Interview questions & answers
Q1. What is a transfer function and what are its limitations?
A transfer function is the ratio of the Laplace transform of the output to the Laplace transform of the input, with all initial conditions set to zero. For example, a simple RC low-pass filter with R=1kΩ and C=1µF has the transfer function H(s) = 1/(1 + 0.001s), directly giving the time constant of 1 ms. The key limitation is that transfer functions only apply to linear, time-invariant systems — a saturating op-amp or a nonlinear actuator cannot be accurately described this way.
Follow-up: How do you handle a system that is not linear and time-invariant?
Q2. What do the poles of a transfer function tell you about the system?
Poles are the values of s where the transfer function becomes infinite, and they directly determine the stability and transient behavior of the system. A pole at s = -2 means the natural response decays as e^(-2t), giving a time constant of 0.5 seconds. A pole in the right-half s-plane means the natural response grows unboundedly, so any real system with an RHP pole will be unstable without feedback.
Follow-up: If a pole and zero cancel each other, is the system still stable?
Q3. What is the difference between poles and zeros?
Poles make the transfer function infinite and govern the system's natural response, while zeros make the transfer function zero and shape how each input frequency is attenuated or amplified. In a notch filter designed to reject 50 Hz mains hum, a pair of zeros is placed at s = ±j314 rad/s, creating a sharp null at exactly that frequency. Zeros do not affect stability but can cause non-minimum phase behavior and undershoot if they sit in the right-half plane.
Follow-up: What is a non-minimum phase system and where does it appear in practice?
Q4. How do you find the transfer function of a system from its differential equation?
You take the Laplace transform of the differential equation with zero initial conditions and then rearrange to form the ratio Y(s)/U(s). For a second-order mechanical system described by mẍ + bẋ + kx = F, the transfer function becomes X(s)/F(s) = 1/(ms² + bs + k), which is identical in form to a series RLC circuit. This algebraic form lets you immediately read off the natural frequency as √(k/m) and the damping ratio as b/(2√(mk)).
Follow-up: What is the significance of the characteristic equation of this transfer function?
Q5. What is the order of a system and why does it matter?
The order of a system equals the highest power of s in the denominator polynomial of its transfer function, and it tells you how many energy-storing elements the system contains. A DC motor with armature inductance and rotor inertia is a second-order system because it has two energy stores — the inductor and the rotating mass. Higher-order systems have more complex transient responses and are harder to stabilize, which is why control engineers often reduce high-order plant models to second-order approximations for controller design.
Follow-up: How does system order affect the complexity of the controller you need?
Q6. What is a first-order system and what is its step response?
A first-order system has a transfer function of the form K/(τs + 1), where τ is the time constant defining how quickly the output responds. A thermocouple in a temperature measurement circuit behaves as a first-order system with τ around 2–5 seconds, meaning it reaches 63.2% of its final reading after one time constant. The step response never overshoots in a first-order system — this is a fundamental property that distinguishes it from second-order and higher systems.
Follow-up: How do you experimentally determine the time constant of a first-order system?
Q7. What is a second-order system and what are its key parameters?
A second-order system has the standard form ωn²/(s² + 2ζωns + ωn²), where ωn is the natural frequency and ζ is the damping ratio that together completely define the transient response shape. A servo motor position loop might have ωn = 100 rad/s and ζ = 0.7, giving a well-damped response with about 4.6% overshoot. These two parameters let you predict peak overshoot, settling time, and resonant frequency without solving the differential equation.
Follow-up: What happens to the step response when the damping ratio is exactly 1?
Q8. How does gain affect the transfer function of a closed-loop system?
Increasing gain in a closed-loop system moves the closed-loop poles, changing the transient response and potentially pushing the system toward instability. In a proportional-only temperature controller for a furnace, doubling the gain reduces steady-state error but can cause oscillation or instability if the gain exceeds the ultimate gain. The root locus method shows exactly how the closed-loop poles migrate as gain is varied from zero to infinity.
Follow-up: What is the maximum gain you can apply before the system becomes unstable?
Q9. What is the difference between open-loop and closed-loop transfer functions?
The open-loop transfer function G(s)H(s) describes the plant and feedback path combined without closing the loop, while the closed-loop transfer function C(s)/R(s) = G(s)/(1 + G(s)H(s)) describes the actual system response. An industrial conveyor speed controller has an open-loop gain that can be measured by breaking the feedback wire, but the closed-loop performance — including disturbance rejection — only appears when the loop is closed. Stability analysis using Bode or Nyquist plots is done on the open-loop function, while time-domain specs are read from the closed-loop.
