Lecture 11
Sensitivity transfer functions
Motivation
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Examples
Sensitivity transfer functions Steady state errors
Conclusions
Motivation
Any control system typically has a number of different inputs (reference inputs, different disturbances)
We want output to be sensitive to the reference, but also insensitive to disturbances entering the loop.
Effects of references and disturbances on signals in the loop captured by a gang of four sensitivity transfer functions.
Example (room temperature control)
Reference temperature Actual temperature
Outside temperature
Control algorithm
Example (room temperature)
Temperature controller
Reference signal = desired temperature
Input disturbance = heat dissipation (outside
temperature, opening doors & windows) Measurement noise
Example (room temperature)
Reference is typically a piecewise constant signal.
Input disturbance is typically
a slowly varying large signal (low frequency, large amplitude).
Measurement noise is typically
a fast varying small signal
(high frequency, small amplitude).
Example (cruise control)
Cruise control
Reference signal = desired velocity Input disturbance = slope of the road Measurement noise
Taken from Astrom and Murray, Feedback systems.
Example (cruise control)
Reference is typically a constant signal.
Input disturbance is typically
a slowly varying large signal (low frequency, large amplitude).
Measurement noise is typically
a fast varying small signal
(high frequency, small amplitude).
Examples suggest that different control systems share similar structure and inputs possess similar properties.
We now consider a general closed loop and analyse the effect of various inputs on the output.
Sensitivity transfer functions
Turn off one input and compute the transfer function in the usual manner for the other input.
This is how we will compute “sensitivity functions”.
Typical block diagram
Coprime polynomials
Transfer function with coprime polynomials
Transfer function with polynomials that are NOT coprime
Possible performance specifications:
It would be good if the output follows the reference (despite disturbances); i.e. the error should be very sensitive to the reference input.
It would be good if disturbance not to affect the error for any value of reference; i.e. the error should be insensitive to the disturbance input.
Transfer functions:
From the block diagram we have
Next we will show that there exist “sensitivity” transfer functions so that we can write:
Transfer functions with Y as output
We substitute (2) into (1) – i.e. eliminate U:
Transfer functions with U as output
We substitute (1) into (2) – i.e. eliminate Y. Please do this as an exercise! We obtain:
𝑈𝑈𝑠𝑠 = 𝐶𝐶(𝑠𝑠) 𝑅𝑅𝑠𝑠 −𝐷𝐷𝑚𝑚 𝑠𝑠 −𝐷𝐷𝑜𝑜 𝑠𝑠 − 𝐶𝐶(𝑠𝑠)𝐺𝐺0(𝑠𝑠) 𝐷𝐷𝑖𝑖(𝑠𝑠) 1+𝐶𝐶 𝑠𝑠 𝐺𝐺0(𝑠𝑠) 1+𝐶𝐶 𝑠𝑠 𝐺𝐺0(𝑠𝑠)
NB: Another approach to getting Y(s) or U(s)
Suppress all inputs except one
Redraw the loop in unity or general feedback form. Use the corresponding
transfer function to get Y(s) or U(s) in terms of that input.
Use linear superposition to sum up the contributions from all inputs.
“Sensitivity” transfer functions:
For nominal plant and controller we can define:
Algebraic relationships:
Sensitivities are algebraically related:
Special case: two inputs
Two inputs:
Important constraint:
C(s) is the only degree of freedom for “shaping” the two transfer functions.
If we shape one transfer function (e.g. for good tracking), this fixes the other (e.g. for disturbance rejection).
This may lead to a trade-off between:
– Good tracking;
– Good disturbance rejection.
Bode diagram
Two degree of freedom controllers
Input-output relationships
Disturbance-input relationships
We are also interested in how disturbances affect the inputs so it is sometimes of interest to consider:
Input and output relationships
We can rewrite the original relationships as
We can obtain sensitivities in the same manner as before using the block diagram. Please do this as an exercise.
Conclusions
We have introduced one and two degree of freedom controllers and have introduced different sensitivity functions.
Sensitivities are algebraically related! This typically leads to trade-offs in design.
We will study in more detail these sensitivities in the next lecture.
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