Control Systems

Control System

A system that manages, commands, directs, or regulates the behavior of other systems or devices to achieve a desired output.

Open-Loop System

A control system in which the control action is independent of the output; it does not use feedback.

Closed-Loop System (Feedback System)

A control system in which the control action is dependent on the output, using feedback to compare the output with the desired input and generate an error signal.

Transfer Function

The ratio of the Laplace transform of the output to the Laplace transform of the input, assuming all initial conditions are zero, representing the system's input-output relationship.

Pole

A value of 's' (complex frequency) for which the transfer function of a system becomes infinite. Poles determine the system's stability and transient response.

Zero

A value of 's' (complex frequency) for which the transfer function of a system becomes zero. Zeros influence the shape of the transient response and frequency response.

Stability

The property of a system to return to its equilibrium state after a disturbance, or for its output to remain bounded for a bounded input.

Steady-State Error

The difference between the desired output and the actual output of a system as time approaches ∞.

Root Locus

A graphical plot showing the locations of the closed-loop poles of a system in the s-plane as a system parameter (typically gain K) is varied from zero to ∞.

Bode Plot

A pair of plots (magnitude in dB vs. log frequency, and phase in degrees vs. log frequency) that graphically represent the frequency response of a system.

Nyquist Plot

A polar plot of the open-loop transfer function G(jω)H(jω) as the frequency ω varies from 0 to ∞, used for stability analysis.

Compensator

A device or network inserted into a control system to modify its open-loop transfer function and improve its performance characteristics, such as stability, transient response, or steady-state error.

State-Space Representation

A mathematical model of a physical system as a set of first-order differential equations relating input, output, and state variables.

Transfer Function

Ratio of Laplace transform of output to Laplace transform of input, with zero initial conditions.

H(s) = C(s)/R(s)

Characteristic Equation

Equation whose roots are the closed-loop poles of the system.

1 + G(s)H(s) = 0

Standard Second-Order System

General form for a second-order system, where ωn is natural frequency and ζ is damping ratio.

G(s) = ωn² / (s² + 2ζωn s + ωn²)

Peak Overshoot (Mp)

Maximum percentage overshoot for a step input in an underdamped system.

Mp = exp(-ζπ / √(1-ζ²)) * 100%

Settling Time (ts) (2% criterion)

Time required for the response to settle within 2% of the final value.

ts ≈ 4 / (ζωn)

Rise Time (tr) (0-100% for underdamped)

Time for the response to rise from 0% to 100% of the final value.

tr ≈ (π - β) / ωd, where ωd = ωn * √(1-ζ²) and β = tan^-1(ωd / (ζωn))

Steady-State Error (e_ss) for Unit Step Input

Applicable for Type 0 systems.

e_ss = 1 / (1 + Kp), where Kp = lim(s→0) G(s)H(s)

Steady-State Error (e_ss) for Unit Ramp Input

Applicable for Type 1 systems.

e_ss = 1 / Kv, where Kv = lim(s→0) sG(s)H(s)

Steady-State Error (e_ss) for Unit Parabolic Input

Applicable for Type 2 systems.

e_ss = 1 / Ka, where Ka = lim(s→0) s² G(s)H(s)

Gain Margin (GM)

Amount of gain that can be added to the system before it becomes unstable.

GM = 1 / |G(jωpc)H(jωpc)|, where ∠G(jωpc)H(jωpc) = -180°

Phase Margin (PM)

Amount of phase lag that can be added to the system before it becomes unstable.

PM = 180° + ∠G(jωgc)H(jωgc), where |G(jωgc)H(jωgc)| = 1

Nyquist Stability Criterion

N = number of encirclements of -1+j0 by Nyquist plot, P = number of open-loop RHP poles, Z = number of closed-loop RHP poles. For stability, Z must be 0.

N = P - Z

State Equation

Describes the dynamics of the system's state variables.

ẋ(t) = Ax(t) + Bu(t)

Output Equation

Relates the system's output to its state variables and input.

y(t) = Cx(t) + Du(t)