Control systems – Match concepts with typical associations List I A. Lag compensation B. Lead compensation C. Stability (continuous-time) D. Instability (discrete-time) List II Exterior of the unit circle |z| > 1 PD control Non-minimum phase Left-half s-plane PI control Choose the correct mapping.

Introduction / Context:Lead/lag compensators are classical tools for shaping transient response and steady-state error. Stability criteria differ for continuous-time (s-plane) and discrete-time (z-plane) systems; recognizing the canonical regions helps avoid design missteps.

Given Data / Assumptions:

• Lag compensation primarily improves steady-state accuracy (integral action) → PI-like behavior.
• Lead compensation primarily improves phase margin and speed (derivative action) → PD-like behavior.
• Continuous-time stability requires poles in the left-half s-plane.
• Discrete-time instability corresponds to poles outside the unit circle in the z-plane.

Concept / Approach:

Map each concept to its most common association: lag ↔ PI; lead ↔ PD; stability ↔ left-half s-plane; discrete-time instability ↔ |z| > 1. Although lag/lead can be implemented in several forms, these pairings capture their standard effects taught in introductory courses.

Step-by-Step Solution:

Verification / Alternative check:

Root-locus and Bode design recipes show lag adding low-frequency gain (integral) and lead adding positive phase (derivative), while stability regions match the standard s- and z-plane results.

Why Other Options Are Wrong:

Swapping PI/PD reverses the compensator roles. Associating stability with the exterior of the unit circle applies to discrete-time instability, not continuous stability.

Common Pitfalls:

Confusing steady-state error reduction (lag/PI) with transient acceleration (lead/PD), and mixing s-plane with z-plane criteria.

Final Answer:

A-5, B-2, C-4, D-1.