Frequency domain control design Flashcards
What is a Bode plot?
A Bode plot visualises the range of different sinusoidal response for different frequencies. It contains two subplots: mag (log) vs freq (log) and phase (linear) vs freq (log)
What is Bode’s gain phase relation?
Bode discovered that if a system has no poles or zeroes in the RHP, then the gain curve completely determines the phase curve. That is, the phase is approximately proportional to the derivative of the gain on a Bode plot - each +/- 20dB/dec slope corresponds to a +/- 90 deg phase shift
What is a Nyquist plot?
A single plot with the phase on the complex plane for all omega, -inf to inf
Why would changes in gain be common?
If the effect of an actuator on the system is uncertain
Why would changes in phase be common?
Due to filtering of signals, time delay in computation etc.
How can you find the gain margin from a Nyquist plot?
How much can the Nyquist plot be scaled up or down before it crosses -1
How can you find the phase margin from a Nyquist plot?
How much would the Nyquist plot have to be rotated in order to cross -1
What are the three frequency domain control objectives?
- Increase crossover freq and hence CL bandwidth. (also leads to faster settling times)
- Increase low freq gain. Leads to reduced SSE to low freq disturbances.
- Increase the phase margin. Larger PM = less overshoot/oscillation and better robustness
What is the relation between zeta and phase margin?
zeta = PM/100
What are the two steps in PID control design?
- Design a PD controller to set the desired transient response, by boosting the overall gain and increasing the phase margin
- Design a PI controller w high freq gain of 1 to boost low freq gain, but with a zero small enough not to effect the PM achieved through PD control.
What is integrator windup?
A very common form of non-linearity is saturation or clipping. The main issue is that the system takes longer to reach the steady state, so error is integrated for longer than it should, and so the control signal builds up and the system overshoots.
Describe the effect of the P component in a PID controller.
It can be considered as a virtual spring. A large P gain usually results in reduced rise time and SSE, but it could also make the system overshoot and oscillate
Describe the effect of the I component in a PID controller.
In ensures zero SSE. A large I gain will also make the system react faster to some constant error or disturbance, but can result in overshoot
Describe the effect of the D component in a PID controller.
It should behave like a virtual damper. Increasing the D gain will reduce overshoot/oscillation. Has no effect on SSE
What is the sensitivity function?
S = 1/1+PC
How is the sensitivity function related to disturbance response?
S < 1: control system attenuates disturbances at that frequency
S = 0: disturbances at that frequency are eliminated
S > 1: feedback system actually makes things worse than they would have been in open loop
What is the complementary sensitivity function?
T = PC/1+PC
How is the complementary sensitivity a limitation?
Looking at the effect of all external signals (d and n) on the tracking error: e = Sr - Sd - Tn. If we want e to be zero, then we want both T and S to be zero, but T+S=1. Therefore cannot eliminate error from both disturbance and noise
How would you shape S and T?
S small at low freq (to attenuate reference signals and disturbances)
T small at high freq (to attenuate high freq noise)
Why is Bode’s gain phase relation a limitation?
A rapid large loop gain roll off will induce a large phase lag, which will have a damaging effect on the PM, leading to oscillation and poor robustness
What is the desired loop shape of L = PC?
Large gain at low freq
Low gain at high freq
Moderate slope at crossover freq
What is the desired loop L?
L = alpha/s (alpha = crossover freq)
What is the limitation surrounding interpolation constraints?
For internal stability:
- S = 1 for all s = RHP zeroes
- S = 0 for all s = RHP poles
Basically we cannot cancel RHP poles and zeroes via feedback, we are stuck with them and their effects
What is Bode’ sensitivity integral and why is it a limitation?
For a stable open loop system:
int(ln(|S(iw)|)dw = 0
This equality shows that if sensitivity to disturbance is suppressed at some frequency range, it is necessarily increased at some other range. (waterbed effect).
