Problem 1
It increases corrective action when error grows, changing the effective time constant and steady-state error, subject to actuator limits.
Faculty use
Feedback, Stability and Control Interpretation
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
Delays, saturation, unmodelled dynamics, and noise can cause oscillation or instability.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind using measurement and actuation to shape dynamic response. Use the formulas u=−Kx or proportional error feedback; closed-loop dynamics change eigenvalues/poles.
Problem 4
A strong error analysis identifies the first step that violates classification, algebra, theorem hypotheses, units, initial/boundary data, or numerical assumptions; then repairs that step and re-verifies the final result.
Problem 5
Application responses vary. Require defined variables, a compatible differential-equation model connected to experimental data, engineering design, control, reliability, environmental systems, biomedical modelling, and research communication, a complete computation, interpretation, and limitations.
Problem 6
Accept two independent checks appropriate to the task: differentiation/substitution, initial or boundary conditions, dimensional analysis, phase-line or stability evidence, residuals, step refinement, conservation/energy checks, data comparison, or a second method.
Scoring guidance
Six problems × 10 points: 5 mathematical accuracy, 2 structure/justification, 2 verification/interpretation, 1 notation and communication.