What Feedback Buys You, and the One Thing It Cannot

Feedback can reduce tracking error, attenuate disturbances, and reduce sensitivity to plant uncertainty. These benefits come from a common denominator in the loop equations. The same equations also show why good tracking and sensor-noise rejection conflict at the same frequency.

The order of analysis matters: derive the interconnection, establish stability, and then calculate its performance. Large loop gain alone is not a stability argument, and a small error reported by the controller need not mean a small physical tracking error.

Core question and definition

Which signals does feedback suppress, which signals does it transmit, and what assumptions make the steady-state claims valid?

We use a scalar, continuous-time LTI negative-feedback loop with zero initial conditions for the transfer calculations.

Label the signals before reducing the loop

Let the physical output be \(y\) and the reference be \(r\). The disturbance \(w\) is added at the plant input, after the controller. Sensor noise \(v\) is added to the output measurement, before the feedback subtraction.

Define the signals as follows:

With controller \(D(s)\) and plant \(G(s)\), the Laplace-domain equations are

The true error is separately defined by

Thus

The controller acts on \(E_m\). It does not directly know \(E\).

Deriving the output response to every input

Substitute the controller equation into the plant equation:

Now substitute the measured error:

Move the term containing \(Y\) to the left:

Factor:

Therefore,

A transfer path is obtained by setting the other independent inputs to zero:

The disturbance path contains an extra factor of \(G\) because the disturbance enters before the plant. The negative sign in the noise path comes from subtracting the noisy measurement.

These answers depend on the injection locations. For example, if a different disturbance \(W_o\) were added directly to the physical output,

then

so its path would be

An output disturbance and a plant-input disturbance do not have the same transfer function.

Sensitivity, complementary sensitivity, and their identity

Define the loop transfer function

Then define

These are the sensitivity and complementary sensitivity functions. The output equation becomes

Adding the two functions gives

Thus

This is structural: it follows from the specified loop equations, without choosing a particular controller. It is an identity of transfer functions wherever their evaluations are finite, not a performance target achieved by tuning.

The original numerical checks found agreement to machine precision at several complex operating points. Those checks validate the calculation; the algebra explains why the identity holds.

The identity also holds algebraically for unstable loop designs. It does not establish stability.

Deriving true error and measured error separately

Start with the true error:

Substitute the output equation:

Using \(1-\mathcal{T}=\mathcal{S}\),

Now subtract the sensor noise to obtain the measured error:

Therefore,

The reference and disturbance terms match. The noise terms do not:

For completeness, the controller output is

because \(DG\mathcal{S}=L/(1+L)=\mathcal{T}\).

The measured output satisfies

All paths can now be collected without ambiguity:

Key concepts

Disturbance rejection: what is divided by sensitivity?

For comparison, remove the feedback and let the controller act directly on the reference:

With the same plant-input disturbance,

The open-loop disturbance path is \(G\). The closed-loop path is \(G\mathcal{S}\). Their ratio is

where the ratio is defined.

Thus small \(\lvert\mathcal{S}\rvert\) means attenuation relative to the open-loop disturbance response. The absolute closed-loop path is still \(G\mathcal{S}\), not just \(\mathcal{S}\).

For large loop magnitude,

and hence

This distinction matters: increasing plant gain and increasing controller gain need not have the same effect on absolute disturbance transmission, even when both increase \(\lvert L\rvert\).

The original cruise-control example, holding 60 km/h despite a hill or wind, corresponds to suppressing the disturbance path in the frequency range occupied by those disturbances.

Why the sensitivity function really is relative sensitivity

A transfer function can have units. Comparing an absolute change in a quantity measured on a scale of \(10^6\) with one measured on a scale of \(1\) is not a useful measure of robustness by itself.

For a transfer function \(F\) depending on the plant \(G\), define the dimensionless relative sensitivity

Here the controller and the evaluation point \(s\) are fixed. The normalized expression requires nonzero nominal \(G\) and \(F\).

