Control Systems

Define an open-loop control system.

System that operates without feedback; output does not affect control action.

Define a closed-loop control system.

System that uses feedback to compare output with desired input and correct errors.

What is a transfer function?

Ratio of Laplace-transformed output to input assuming zero initial conditions: G(s)=\frac{Y(s)}{U(s)}

What is the overall transfer function of a simple feedback system?

T(s)=\frac{G(s)}{1+G(s)H(s)}

What is the error transfer function of a unity feedback system?

E(s)=\frac{1}{1+G(s)}R(s)

Define open-loop transfer function.

Product of all transfer functions in the forward path of a feedback loop.

What is block diagram reduction?

Simplifying multiple blocks and feedback loops into an equivalent single transfer function.

State the rule for blocks in series.

Multiply their transfer functions:

G{eq}=G1 G_2

State the rule for blocks in parallel.

Add their transfer functions:

G{eq}=G1+G_2

What is feedback in a control system?

Process of comparing output with desired input and adjusting input to reduce error.

Define system type.

The number of pure integrators (poles at the origin) in the open-loop transfer function.

Define steady-state error.

Difference between input and output as t \to \infty.

What is the steady-state error for a step input?

e{ss}=\frac{1}{1+Kp} where Kp=\lim{s\to 0}G(s)H(s)

What is the steady-state error for a ramp input?

e{ss}=\frac{1}{Kv} where Kv=\lim{s\to 0}sG(s)H(s)

What is the steady-state error for a parabolic input?

e{ss}=\frac{1}{Ka} where Ka=\lim{s\to 0}s^2G(s)H(s)

Define proportional control.

Control law: u(t)=K_p e(t)

Define integral control.

Control law: u(t)=K_i\int e(t)\,dt

Define derivative control.

Control law: u(t)=K_d\frac{de(t)}{dt}

What is a PID controller?

Combines proportional, integral, and derivative terms:

u(t)=Kp e(t)+Ki\int e(t)dt+K_d\frac{de(t)}{dt}

Define damping ratio \zeta.

Measures how oscillations decay in a second-order system.

Define natural frequency \omega_n.

Frequency of undamped oscillation in a second-order system.

State the standard second-order transfer function.

T(s)=\frac{\omega_{n}^2}{s^2+2\zeta\omega_{n}s+\omega_{n}^2}

Define percent overshoot (PO).

PO=100\,e^{-\pi\zeta/\sqrt{1-\zeta^2}}\% for a step response.

Define settling time (T_s).

Approximate time for output to stay within 2% of final value:

Ts\approx\frac{4}{\zeta\omega_{n}}

Define peak time (T_p).

Time to reach first maximum:

Tp=\frac{\pi}{\omega_{n}\sqrt{1-\zeta^2}}

Define rise time (T_r).

Time for response to go from 10% to 90% of final value (≈ 1.8/\omega_n for underdamped system).

What is the condition for stability (Routh-Hurwitz)?

All coefficients of the first column of the Routh array must be positive.

Define Bode plot.

A logarithmic plot of magnitude (in dB) and phase versus frequency.

How is gain margin (GM) defined?

The factor by which gain can increase before instability: GM=\frac{1}{|G(j\omega_{180})|}.

How is phase margin (PM) defined?

The additional phase lag at the gain crossover frequency to reach −180°: PM=180°+\angle G(j\omega_{gc}).

What is gain crossover frequency?

Frequency where magnitude |G(j\omega)|=1 (0 dB).

What is phase crossover frequency?

Frequency where phase \angle G(j\omega)=-180°.

Define Nyquist stability criterion.

Number of clockwise encirclements of (−1,0) equals number of open-loop right-half-plane poles for marginal stability.

What is root locus?

Graphical method showing how system poles move in the s-plane as gain varies.

What are the asymptotes in root locus?

Lines that show pole movement direction as gain → ∞, intersect at centroid \sigma=\frac{\sum p - \sum z}{n-m}.

What is dominant pole approximation?

System response is primarily determined by poles closest to the imaginary axis.

Define bandwidth.

Frequency range where system gain ≥ −3 dB of its low-frequency value.

What is system sensitivity?

Change in output due to change in system parameter: S=\frac{\Delta T/T}{\Delta G/G}.

Define transient response.

System behavior before reaching steady state.

Define steady-state response.

Final constant response of a system after transients die out.

What does increasing K_p do in a PID system?

Decreases rise time, increases overshoot, decreases steady-state error.

What does increasing K_i do in a PID system?

Eliminates steady-state error but increases overshoot and settling time.

What does increasing K_d do in a PID system?

Reduces overshoot and improves stability but increases noise sensitivity.

Define type 0 system.

No integrators in open loop; finite steady-state error for step input.

Define type 1 system.

One integrator in open loop; zero steady-state error for step input.

Define type 2 system.

Two integrators in open loop; zero steady-state error for ramp input. #