Application of Mathematics in Real-World Usage · Guide 30 · BTT Mathematics Hub
A thermostat compares a desired temperature with a measured temperature. A cruise controller compares target speed with actual speed. A robot compares where it should be with where its sensors say it is. The recurring mathematical structure is feedback: measure, compare, act, observe the new state, and repeat.
This guide uses simplified linear systems and fictional controllers for Mathematics teaching. It is not control-system design advice for aircraft, vehicles, medical devices, industrial plants or other safety-critical systems. Real engineering requires validated models, hardware limits, uncertainty analysis, testing and domain-specific safety review.
Feedback begins with an error signal
Let reference r be the desired output and y the measured output. A common negative-feedback error is e=r−y.
If target speed is 20 units and measured speed is 17, error is +3. If measured speed is 22, error is −2. The sign tells the controller which side of the target the measured output lies on under the chosen convention.
A proportional controller multiplies error by gain
A simple proportional law is u=Kp e. With Kp=4 and e=3, control command u=12 model units.
Doubling Kp doubles the immediate command for the same error, but larger gain does not automatically mean a better real system. Saturation, noise amplification, overshoot and stability limits can matter.
Closed-loop gain differs from open-loop gain
For a simple scalar negative-feedback loop with forward gain G and feedback gain H, closed-loop gain is G/(1+GH) when the algebraic assumptions hold.
With G=4 and H=1, the closed-loop gain is 4/5=0.8. A unit reference produces output 0.8 in this static teaching model, leaving steady error 0.2.
High loop gain can reduce steady error in a simple model
If G is replaced by 10 with H=1, the closed-loop gain is 10/11≈0.9091. The unit-reference error falls from 0.2 to about 0.0909.
This benefit is conditional on stability and the validity of the model. Infinite gain is not a free improvement because dynamics, delay and physical limits eventually matter.
Sensor calibration is an inverse-function problem
Suppose a fictional sensor reports m=2.5x+1, where x is the physical quantity. Recovering x requires x=(m−1)/2.5.
If m=16, estimated x=6. A control loop using the raw value 16 as though it were x would compute the wrong error even if the controller equation itself were correct.
Offset error and scale error are different
A sensor m=x+2 has a constant offset of two units. A sensor m=1.05x has a 5% scale factor. At x=100 both report 102 and105 respectively, but their errors evolve differently as x changes.
Calibration therefore needs a model of how measurement error changes with the true quantity, not merely one comparison point.
First-order systems approach a target exponentially
A standard unit-step first-order response can be written y(t)=1−e^(−t/τ), where τ is the time constant.
At t=τ, y≈0.6321. At t=3τ, y≈0.9502. The response moves quickly at first and then approaches the final value asymptotically.
Time constant is a scale, not a finishing time
At one time constant the response is only about 63.2% complete. At four time constants it is about 98.17% complete.
Because the exponential tail never reaches zero error exactly in finite time, practical systems use tolerance bands such as “within 2%” rather than exact equality.
Discrete feedback can be studied through recurrences
Consider xₖ₊₁=0.8xₖ+1. Its equilibrium x* satisfies x*=0.8x*+1, so x*=5.
Subtracting equilibrium gives xₖ₊₁−5=0.8(xₖ−5). The deviation shrinks by factor0.8 each step because |0.8|<1.
A recurrence can be unstable when deviations grow
If the deviation instead obeys dₖ₊₁=1.1dₖ, each step multiplies the deviation by1.1. A small disturbance grows rather than decays.
For a one-dimensional linear discrete model dₖ₊₁=adₖ, asymptotic stability requires |a|<1.
Second-order poles connect damping to oscillation
A standard second-order characteristic polynomial is s²+2ζωₙs+ωₙ². With damping ratio ζ=0.5 and natural frequency ωₙ=4 rad/s, the poles are −2±j3.464.
The negative real part indicates decaying oscillation in this linear continuous-time model. Moving poles across the imaginary axis changes the stability conclusion.
Percent overshoot is nonlinear in damping ratio
For an ideal underdamped second-order step response, percent overshoot can be written 100e^[−πζ/√(1−ζ²)].
