Technology / Technical guide
Control Transfer Functions: Poles, Zeros & Initial State
A transfer function describes how a stated linear model maps an input into an output. Its algebra can reveal time scales and frequency response, but it does not identify every internal variable. Input–output behaviour, stored initial conditions and the physical implementation are different parts of the explanation.
An input–output relationship needs a model boundary.
Start with the represented variables and assumptions, then distinguish a polynomial factor from a complete description of the machinery.
01 / Model boundary
A transform ratio belongs to a declared linear model.
For a linear time-invariant model with zero initial conditions, G(s) = Y(s)/U(s) relates Laplace transforms.
For a linear time-invariant model with zero initial conditions, G(s) = Y(s)/U(s) relates Laplace transforms. It is not the ratio of two instantaneous meter readings. The Laplace variable s describes complex frequency and is distinct from the time variable.
MathWorks describes polynomial and factored transfer representations. Input and output units still matter: a speed-per-voltage model has a dimensional gain, whereas a ratio between quantities with the same units can be dimensionless. Neither form identifies a real controller by itself.
02 / Initial state
Stored conditions add a separate response.
In a simple model τ dy/dt + y = Ku, transforming the derivative introduces the initial value y(0).
In a simple model τ dy/dt + y = Ku, transforming the derivative introduces the initial value y(0). Setting that value to zero gives the input-driven ratio K/(τs+1). A nonzero initial value contributes an additional term to the output.
The same input can therefore produce different early outputs from different stored conditions, without changing G. Initial rotation, a mathematical integrator state and a logical operating label also describe different quantities; the latter is not automatically the dynamic state in this equation.
03 / Worked model
One first-order example links a pole to a time scale.
Hypothetical model: let τ = 2 seconds and K = 3 for input and output in the same arbitrary unit.
Hypothetical model: let τ = 2 seconds and K = 3 for input and output in the same arbitrary unit. G(s) = 3/(2s+1) has a pole at −0.5 per second. These invented coefficients describe a mathematical system.
For a unit step from zero initial state, y(t) = 3[1−exp(−t/2)] for t at least zero. At two seconds the output is approximately 1.90 units, approaching three over time. No turbine response time or controller gain follows from this illustration.
04 / Poles and zeros
Factor locations describe the reduced relationship.
In a rational representation, zeros are roots of the numerator and poles roots of the denominator after common factors are considered.
In a rational representation, zeros are roots of the numerator and poles roots of the denominator after common factors are considered. Their locations affect the represented response. A zero is not simply a time instant when an output trace happens to cross zero.
For a proper continuous-time rational transfer model, poles strictly in the left half-plane give a stable input–output relation with decaying dynamic modes; a direct-feedthrough term can also be present. Other pole locations require their corresponding interpretation. This criterion concerns that reduced model, not every possible internal realization.
05 / Hidden dynamics
Cancellation does not demonstrate a healthy internal mode.
An algebraic factor can disappear from a reduced input–output ratio.
An algebraic factor can disappear from a reduced input–output ratio. That reduction does not show whether a physical realization retains an internal mode that is unobservable or otherwise hidden at the selected endpoints. Internal stability is a separate system-level question.
The observability guide examines this information boundary. Two models with identical endpoint response can differ in internal coordinates or hidden behaviour; a compact equation should not be mistaken for a photograph of the mechanism.
06 / Scope
Linear response and constrained operation differ.
A local linear model can approximate variation around a defined condition while omitting saturation, changing parameters or nonlinear motion.
A local linear model can approximate variation around a defined condition while omitting saturation, changing parameters or nonlinear motion. Time delays can also require a non-polynomial factor. Those omissions matter when interpreting a frequency-response calculation.
No MARSTON transfer function, pole, zero or response is assigned here. The margin guide develops the feedback question from a declared model, without providing field tuning or operating changes.
Related technical guides: Control Gain & Phase Margins: Conditions and Meaning · Dynamic States & Observability: What Outputs Can Reveal · Feedback Delay: Phase Lag and Stability Boundaries.
General background: MathWorks: Polynomial and factored linear transfer representations. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| G(s) | Transform-domain input–output ratio | Linear time-invariant model and zero initial conditions. |
| Initial-state response | Contribution from stored conditions | Separate from the response generated by u. |
| Pole | Reduced denominator root | Its domain and internal-mode boundary matter. |
| Zero | Reduced numerator root | Not an ordinary trace zero crossing. |
| K and τ | Gain and time-scale parameters | Units and the illustrative model are explicit. |
Engineering questions.
Is G the ratio of instantaneous readings?
No. It relates transforms under a defined linear model.
Can initial state change the early response?
Yes, while the input–output transfer function remains unchanged.
Does a pole always identify a physical part?
No. It belongs to a mathematical representation.
Does cancellation prove internal stability?
No. Hidden realized modes require their own interpretation.
Is the two-second example a product response?
No. Its coefficients and units are invented.
Does this page prescribe controller settings?
No. It explains representation and its assumptions.
Technical enquiry
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A transfer function describes how a stated linear model maps an input into an output. Share the model and applicable operating or design information for an engineering discussion.