Control System Stability: Margins, Criteria, and Performance Metrics
Gain Margin
1) It is defined as the margin in gain allowable by which gain can be increased till the system reaches the weige of instability.
2) It is the amount by which system gain can be increased before it becomes unstable.
3) It is measured at the frequency where the phase shift is 180°.
4) The greatest difference between open-loop gain (AR) at crossover frequency and AR = 1 determines stability — the more stable the system, the higher the gain margin.
5) This concept is known as Gain Margin.
Phase Margin
1) The difference between 180° and phase lag at the frequency for which the open-loop gain (AR) is unity is called the Phase Margin.
2) The phase margin represents the additional amount of phase lag required to make the system unstable.
3) It measures frequency at open-loop gain = 1.
4) Practically, the phase margin should be 30° to 60° for a stable system.
5) Phase margin tells how much gain can be added before the system becomes unstable.
Discuss Bode Stability Criterion
1) Consider an open loop transfer function GoL = Gp Gv Gc Gm
2) It should not have any poles on the right side of the imaginary s-axis, meaning the system should not be already unstable.
3) There can be only one pole at origin.
4) Let the system have only one critical frequency (ωc), the point where phase shift = –180°.
5) And one gain crossover frequency where magnitude becomes 1.
6) It provides necessary and sufficient conditions for stability of closed-loop system.
7) Unlike Routh Criterion, Bode Stability Criterion is applicable to systems that contain time delays.
8) Bode Stability Criterion is applicable to a wide variety of process control problems; however, for any GoL(s) which does not satisfy required conditions, Nyquist Stability Criterion is useful.
Feedback Control System
1) Control measures variable once measured
2) It takes action after change in control variable
3) Adjusts input based on the error
4) Simple to implement
5) Doesn’t require detailed knowledge of disturbance
Feed Forward Control
1) Disturbance variables are measured
2) It takes action before change in control variable
3) Adjusts input based on the disturbance
4) It requires a good process model which makes it more complex
5) It requires accurate knowledge of system & disturbance
Open Loop
1) A system in which the output has no effect on control action
2) No feedback signal
3) Less accurate due to disturbance & variations
Example:
Electric heater without thermostat.
Closed Loop
1) A system in which the output is continuously measured & compared
2) Output signal is fed back
3) More accurate & stable — can correct disturbances
Example:
Electric heater with thermostat.
Performance Criteria for Closed Loop Systems
1) Steady-State Error (Ess):
Difference between desired output and actual output after the system settles.
→ Smaller error means better control system performance.
2) Closed-Loop Stability:
The control system must remain stable for all expected operating conditions.
→ An unstable loop gives diverging output.
3) Good Disturbance Rejection:
The system should minimize the effect of load, keeping the controlled variable close to its set point.
4) Avoid Excessive Control Action:
Controller output (manipulated variable) changes should be moderate.
→ Large or frequent adjustments wear out valves and actuators.
5) Robustness: The control system must perform satisfactorily even when process parameters change
Frequency Domain Criteria: 1)
Maintain |T(jω)| ≈ 1 up to high frequency — ensures fast response.
2) Choose peak magnitude:
1.0 < Mr < 1.5 → limits overshoot.
3) Sensitivity magnitude should satisfy: 1.2 < Ms < 2.0 → acceptable disturbance rejection & robustness.