Why Integral Feedback Can Remove Steady-State Error
Negative feedback can reduce the effect of a sustained change in input, but a reduced response is not necessarily a response that returns exactly to its original level. Integral feedback adds a controller state that accumulates the difference between the desired output and the observed output. If the controller can reach an equilibrium, that accumulated error can stop changing only when the output equals its target.
This property is called exact adaptation: following a sustained input change, the output returns to a specified set point at steady state. It does not mean that the output never deviates during the transient, nor does the equilibrium condition alone guarantee that the system converges to the equilibrium.
The central condition follows directly from the controller equation. At an equilibrium, the controller state is no longer changing, so \(\dot z=0\). With \(k_i>0\), the equation \(\dot z=k_i(r-y)\) then requires \(y=r\). The integrator does not achieve this by knowing the disturbance in advance; it changes its control state in response to the error until the error is zero.
Equilibrium Conditions and Their Limits
Consider a first-order pathway output \(y\) controlled by \(z\), with a sustained disturbance \(d\):
Here \(\tau>0\) is the output time constant, \(g>0\) is the effect of controller state on output, and \(b\) is the effect of the disturbance. The controller state \(z\) is assumed to be unconstrained, and the disturbance \(d\) is constant when assessing the post-change equilibrium. At an equilibrium, both derivatives are zero. The controller equation gives \(y^*=r\); substituting that into the pathway equation gives the required controller state:
Thus a feasible equilibrium exists in this model for any sustained \(d\) if the controller can attain the required \(z^*\). This is an equilibrium result, not a claim about how quickly the output returns or whether a more complicated feedback system will converge. The output can have a transient error after the input changes. Stability and convergence must be checked separately for the particular closed-loop dynamics.
Worked Example: The Required Controller Shift After an Input Change
Suppose a hypothetical pathway follows \(\tau\dot y=-y+gz+bd\), with target \(r=5\) nM, \(g=2\) nM per control unit, and \(b=0.5\) nM per input unit. The sustained input changes from \(d=1\) to \(d=3\). Assume the controller state is not bounded.
Before the change, the controller state needed for an equilibrium at the target is \(z^*=(r-bd)/g=(5-(0.5)(1))/2=2.25\) control units. After the change, the required state is \(z^*=(5-(0.5)(3))/2=1.75\) control units.
Substitution into the steady-state pathway equation confirms the output for the new input: \(y^*=(2)(1.75)+(0.5)(3)=3.5+1.5=5\) nM. The input change therefore requires a controller-state shift of \(-0.50\) control units to make a target-level equilibrium possible. If the system converges to that equilibrium, its steady-state output is 5 nM; the equilibrium calculation by itself does not specify the transient.
The result depends on important assumptions. The integral gain must be nonzero, the controller must integrate the actual error without leakage, and the disturbance must be sustained long enough for a post-change steady state to be meaningful. The pathway and controller must also have a feasible equilibrium at \(y=r\). If the controller has bounds or the pathway cannot produce the target for the new input, the integrator condition cannot make the impossible equilibrium possible.
Why Leakage Leaves a Residual Error
A controller may only approximate an integrator. For example, the controller state may decay toward zero as well as respond to output error. This is called leaky integration. To calculate its steady-state effect, use a deviation model in which the target output is zero:
Here \(B\) is the sustained disturbance’s direct effect on the output, measured in nM, and \(\lambda>0\) is the leak rate in inverse time units. The controller is integrating the error relative to the zero deviation target. At equilibrium, the controller equation gives \(z^*=-k_i y^*/\lambda\). The output equation gives \(y^*=gz^*+B\). Combining these relations yields:
For positive \(B\), \(g\), \(k_i\), and \(\lambda\), the residual output deviation is positive. Increasing \(k_i\) reduces the residual in this model, but any positive leak leaves nonzero error when \(B\ne 0\). In the ideal-integrator limit \(\lambda=0\), the equilibrium controller condition instead requires \(y^*=0\), if that equilibrium is feasible.
Worked Example: Calculating Residual Error With a Leaky Integrator
Suppose a hypothetical pathway has a sustained disturbance effect \(B=2\) nM, \(g=0.5\) nM per control unit, \(k_i=3.8\) control units per nM per minute, and leak rate \(\lambda=0.2\) per minute. Assume the deviation model above applies and the system reaches its equilibrium.
The residual is \(y^*=\lambda B/(\lambda+gk_i)\). Substituting the values gives \(y^*=(0.2)(2)/(0.2+(0.5)(3.8))=0.4/(0.2+1.9)=0.4/2.1=0.190476\ldots\) nM.
The equilibrium output is therefore about 0.1905 nM above the target. It is not exactly at the set point: the controller state must balance both the error-driven term and its leak, so a nonzero error is needed to maintain the required controller state. This is a steady-state calculation, not a prediction of the time course.
