track a ramp input — it will have infinite steady-state error.

Understanding this problem

Learning Insight

System type tells you the highest-order input a system can track with finite error. A Type 0 system tracks only step inputs (with finite error), a Type 1 tracks ramps, and a Type 2 tracks parabolas. The number of integrators (poles at s=0) in the open-loop TF determines the type, because each integrator provides an automatic correction for one higher order of input. The outer feedback H(s)=6s acts like a derivative term, which actually removes the integrator's effectiveness for ramp tracking.

Quick Tip

Quick pattern: count the bare 's' factors in the denominator of G_eq(s). That count IS the system type. Then: K_p needs Type≥0, K_v needs Type≥1, K_a needs Type≥2. Any constant below the required type is automatically zero, giving infinite steady-state error for that input.

Common Mistake

Students often forget to convert to an equivalent unity-feedback open-loop TF when H(s)≠1. They mistakenly use G_forward(s) alone and count the integrator in it (giving Type 1), then wrongly compute a finite K_v. The outer feedback H(s)=6s cancels the integrator's benefit, making the system Type 0.