Transfer functions, poles and zeros
Poles describe the system's natural modes; zeros reshape how inputs excite them.
Find a closed-loop pole from a simple plant and proportional gain, then separate pole location from final gain.
The first-order transfer function had one pole that set its decay rate. More complex transfer functions can have several poles and zeros.
Review the previous lesson: Why use the Laplace transform?First, picture it
Think of a bell: its ring-down is a natural mode. A pole tells you how quickly that mode dies and whether it oscillates. A zero can cancel or redirect part of the input-to-output response, but it does not necessarily remove an internal physical mode.
Build the idea step by step
A transfer function is the transformed output divided by the transformed input, with zero initial state. Write it as a fraction of polynomials in s. A value that makes the numerator zero is called a zero. A value that makes the denominator zero is called a pole, after common factors are handled. Poles tell us about exponential growth, decay, and oscillation in the response. When we close a feedback loop, we must form the closed-loop fraction first: its poles generally differ from the plant's poles.
Read the equations
For unity negative feedback, the denominator of this closed-loop relation determines visible closed-loop poles after checking any common factors.
What the symbols mean
- plant transfer function
- proportional gain
- reference-to-output closed-loop transfer function
- complex frequency, in inverse seconds
Here the sensor returns the output without scaling it, and we subtract it from the target. Canceling common factors can hide an internal motion, so a simplified input-output fraction alone is not a complete internal-stability check.
A pole's real part sets growth or decay; its imaginary part sets oscillation frequency. The boundary itself needs separate analysis.
Gain changes speed and final value
Take a normalized first-order plant and proportional gain 2.
Substitute into the feedback formula and simplify.
The pole moves to minus 3 inverse seconds, so the response decays faster. The final output for a unit step is still only two-thirds because proportional control leaves error.
More detail and model assumptions
Build the loop before reading its poles
For unity negative feedback, e=r−y, u=Ce and y=Pu. Substituting y=PC(r−y) gives (1+PC)y=PCr and T=y/r=PC/(1+PC). The factor 1+PC, not P alone, sets the closed-loop characteristic equation. This derivation also shows exactly where the minus sign entered.
A transfer function maps a specified input to a specified output under zero initial conditions. The motor's voltage-to-speed and load-torque-to-speed transfers are different, even though they belong to one motor. First name both ports; only then interpret a zero or a steady-state gain.
What poles and zeros do not tell you
In a minimal linear model, poles are natural modes and zeros shape the chosen input–output response. A pole near the imaginary axis often decays slowly; a complex pair can oscillate. A zero can make one measurement appear to hide a mode, so a canceled unstable factor must be checked in the underlying state model.
A linear transfer function also omits hard voltage limits and many parameter changes. A beautiful pole plot cannot guarantee that a motor asked for 20 V will behave as modeled when only 12 V is available.
Connect the example to the lab
For L=K/(s+1), the closed-loop pole is −(1+K). Increasing positive K moves this one pole left, but more complex plants need not behave so simply.
Make a prediction before moving a control.
- PREDICTWhich denominator matters after closing a loop?
- CHANGE ONE THINGSet P(s)=1/(s+1), C(s)=K, and derive T(s) by substitution.
- OBSERVECompare the open-loop pole −1 with the closed-loop pole −(1+K).
- EXPLAINState why this simple gain trend cannot be generalized to every plant or to saturation.
A pole-zero cancellation in an input-output formula can hide an unstable internal mode; do not treat cancellation as proof of physical safety.
Check your understanding+
Which equation determines the closed-loop poles of unity negative feedback?
The characteristic equation 1+L(s)=0.Why the next lesson follows
Real diagrams often have several blocks and a sensor path. We need signal algebra before studying their time responses.