Open-loop structure

The reference transfer function is

Differentiating with \(D\) fixed,

Therefore,

Thus

Closed-loop structure

The reference transfer function is

Apply the quotient rule:

Now form the relative sensitivity:

Hence

The name is literal: the sensitivity function is the relative differential sensitivity of the closed-loop reference transfer function to the plant.

Preserved sensitivity checks

The original numerical differentiation results are reproduced unchanged:

 G=2.0 D=1.0 : open-loop sens=1.000000   closed-loop sens=0.333333
 G=2.0 D=50.0: open-loop sens=1.000000   closed-loop sens=0.009901
 G=2.0 D=0.02: open-loop sens=1.000000   closed-loop sens=0.961538

The exact closed-loop values follow directly from the formula:

These are the exact fractions behind the recorded rounded decimals.

Infinitesimal sensitivity versus a finite plant change

The differential result describes an infinitesimal perturbation. To calculate a finite change, write

where \(\delta\) is the relative plant change, and keep \(D\) fixed. Then

and

Subtract the nominal transfer function:

Divide by the nominal \(\mathcal{T}=L/(1+L)\):

Factoring \(1+L\) out of the denominator gives

Only when the additional denominator is close to one can this be approximated by

For open loop, in contrast,

so

exactly, including for finite changes. A 10% plant change gives a 10% open-loop transfer change under this fixed-controller comparison.

The original statement that loop gain \(100\) turns a \(10\%\) plant error into approximately \(0.1\%\) is a rounded, first-order estimate. Its linearized fractional change is

As a percentage, this is

For a finite positive 10% perturbation, the exact fractional change is instead

or exactly

No new decimal approximation is needed to see the distinction.

The denominator also identifies a limitation:

For dynamic uncertainty, the perturbed loop’s stability still needs to be checked. A nominal small differential sensitivity is not a blanket guarantee for arbitrary finite perturbations.

The structural trade-off is between complex functions

From the derived paths,

Good reference tracking asks for small \(\lvert\mathcal{S}\rvert\). Sensor-noise rejection asks for small \(\lvert\mathcal{T}\rvert\).

But

In particular, if \(\lvert\mathcal{S}\rvert\le\epsilon\) at a frequency, then

and the reverse triangle inequality gives

The two functions cannot both be arbitrarily small at that frequency. More generally,

However,

is not generally equal to one. The identity is a complex sum, not a conserved budget of two nonnegative magnitudes.

For example, at a point where \(L=-1+\epsilon\) with nonzero real \(\epsilon\),

Both magnitudes can be large while their complex sum remains one. This is why a loop near the critical value \(-1\) can amplify errors and noise rather than offering a useful trade-off.

Why a small measured error can hide a large true error

Set \(R=W=0\) and consider the sensor-noise response:

In a frequency range with large loop gain,

Consequently,

The controller drives the measured output toward its reference by moving the physical output against the sensor error. A small measured error can therefore coexist with a physical tracking error approximately equal to the sensor error.

This is especially clear for a constant sensor bias. Integral action can remove the controller’s measured error while preserving the bias in the true output. The final-value calculation below makes that statement exact.

Separating requirements across frequency

When reference changes and important disturbances occupy lower frequencies than sensor noise, the usual design aim is

and

The corresponding approximations follow from the definitions:

For plant-input disturbances, also inspect \(G\mathcal{S}\). For actuator effort, inspect \(D\mathcal{S}\).

Frequency separation does not defeat the identity. It places different objectives in different frequency ranges. Overlap, transition-band amplification, and stability still constrain the design.

Feedback changes poles through the characteristic equation

Write

Then

The closed-loop characteristic polynomial is therefore

for the corresponding well-defined interconnection, retaining internal factors when assessing internal stability.

The plant and controller coefficients both enter this polynomial. Feedback gives a way to change pole locations; it does not guarantee that they move into the LHP. The stable gain windows in the Routh–Hurwitz note are concrete examples of the resulting inequalities.

