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Key formulas
| Name | Formula | Variables / Notes |
|---|---|---|
| Closed-loop transfer function | T(s) = G(s) / (1 + G(s)H(s)) | T(s): closed-loop transfer function; G(s): forward path gain; H(s): feedback path gain; denominator 1+GH is the characteristic equation |
| Steady-state error (unity feedback) | e_ss = lim as s->0 of s E(s) = R / (1 + Kp) for step | e_ss: steady-state error; Kp: position error constant = lim G(s) as s->0; R: step magnitude; for ramp use velocity constant Kv |
| Second-order system natural frequency | omega_n = sqrt(K / (J + m)) | omega_n: undamped natural frequency (rad/s); K: stiffness or gain; J: inertia; m: mass; generalized from standard 2nd order denominator s^2 + 2 zeta omega_n s + omega_n^2 |
| Damping ratio | zeta = sigma / omega_n = c / (2 sqrt(Km)) | zeta: damping ratio; sigma: real part of poles; omega_n: natural frequency; c: damping coefficient; K: spring constant; m: mass |
| Phase margin | PM = 180 deg + angle(G(jw_gc)) | PM: phase margin (degrees); w_gc: gain crossover frequency where |G(jw)| = 1 (0 dB); positive PM means stable; typical design target is PM = 45 to 60 deg |
| Gain margin | GM = -20 log10 |G(jw_pc)| dB | GM: gain margin; w_pc: phase crossover frequency where angle(G(jw)) = -180 deg; GM > 0 dB means stable for minimum-phase systems |
Key concepts
Routh-Hurwitz criterion
A polynomial is Hurwitz stable (all roots in left half-plane) if and only if all elements of the first column of the Routh array are positive. The number of sign changes in the first column equals the number of right half-plane poles.
Root locus rules
Branches start at open-loop poles (K=0) and end at open-loop zeros (K=infinity). Number of branches equals number of poles. Branches on the real axis exist where an odd number of real poles and zeros lie to the right. Asymptote angles: (2k+1)*180/P-Z for k=0,1,...,P-Z-1.
Type number and steady-state error
Type 0 system: finite error to step, infinite error to ramp. Type 1 system: zero error to step, finite error to ramp, infinite error to parabola. Type 2: zero error to step and ramp, finite error to parabola. Each integrator in the open loop increases type number by 1.
PID controller action
Proportional: reduces steady-state error but cannot eliminate it for type 0 systems. Integral: adds a pole at origin, increases type number, eliminates steady-state error for one input class lower. Derivative: improves transient response and phase margin but amplifies noise.
Gain and phase crossover
Gain crossover frequency w_gc is where |G(jw)H(jw)| = 1. Phase crossover frequency w_pc is where angle equals -180 deg. Gain margin and phase margin are the distances from these points to the instability boundary.
Final value theorem
The steady-state value of y(t) as t approaches infinity equals lim s->0 of s Y(s), provided all poles of s Y(s) are in the left half-plane. Directly apply in Laplace domain to find final output without inverse transforming.
Tables
Second-Order System Response vs Damping Ratio
| Damping (zeta) | Response type | Overshoot | Settling |
|---|---|---|---|
| zeta = 0 | Undamped | Sustained oscillation | Never |
| 0 < zeta < 1 | Underdamped | Yes (decaying) | Fast if zeta near 0.7 |
| zeta = 1 | Critically damped | None | Fastest no-overshoot |
| zeta > 1 | Overdamped | None | Slow (two real poles) |
Error Constants and Steady-State Errors (Unity Feedback)
| Input | Error constant | Type 0 e_ss | Type 1 e_ss | Type 2 e_ss |
|---|---|---|---|---|
| Step R(s)=1/s | Kp = lim G(s) s->0 | 1/(1+Kp) | 0 | 0 |
| Ramp R(s)=1/s^2 | Kv = lim sG(s) s->0 | infinity | 1/Kv | 0 |
| Parabola R(s)=1/s^3 | Ka = lim s^2 G(s) s->0 | infinity | infinity | 1/Ka |
PID Tuning Effects
| Action | Rise time | Overshoot | Settling time | SS error |
|---|---|---|---|---|
| Increase Kp | Decrease | Increase | Small change | Decrease |
| Increase Ki | Decrease | Increase | Increase | Eliminate |
| Increase Kd | Minor change | Decrease | Decrease | No effect |
Quick facts
- For a second-order underdamped system, percent overshoot = e^(-pi*zeta / sqrt(1-zeta^2)) * 100.
- Peak time t_p = pi / omega_d where omega_d = omega_n sqrt(1 - zeta^2) is the damped natural frequency.
- Settling time (2% criterion) is approximately 4 / (zeta omega_n) for underdamped systems.
- A gain margin greater than 6 dB and a phase margin of 45 to 60 degrees are common design targets.
- The number of asymptotes on a root locus equals the number of poles minus the number of zeros.
- For a type 1 system with open-loop gain K, the velocity error constant Kv = K.
- Adding a zero to the left half-plane improves transient response and moves root locus toward it.
- The characteristic equation is 1 + G(s)H(s) = 0. Its roots are the closed-loop poles.
Exam shortcuts
- Routh array shortcut for 3rd order: for s^3 + a s^2 + b s + c = 0, the conditions for stability are a > 0, c > 0, and ab > c. No need to build the full array for third-order polynomials, just apply this test directly.
- Phase margin from Bode plot: read the phase at the gain crossover frequency (where the gain magnitude curve crosses 0 dB). Add 180 degrees to that phase value. If result is positive, the system is stable.
- Root locus breakaway point: differentiate the characteristic equation or set d/ds[N(s)/D(s)] = 0. For simple cases with two poles, the breakaway is at the midpoint on the real axis between them.
- Steady-state error shortcut: count the number of integrators in the open-loop G(s)H(s) to find the system type. Then match the input type: if both match (Type 1 and ramp), compute the error constant directly as lim s G(s) as s->0.
- Dominant pole approximation: if one pair of complex poles is much closer to the jw axis than all others (by at least a factor of 5), treat the system as second-order using only those poles. Then apply standard overshoot and settling time formulas directly.