At ζ=0.5, the result is about16.3%. Doubling ζ does not simply halve overshoot because the relationship is exponential and includes a square root.
Settling-time formulas are approximations with stated criteria
A common 2% settling-time approximation for an underdamped second-order model is Ts≈4/(ζωₙ). With ζ=0.5 and ωₙ=4, Ts≈2 seconds.
The numerical answer depends on the settling-band convention and the validity of the second-order approximation.
Delay creates phase lag
A pure time delay T contributes phase −ωT radians at angular frequency ω. For T=0.1 s and ω=5 rad/s, phase contribution is −0.5 rad≈−28.65°.
As frequency rises, the same time delay creates greater phase lag. Delay can therefore reduce stability margin even though its static gain is one.
Integral action accumulates error
A simplified integral term is Ki∫e(t)dt. A persistent small error can therefore build a growing correction until the modelled steady error is reduced.
But integral action can also accumulate while an actuator is saturated, producing windup. Real implementations often include anti-windup logic.
Derivative action reacts to change in error
A derivative term Kd de/dt responds to the rate of change of error. If error is changing rapidly, derivative action can produce a large response even when the current error is modest.
Differentiation also amplifies high-frequency measurement noise, so practical systems often filter derivative action.
Saturation turns a linear model into a nonlinear system
If a controller computes u=150 but the actuator is limited to ±100, actual command is100. The relation between requested and delivered command is no longer linear beyond the limit.
Stability and transient predictions based purely on the unsaturated linear model may therefore fail during large excursions.
Feedback can reject disturbances without measuring them directly
If an external disturbance pushes the output away from the reference, negative feedback changes the error and therefore the control action.
The loop can compensate for some unmodelled effects because it measures the consequence rather than needing to predict every disturbance in advance.
A complete control analysis states the model boundary
State what is measured, what is controlled, the sign convention for error, controller law, actuator limits, sampling period, delay, disturbance assumptions and the stability criterion being used.
The mathematical return path is reference → measurement → error → controller → plant → new measurement. Closing that loop is what makes feedback a system rather than a one-off calculation.
Practice: twenty control-system Mathematics questions
- Reference20, output17. Find error r−y.
- Reference20, output22. Find error.
- Kp=4 and e=3. Find u.
- For G=4,H=1, find closed-loop gain G/(1+GH).
- For unit reference in question4, find static error.
- For G=10,H=1, find closed-loop gain.
- Sensor m=2.5x+1 reports16. Find x.
- For y=1−e^(−t/τ), find y at t=τ.
- Find y at t=3τ.
- For xₖ₊₁=0.8xₖ+1, find equilibrium.
- Is dₖ₊₁=0.8dₖ asymptotically stable?
- Is dₖ₊₁=1.1dₖ asymptotically stable?
- For ζ=.5,ωₙ=4, find real part of the second-order poles.
- Find imaginary magnitude of those poles.
- Find ideal percent overshoot for ζ=.5.
- Use Ts≈4/(ζωₙ) to estimate 2% settling time.
- For delay0.1s at5rad/s, find phase lag in radians.
- Convert question17 to degrees.
- A computed command150 is limited to100. What command is delivered?
- Why can a controller with correct static algebra still be unstable?
Worked answers
- +3.
- −2.
- 12.
- 0.8.
- 0.2.
- 10/11≈0.9091.
- 6.
- About0.6321.
- About0.9502.
- 5.
- Yes. |0.8|<1.
- No. |1.1|>1.
- −2.
- About3.464.
- About16.3%.
- About2 s.
- −0.5 rad.
- About−28.65°.
- 100.
- Because dynamics, delay and feedback can amplify disturbances or produce growing oscillations even when the static equation looks sensible.
Sources and connected applications
For feedback, state-space, stability, Bode and digital-control foundations, see MIT OpenCourseWare: Feedback Control Systems and MIT Signals and Systems: Feedback and Control. All controller values and plant examples here are fictional teaching models.
Continue with Reliability Engineering, Failure Rates, Redundancy and Availability; Signal Processing, Sampling, Filtering, Noise and Fourier Analysis; and Urban Planning, Land Use, Density, Accessibility and Networks. Return to the BTT Mathematics Hub.