4: Control objectives (stability/accuracy/speed)

Control systems are engineered to achieve specific performance goals, with stability, accuracy, and speed forming the fundamental triad of objectives. These criteria dictate a system's ability to perform reliably and effectively in dynamic environments. Understanding each objective is essential for analyzing and designing control systems across engineering disciplines.

Stability

Stability is the paramount objective, acting as a prerequisite for any functional control system. It ensures the system's output converges to a bounded value following a disturbance or reference change, rather than oscillating uncontrollably or diverging. Mathematically, stability is determined by the location of the system's poles in the complex plane (s-plane): all poles must lie in the left-half plane (have negative real parts) for asymptotic stability. In practical terms, an unstable system is hazardous and unusable — imagine an aircraft pitch control causing uncontrollable oscillations. Stability is assessed through methods like:

• Routh-Hurwitz
• Root locus
• Nyquist plots

and is often ensured via feedback mechanisms that actively adjust system dynamics.

Accuracy

Accuracy quantifies how closely the system's steady-state output matches the desired reference value. It is measured by steady-state error, the persistent deviation after transients settle. Key accuracy metrics include:

• Position error constant ($K_p$) for step inputs
• Velocity error constant ($K_v$) for ramps
• Acceleration error constant ($K_a$) for parabolic inputs

Higher constants typically reduce steady-state error. For instance, a robot arm must position itself precisely (low step error) or track a moving conveyor (low ramp error). Integral control is frequently employed to enhance accuracy by eliminating steady-state error through increasing system type, though this can impact stability margins.

Speed

Speed describes how rapidly the system responds to changes and reaches its desired state. It is characterized by transient response metrics like:

• Rise time (time to reach 90% of final value)
• Settling time (time to stay within a tolerance band, e.g., 2%)
• Peak time

A fast system reacts quickly to commands or disturbances — critical for applications like anti-lock brakes. However, excessive speed often conflicts with stability and accuracy; aggressive responses may cause overshoot or oscillations. Designers balance speed with other objectives using parameters like damping ratio ($\zeta$) and natural frequency ($\omega_n$). Higher $\omega_n$ generally speeds up response, while $\zeta$ controls oscillation:

• Underdamped: $\zeta < 1$
• Overdamped: $\zeta > 1$