Questions: Reference Tracking and Servo System Design

Question 1

A type-1 servo system (one integrator in the open loop) is commanded to track a ramp reference (constant-velocity input). What is the steady-state tracking behavior?

• A. Zero steady-state error — one integrator is sufficient to track any reference signal asymptotically
• B. A finite, nonzero constant tracking error that persists indefinitely as the ramp continues
• C. Unstable behavior — a type-1 system cannot handle ramp inputs without going unstable
• D. Zero steady-state error only if the loop gain is sufficiently high

Answer: (B)

System type determines which reference classes can be tracked with zero steady-state error. A type-1 system (one open-loop integrator) achieves zero steady-state error to a step reference and finite constant error to a ramp reference. To track a ramp with zero steady-state error, you need at least type-2 (two integrators). The finite ramp error is called the velocity error constant, and it depends on the loop gain — higher gain reduces it but cannot eliminate it without adding another integrator. Option A confuses type-1 sufficiency for steps with sufficiency for ramps.

Question 2

A servo designer increases loop gain to improve steady-state accuracy. What is the most likely consequence for transient tracking performance?

• A. Transient performance improves proportionally — higher gain makes the system respond faster with less overshoot
• B. Transient performance is unchanged because steady-state error and transient response are independent
• C. Phase margin decreases, leading to increased overshoot, longer settling time, and potential instability
• D. Rise time increases because the higher gain slows the initial response to reference changes

Answer: (C)

This is the fundamental tension in servo design. Increasing loop gain improves steady-state accuracy by raising the error constants, but it also reduces phase margin — the phase buffer before instability. Lower phase margin means more oscillatory step responses, greater overshoot, longer settling times, and at the extreme, instability. The gain that zeroes steady-state error for a given reference class may produce intolerable oscillations in the transient response. Good servo design uses integral action to raise system type (for steady-state accuracy) and lead compensation or filters to preserve phase margin (for transient performance).

Question 3

A servo system that achieves zero steady-state error to a step reference input will automatically also achieve zero steady-state error to a ramp reference input.

• T. True
• F. False

Answer: False

Steady-state tracking error depends on system type, not just gain. Zero steady-state error to a step requires at least type-1 (one open-loop integrator). Zero steady-state error to a ramp requires at least type-2 (two open-loop integrators). A type-1 system tracks steps perfectly but produces a finite, persistent error to ramp inputs. Knowing the error to one class of reference tells you nothing about error to a higher-order class — you must match system type to the most demanding signal your application requires.

Question 4

The system type — the number of open-loop integrators — determines whether steady-state tracking error to a given class of reference signal can be driven to zero.

• T. True
• F. False

Answer: True

System type is the key structural property for steady-state performance. Type-0: constant error to step, unbounded error to ramp. Type-1: zero error to step, finite constant error to ramp, unbounded error to parabola. Type-2: zero error to both step and ramp, finite error to parabolic input. This hierarchy follows from the internal model principle: to track a reference with zero steady-state error, the controller must contain a model of the reference signal's dynamics — hence integrators for steps and ramps. Loop gain affects the magnitude of finite errors but cannot change the fundamental type classification.

Question 5

Explain the fundamental tension between steady-state accuracy and transient tracking performance in servo design, and how a well-designed controller addresses both simultaneously.

Model answer: Steady-state accuracy requires sufficient system type (integrators) and loop gain. But adding integrators reduces phase margin, and increasing gain further reduces it — both tend to make the closed-loop response oscillatory, increasing overshoot and settling time (poor transient performance). The tension is that the tools for eliminating steady-state error degrade transient behavior. A well-designed servo controller separates these concerns: integral action raises the system type to meet steady-state requirements, while lead compensation or bandwidth-limiting filters are added to restore phase margin and shape the transient response to meet rise time, overshoot, and settling time specifications.

The practical approach starts by specifying both requirement classes independently before choosing a controller structure. Steady-state specs determine minimum system type and low-frequency gain. Transient specs determine required bandwidth and phase margin. A controller (often a PI or PID with lead compensation) must be designed to satisfy both simultaneously. When they genuinely conflict, the engineer faces a deliberate design tradeoff and must document which requirement was relaxed and why — rather than discovering the conflict after the fact during testing.