Open-loop versus closed-loop
Motivation
Static example (car cruise control) Dynamic systems – general case Operational amplifier
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Conclusions
Motivation
This subject is mainly about closed loop (feedback) control, but open loop control is sometimes useful.
The considered examples capture the main differences between open and closed loop.
More details will be given in the following lectures.
Open loop and closed loop
Reference (desired output)
Controller
Reference (desired output) +
OPEN LOOP CONTROL
CLOSED LOOP CONTROL
Controller
Open or Closed Loop?
Electric Kettle
Hand drier
Washing machine Dishwasher
Power Steering
Open versus closed loop
Cruise control problem (ignore dynamics):
Model is found to be y = 10 (u – 0.5 w)
u [degrees] is the throttle angle
From Feedback Control of Dynamic Systems, G.F. Franklin et al
Block diagram
We have the following block diagram:
u +- y y=10(u-0.5w)
Comparator: v
Open loop controller
We have the following block diagram:
Comments on open loop
When disturbance is zero, we have perfect tracking r=y or in other words the error is zero.
When w=1, r=65, then y=60 and we get 7.69% error in speed.
When w=2, r=65, then y=55 and we get 15.38% error in speed.
We can say that the scheme is not robust with respect to disturbance.
Closed loop controller
We have the following block diagram: Controller
Comments on closed loop
When disturbance is zero, we do not have perfect tracking! The error is 0.643%
If w=1, r=65, the error is 0.693%. Note that it is 10 times smaller than in the open loop case!
We can say that feedback improves robustness of the system in general.
What if we measure disturbance?
y = 10(u -0.5w ) = 10 [1/10(r+5w) – 0.5w] = r
General case open loop (we measure disturbance)
where is a causal mapping and is its inverse.
Open loop control gives perfect tracking in a perfect scenario (no disturbance, perfect model).
Open loop control requires a lot:
The disturbance needs be measured
The model needs to be inverted exactly
The inverse may not be realisable
Model and inverse need to be “BIBO stable”
An alternative implementation
(no disturbance)
e = r – 10u u+1000u=100r
u=100 r/(1+1000) ≈ r/10
y=10u =10(100/(1+1000)r)≈10(1/10r)=r
Approximate open-loop inversion..
We have here
High-gain closed-loop control!
This scheme can be viewed as an alternative to approximate open loop control
High-gain feedback controllers respond aggressively to small errors. May destabilise the system.
But feedback can help make the system robust to disturbances and uncertainty
Example: Operational amplifier in feedback configuration
High gain amplifiers are crucial in long- distance telecoms
Major issue: very uncertain gain
Feedback mitigates this uncertainty
Too high a gain may destabilise the system.
But too low a gain can lead to poor
robustness against disturbances
Nyquist and Bode in Bell Labs formalised these trade-offs from 1920-40, sparking the “Golden Age of Invention” in the US, and the birth of feedback control theory.
Opamp model
comparator Two port model
Simplified analysis
Open loop control gives perfect tracking when there are no disturbances and model uncertainties. However, it is NOT robust to disturbances and modelling errors.
Closed loop (feedback) may give imperfect tracking when there are no disturbances and model uncertainties. However, the scheme is robust to disturbances and modelling errors.
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