Steady-state error

Final value theorem: what must be checked first

For reference tracking alone, set

Then true and measured errors coincide:

The final value theorem gives

provided the rational error transform has the required pole structure: all poles of the reduced expression \(sE(s)\) lie strictly in the open LHP.

Why this condition? It means \(E\) has at most a simple pole at the origin, with all its other poles strictly in the LHP. Its partial fractions have the form

Inverting,

Every term in the sum decays, leaving \(c\). Multiplying the transform by \(s\) and taking its limit at zero also gives \(c\).

This is why a simple pole of \(E\) at the origin is allowed, while a pole of \(sE\) at the origin is not. A repeated origin pole in \(E\) produces polynomial growth rather than a finite final value.

Closed-loop stability must be established, but it does not alone guarantee a finite error for every reference. A ramp or parabola can leave poles at the origin in the error transform even when all closed-loop natural modes are stable.

Why polynomial references are written as \(t^k/k!\)

For integer \(k\ge0\), consider

Integration by parts gives

starting from

Repeated substitution yields

Therefore,

The factorial normalization removes an unnecessary coefficient:

Deriving the error constants

With reference input only,

Step input

For \(R=1/s\),

Define the position error constant

When the final value theorem applies,

If the loop gain tends to infinity at DC and the closed loop is stable, this limit is zero.

Here \(K_p\) is an error constant, not necessarily the proportional controller gain.

Ramp input

For \(R=1/s^2\),

Define

If \(K_v\) is finite and nonzero and the pole condition holds,

A diverging \(sL(s)\) can instead give zero error. If \(K_v=0\), the finite-final-value condition must be examined; writing “\(1/0\)” is not a valid use of the theorem.

Parabolic input

For \(R=1/s^3\),

Define

Under the corresponding finite-value conditions,

The factors of \(s\) and \(s^2\) in these definitions are not arbitrary: they come directly from multiplying the reference transform by \(s\) in the final value calculation.

Deriving the system-type triangle

The system type is the number of uncancelled poles at the origin in the loop transfer function. Write its low-frequency form as

where \(A\) is analytic near the origin.

For the reference \(t^k/k!\),

First take \(n\ge1\). Then

and the denominator tends to \(A_0\).

More integrators than reference order: \(n>k\)

The numerator tends to zero:

With stable closed-loop dynamics and the final-value pole condition,

Equal integrator and reference orders: \(n=k\ge1\)

The numerator is one:

Therefore,

This is the diagonal entry. For type 1, \(A_0=K_v\); for type 2, \(A_0=K_a\).

Too few integrators: \(n<k\)

Near the origin,

Using the polynomial-reference transform just derived,

Lower-order polynomial terms and decaying transients can also be present, but the displayed term determines the growth.

This is not an application of the finite final value theorem: \(sE\) retains a pole at the origin. The error is unbounded in magnitude.

Why the type-0 step entry is different

For type 0,

so the sensitivity is

There is no diverging integrator term that makes the one negligible. For a step,

and therefore

The asymmetry is exactly the difference between

and

For higher-order references in a type-0 loop,

so

Assuming stable closed-loop dynamics and the positive low-frequency constants used in the examples, the familiar table follows:

Here \(\infty\) denotes unbounded error magnitude. If a leading coefficient has a different sign, the direction of growth follows the derived polynomial term.

One additional integrator moves the first nonzero tracking-error case up by one polynomial order. The claim concerns the loop’s uncancelled origin poles and presumes that the resulting closed loop remains stable.

Worked example: proportional control of a first-order plant

Use the original plant

with proportional controller

Then

and

The closed-loop pole is

There is no loop integrator, so this is type 0:

For a unit step,

The full error response can also be obtained by partial fractions. Define

Then

Matching numerators gives

Therefore,

so

Thus

The steady-state error is the constant term:

For the original parameter values,

A higher proportional gain reduces the step error, but finite gain does not make it zero.