Follow-up: Why do we analyze stability using the open-loop transfer function instead of the closed-loop?
Q10. What is steady-state error and how is it related to the system type?
Steady-state error is the difference between the desired output and the actual output after all transients have died out, and it is determined by how many pure integrators the open-loop transfer function contains. A Type-0 system like a proportional position controller has a finite steady-state error to a step input, while a Type-1 system with one integrator — such as a motor velocity controller with an integrator in the compensator — achieves zero steady-state error for step inputs. Adding more integrators reduces error to higher-order inputs but makes stability harder to maintain.
Follow-up: How do you calculate the steady-state error to a ramp input for a Type-1 system?
Q11. What is the significance of the DC gain of a transfer function?
DC gain is the value of the transfer function at s = 0, and it represents the ratio of steady-state output to a constant (DC) input. For the transfer function 10/(s + 2), the DC gain is 10/2 = 5, meaning a constant input of 1V will produce 5V at the output once all transients settle. In power supply design, the DC gain of the control loop determines how accurately the output voltage tracks the reference under steady load conditions.
Follow-up: How does the DC gain change if you add an integrator to the system?
Q12. How do you simplify a block diagram to find the overall transfer function?
You reduce the block diagram using three rules: series blocks multiply, parallel blocks add, and a feedback loop with forward gain G and feedback gain H gives G/(1 + GH). In a missile guidance system, the seeker, autopilot, and airframe dynamics are three series blocks that multiply to give the open-loop plant, before applying the feedback reduction. Always simplify innermost loops first when dealing with nested feedback structures.
Follow-up: What happens to the overall transfer function if the feedback gain H is greater than 1?
Q13. What is a minimum phase system?
A minimum phase system has all its poles and zeros in the left-half s-plane, and its phase response is the minimum possible for a given magnitude response — the Bode magnitude plot alone completely determines the phase. An audio amplifier with only RC networks in the signal path is minimum phase, meaning you can recover the exact phase curve from the magnitude curve using the Hilbert transform. Non-minimum phase systems, like a process with a transport delay or a right-half-plane zero, have more phase lag than the minimum, making them harder to control.
Follow-up: Give a practical example where a non-minimum phase response causes a problem in control.
Q14. How does time delay appear in a transfer function and why is it problematic?
A pure time delay of T seconds is represented by the factor e^(-sT) in the transfer function, which is not a rational polynomial and adds unbounded phase lag at high frequencies. In a networked control system where sensor data travels over a 100 ms communication link, the delay e^(-0.1s) reduces the phase margin and can destabilize an otherwise well-tuned loop. Unlike a pole or zero, a delay cannot be exactly cancelled by a finite-order compensator, so approximations like Pade are used for analysis.
Follow-up: What is the Pade approximation and when would you use it?
Q15. What is the relationship between transfer function poles and system time constants?
Each real pole at s = -a contributes a time constant of τ = 1/a seconds to the system response, and the dominant poles — those closest to the imaginary axis — determine the slowest, most visible part of the transient. A control system with poles at s = -1 and s = -20 will appear nearly first-order because the component with τ = 0.05 s dies out 20 times faster than the one with τ = 1 s. In practice, poles more than five times further from the imaginary axis than the dominant poles are often ignored in simplified design.
Follow-up: When is it valid to ignore non-dominant poles in your design?
Common misconceptions
Misconception: Cancelling a pole with a zero always makes the system behave as if those dynamics don't exist.
Correct: Pole-zero cancellation removes the mode from the input-output transfer function but the cancelled pole still appears in the state equations, meaning an unstable cancelled pole still grows internally and will eventually cause failure.
Misconception: A higher-order system is always more accurate than a first-order approximation.
Correct: A higher-order model is more complete, but for controller design purposes an accurate second-order dominant-pole approximation is often more useful because it keeps the math tractable and the behavior predictable.
Misconception: If the transfer function has all negative real poles, the system is definitely stable in closed loop.
Correct: Open-loop stability does not guarantee closed-loop stability; adding feedback with sufficient gain can push closed-loop poles into the right-half plane even if every open-loop pole is stable.
Misconception: The transfer function fully describes any physical system.
Correct: Transfer functions only apply to linear, time-invariant, single-input single-output systems and lose information about internal states, making them unsuitable for nonlinear, time-varying, or MIMO systems without additional tools.