Comparison With Proportional Feedback
A proportional controller changes its command in proportion to the current error but does not accumulate past error. For a simple steady-state deviation model, let the controller command be \(u=-k_p y\), where \(k_p>0\), and let the output satisfy \(y=gu+B\). Then:
For finite \(gk_p\) and nonzero \(B\), the proportional controller reduces but does not eliminate the steady-state deviation in this model. Integral feedback can remove that error under its feasibility and convergence conditions; leaky integration generally leaves a residual. This comparison concerns steady-state error, not which controller gives a preferable transient or is more robust in a particular pathway.
Worked Example: Proportional Feedback Leaves an Offset
For a hypothetical pathway with disturbance effect \(B=2\) nM, gain \(g=0.5\) nM per control unit, and proportional gain \(k_p=2\) control units per nM, the steady-state deviation is \(y=B/(1+gk_p)=2/(1+(0.5)(2))=2/2=1\) nM.
The proportional controller reduces the disturbance’s direct 2 nM effect to a 1 nM steady-state deviation, but does not return the output to the target. An ideal integral controller could make the steady-state deviation zero in a feasible, convergent system. The contrast is not that proportional control never produces zero error under any circumstance; it is that it generally leaves an offset for a sustained disturbance in this model.
Feasibility, Diagnostics, and Common Mistakes
Exact adaptation requires more than writing an integral term in a model. Check whether the steady-state controller value required to produce \(y=r\) is attainable. In the linear model, that value is \((r-bd)/g\). If it lies outside the actuator’s permitted range, the pathway cannot meet the target under the new input. If it is attainable, the equilibrium condition still does not establish convergence: the observed output and controller state must be examined over time, and the dynamics must be suitable for reaching the equilibrium.
A practical diagnostic is to inspect both output error and controller-state change. Under ideal integration, \(\dot z=k_i(r-y)\): a persistent nonzero error implies a continuing change in \(z\), not a stationary controller state. If the output plateaus away from the target, possible explanations include leakage, actuator saturation, a changing rather than sustained input, or failure to reach a stable equilibrium. Simulations should be run long enough to distinguish a true plateau from a slow transient, and the controller state should be checked for continued drift or contact with a bound.
Worked Example: An Unreachable Target Under an Actuator Bound
Consider the pathway \(\tau\dot y=-y+gz+bd\) with \(r=5\) nM, \(g=2\) nM per control unit, \(b=0.5\) nM per input unit, and a sustained input \(d=12\). Suppose the actuator is restricted to \(z\ge 0\).
The controller value required for a target-level equilibrium is \(z^*=(r-bd)/g=(5-(0.5)(12))/2=(5-6)/2=-0.5\) control units. This value violates the actuator bound. At the smallest permitted value, \(z=0\), the steady-state output from the pathway equation is \(y=0+(0.5)(12)=6\) nM, already 1 nM above the target. Increasing \(z\) can only increase the output because \(g>0\).
Therefore no admissible controller value produces \(y=5\) nM at this input. The integral-feedback condition does not overcome this physical limit. If the controller continues integrating while the actuator is held at its bound, its internal state may keep changing even though the applied command cannot; this is a form of integrator windup. The equilibrium calculation diagnoses infeasibility rather than a successful return to the set point.
A common mistake is to say that an integral controller guarantees exact adaptation under every condition. The incorrect reasoning is: “The controller integrates error, so the error must eventually become zero.” Integration only implies that a stationary controller state requires zero error in the ideal, unconstrained controller equation. It does not guarantee that the required state is attainable or that the closed-loop system converges. Another mistake is to confuse exact adaptation with an error-free transient: after a sustained input change, the output can depart from its target before any adapted equilibrium is reached.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- For \(\dot z=k_i(r-y)\) with \(k_i>0\), what output value is required at an equilibrium of the controller state?
- In the linear pathway model, use \(r=4\) nM, \(g=1\) nM per control unit, \(b=0.5\) nM per input unit, and \(d=2\). What controller state is required for a target-level equilibrium?
- For the leaky model, let \(B=1\) nM, \(\lambda=0.1\) per minute, and \(gk_i=0.9\) per minute. Calculate the residual deviation.
- Why does an actuator lower bound of \(z\ge0\) prevent exact adaptation if the required equilibrium value is \(z^*=-0.5\)?
- Does the equilibrium condition \(y=r\) prove that the output returns to the set point after a disturbance? Explain.
Solutions. (1) At equilibrium \(\dot z=0\), so \(k_i(r-y)=0\); because \(k_i>0\), \(y=r\). (2) The required state is \(z^*=(r-bd)/g=(4-(0.5)(2))/1=3\) control units. (3) \(y^*=\lambda B/(\lambda+gk_i)=(0.1)(1)/(0.1+0.9)=0.100\) nM. (4) The required value is outside the permitted range, so no admissible controller state can satisfy the target-level equilibrium condition. (5) No. The condition identifies the output at a feasible equilibrium; convergence to that equilibrium must be established separately.