Worked example: adding integral action

Use the same plant with the PI controller

The loop transfer function is

The characteristic polynomial is

Its quadratic Routh first column is

which is strictly positive under the stated assumptions. The closed loop is stable.

The sensitivity function is

There is one uncancelled loop integrator, so the system is type 1. Its velocity error constant is

For a unit step,

so

For a unit ramp,

so

For the original values,

this gives

The original simulation records are reproduced unchanged:

 P control on a/(tau s+1)  [TYPE 0]
   a=1 k1=2: Kp=2  1/(1+Kp)=0.333333  simulated=0.333333
   a=2 k1=2: Kp=4  1/(1+Kp)=0.200000  simulated=0.200000
 PI control on the same plant [TYPE 1]
   a=1 k1=2 k2=2: Kv=2  1/Kv=0.500000  ramp sim=0.500000  step sim=1.4e-14

The recorded step residual \(1.4e-14\) is on the order of \(10^{-14}\). The analytical step error is zero under the model assumptions; the residual belongs to the numerical calculation.

System type describes reference tracking, not every disturbance path

The triangle was derived with \(W=V=0\). It does not automatically classify the other transfer paths.

For constant inputs with transforms

apply the final value theorem to the full true-error expression:

When the relevant pole conditions hold,

The product limit is written explicitly because \(G(0)\) may be unbounded even when \(G\mathcal{S}\) has a finite DC limit.

For the measured error,

Two consequences follow.

Sensor bias survives perfect measured tracking. With a stable integrating loop, \(\mathcal{S}(0)=0\) and \(\mathcal{T}(0)=1\). If \(w_0=0\),

An integrator in the plant need not reject a plant-input disturbance. For example, take

Then

The loop is type 1 and has the stable closed-loop pole \(-k\). It has zero step-reference error, but a constant plant-input disturbance produces

when sensor noise is absent.

The injection point and the location of the integrator matter. Always derive the path for the signal whose rejection is being claimed.

What integral action guarantees structurally

In the PI example, the zero step-reference error survives changes in the finite plant gain because

as long as the loop integrator remains uncancelled and the closed loop remains stable. The exact value of \(ak_2\) changes the finite ramp error, but not the zero step-error result.

The corresponding open-loop claims must be separated:

• Exact tracking of arbitrary reference signals would require \(DG\equiv1\), or an appropriate plant inverse.
• Zero steady-state error to a step only requires \(D(0)G(0)=1\).

The second condition is less demanding than exact inversion. Indeed, for open loop with reference input only,

so a step gives

when the final value exists.

That DC matching can be achieved nominally, but plant drift changes the match. Integral feedback obtains the step-tracking result through the loop’s low-frequency structure instead. This guarantee belongs to the stable linear model with the stated signal locations; it does not imply rejection of sensor bias or preservation under actuator saturation.

Revision checklist

Why it matters for my work

The useful habit is to name the measured quantity and the true quantity separately before claiming that a correction loop is working. Reducing a measurement residual is not the same as reducing every physical error.

A second habit is to distinguish a structural identity, a local sensitivity, and a finite-perturbation guarantee. They answer different questions even when they share the same algebra.

What I have not resolved

I want to formulate the design problem when reference, disturbance, and sensor-noise spectra overlap, including the cost of controller effort. I also want to understand which analogous limitations can be stated for nonlinear feedback without treating nonlinear operators as scalar transfer functions.

Sources: Ajou University lecture notes and the standard classical-control treatment in Franklin, Powell, and Emami-Naeini, Feedback Control of Dynamic Systems. The loop transfer paths, sensitivity identity, relative-sensitivity derivatives, final-value conditions, error constants, and system-type classification are derived above. The original numerical records are retained: checks of the sensitivity identity at complex operating points, numerical differentiation of open- and closed-loop relative sensitivity, and the proportional/PI step- and ramp-error simulations. New calculations are symbolic; the finite-perturbation example is kept in exact fractional form, and no new simulation results are claimed.