Q1Control System

Background — steady-state error and system type. For a unity-feedback system with open-loop transfer function G(s)H(s), the error transform is E(s) = R(s)/[1+G(s)H(s)], and by the final value theorem the steady-state error is ess = lim(s→0) sE(s) = lim(s→0) sR(s)/[1+G(s)H(s)]. The 'type' of a system is defined as the number of poles of G(s)H(s) located exactly at the origin s = 0, since this determines how many free integrators exist in the forward path. Each integrator can 'absorb' one additional order of a polynomial input (step, then ramp, then parabola) with zero resulting steady-state error; beyond that order, the error saturates to a finite, non-zero constant, and beyond one further order it grows unbounded. The static error coefficients Kp, Kv, Ka formalize this: they are computed as the limiting value of G(s)H(s), sG(s)H(s), s²G(s)H(s) respectively as s→0, and the corresponding steady-state errors for step, ramp and parabolic inputs of unit magnitude are ess = 1/(1+Kp), 1/Kv, and 1/Ka respectively.

Steady-state error by system type. The general relationship between system type and steady-state error for the three standard test inputs (unit step, unit ramp, unit parabola/acceleration) is summarized below, and is essential background for correctly interpreting the specific Type-2 calculation that follows:

• Type-0 system (no pole at origin): Kp finite, Kv = 0, Ka = 0. Step input gives finite error 1/(1+Kp); ramp and parabolic inputs both give infinite (unbounded) error — a Type-0 system cannot track a ramp or acceleration command at all.
• Type-1 system (one pole at origin): Kp = ∞, Kv finite, Ka = 0. Step input gives zero error; ramp input gives finite error 1/Kv; parabolic input gives infinite error — a Type-1 system tracks position perfectly but cannot follow a constantly accelerating target without growing error.
• Type-2 system (two poles at origin, as in this problem): Kp = ∞, Kv = ∞, Ka finite. Both step and ramp inputs give zero steady-state error; only the parabolic (acceleration) component produces a finite, non-zero error 1/Ka.
• Type-3 and higher: all three coefficients become infinite and even parabolic inputs are tracked with zero steady-state error, but such high-type systems become progressively harder to stabilize since each additional integrator adds a further -90° of open-loop phase lag at all frequencies.

Part (a): Static error coefficients. Given G(s)H(s) = k/[s²(s+20)(s+30)], this is a Type-2 system (two poles at the origin, from the s² factor).

Position error constant:

Velocity error constant:

Acceleration error constant:

So Kp = ∞ (zero steady-state error for step input), Kv = ∞ (zero steady-state error for ramp input), and Ka = k/600 is finite (non-zero, bounded steady-state error for parabolic/acceleration input), which is the expected pattern for a Type-2 system.

Part (b): Finding K for ess = 10 units. The input is r(t) = 1 + 10t + 20t², a combination of step (magnitude 1), ramp (magnitude 10) and parabolic (coefficient 20, meaning the acceleration input magnitude is 2×20 = 40 since parabolic term is conventionally written as (A/2)t² with A being the acceleration magnitude). For a Type-2 system, the step and ramp components contribute zero steady-state error (since Kp, Kv = ∞); only the parabolic component contributes non-zero steady-state error, given by:

Setting ess = 10:

Since Ka = k/600:

Hence the required gain is K = 2400, which limits the steady-state error due to the parabolic component of the given composite input to the specified 10 units, while the step and ramp components inherently produce zero steady-state error in this Type-2 system regardless of the value of K.

Physical interpretation. This result highlights the characteristic trade-off in Type-2 servo systems: because the system already contains two free integrators, it perfectly tracks both constant-position and constant-velocity commands (step and ramp) with zero residual error irrespective of the loop gain K, which is a major advantage over Type-0 and Type-1 systems for tracking applications such as radar antenna pointing or machine-tool position control. However, this benefit does not extend to acceleration (parabolic) commands — the acceleration error constant Ka remains finite and directly proportional to K, so a designer wishing to tighten the acceleration-tracking accuracy of such a servo can only do so by increasing the gain K, which in turn (as seen in other parts of this paper) tends to reduce the phase margin and can push the system toward instability. This is exactly why practical Type-2 designs (e.g., in guided-missile or radar-tracking servos) are always accompanied by careful compensator design — the required Ka for acceptable acceleration-tracking error must be balanced against the stability margin the resulting K allows, often necessitating lead or lag-lead compensation rather than a bare increase in proportional gain.

Worked verification. As a sanity check on the final gain value, substitute k = 2400 back into the acceleration error constant formula: Ka = k/600 = 2400/600 = 4, and then ess (parabolic component only) = A/Ka = 40/4 = 10 units, exactly matching the specified allowable steady-state error and confirming the algebra. It is also useful to note that if the problem's parabolic input coefficient were instead interpreted as the full acceleration magnitude A = 20 directly (rather than requiring the factor-of-2 conversion from the (A/2)t² convention), the required gain would instead work out to k = 1200; the factor-of-2 convention used above (writing r(t) = 1 + 10t + 20t² as containing a (40/2)t² acceleration term) follows the standard textbook definition of the parabolic test signal r(t) = (A/2)t²U(t), and is the interpretation consistently applied throughout RTU's control systems curriculum and reference texts (Nagrath & Gopal, Ogata).

Extending the analysis. If instead the design specification demanded zero steady-state error for the parabolic component as well, the only recourse would be to raise the system to Type-3 by adding a third free integrator to G(s)H(s); however, as noted above, each additional integrator contributes a further, constant -90° of open-loop phase lag at every frequency, making the loop progressively harder to stabilize with adequate phase margin. This is why real acceleration-tracking servo systems (radar trackers, missile guidance loops) are rarely built as bare Type-3 systems; instead, designers accept a small, bounded steady-state error at Type-2 (as computed here) and rely on feedforward or command-shaping techniques to further reduce tracking error for known, predictable acceleration profiles, rather than pushing the feedback loop itself to a higher, less stable type.