Revision of Laplace Transforms Algebra of Block Diagrams
Motivation
Complex numbers
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Definition of Laplace Transform Properties of Laplace Transform Partial fraction expansion
Conclusions
Why we love Laplace transforms
Time derivatives, integrals and convolutions become algebraic operations in s-domain Much, much simpler to analyse and design systems – and write them down concisely.
Different functions in time-domain yield different functions in s-domain No loss of information going from one domain to the other
Leads to crucial concepts of transfer functions, poles, zeros etc
Complex numbers
Complex numbers consist of real and imaginary parts:
Complex numbers in complex plane
Operations with complex numbers
Addition of complex numbers:
Multiplication of complex numbers: Exponential of a complex number:
Laplace transform
Motivation (possible design steps)
Time domain: Complex (Laplace) domain.
Step 1: Modelling
Step 2: Design specifications
Step 4: analysis/design
Overshoot, Undershoot, Rise time, Setting time, Steady-state error
Locations of poles
and zeroes of transfer function in the complex plane
Locations of poles in the complex plane
Step 6: Verification, interpretation
Laplace transform
Laplace transform for signal is
Inverse Laplace transform of Y(s) is
Region of convergence is for which where ,
Taken from “Control Systems Design”, Goodwin, Graebe & from “Control Systems Design”, Goodwin, Graebe & :
It is useful to understand at least a few proofs of items in the tables of Laplace transforms.
You need to understand all proofs that are given in slides at the end of this lecture.
It is essential to know how to use Laplace tables/rules in solving problems.
Determine the unit step response of the following system, assuming the initial condition y(0)=10:
Step 1: take Laplace transform of both sides:
(we used the linearity of LT and formulas for the derivative and for the unit step signal)
Step 2: solve for Y(s) – this involves algebraic calculations!
Step 3: rewrite Y(s) noting that y(0)=10 and using “partial fraction expansion”:
Step 4: take the inverse Laplace transform
Example continued
Suppose that assuming the same input and initial condition, we now want to calculate
One way to do this is to go through all previous steps, obtain y(t) and then calculate:
Example continued
But this is a place where we can use “Final Value Theorem” and calculate the limit directly in complex domain:
Note that you always need to first verify that the limit exists when using Final Value Theorem (e.g. stability).
Consider the system
Assuming zero initial conditions, find:
– Response to unit step signal
– Response to unit impulse signal
– Response to a sine signal with amplitude 2
– Limits as time goes to infinity for all the above
Example revisited
Recall that in the example in we needed to find the inverse Laplace transform of:
This term can not be found in LT tables. We need to transform it to its “partial fraction form”:
Note that in new form, we can find all terms in LT tables. We show how to do this in general.
Partial fraction expansion (non-repeated poles)
We can represent strictly proper transfer functions with different poles as follows:
where “residues” are computed as follows Using LT tables, we have
If we consider
its poles are s=0 and s=-1. Using the
formulas we have just given, we obtain
which confirms what we already shown earlier by direct computation.
Complex conjugate poles
Consider a typical term in FPE:
When the pole is complex, then the residue is
too and they appear in conjugate pairs: (if system impulse response is real-valued).
Complex conjugate pole
Direct calculations yield:
In the last line we have a real function!
Partial fraction expansion (repeated poles)
Consider now a transfer function with repeated poles that can be written as follows:
PFE can be written as follows:
See Ogata “Modern Control Engineering”
Partial fraction expansion via Matlab
We can write in Matlab:
num=[2 5 3 6]
den=[1 6 11 6] [r,p,k]=residue(num,den)
r =-6.0000 -4.0000 3.0000
p =-3.0000 -2.0000 -1.0000
Example 1 (exponential):
Example 2 (powers):
Property 1 (integration):
Property 2 (time shift):
Property 3 (exponential weighting):
Property 4 (differentiation):
Property 5 (linearity)
Properties 6 & 7 (limits)
NOTE: You need to always check that limits exist before applying formulas! We will see that existence of these limits is related to the notion of “stability”.
Transfer function
Consider an input-output model: Assuming zero initial conditions: This yields:
Property 8 (convolution)
Transfer functions obtained assuming zero initial conditions.
Transfer function = Laplace transform of system impulse response.
Often easier to find the transfer function than the impulse response
Convolution is difficult to do. But if we go to s- domain it becomes multiplication: easy.
Algebra of block diagrams
Series connection
Parallel connection
Unity feedback connection
Feedforward transfer function Comparator
General feedback connection
Feedforward connection
Comparator + feedback
Two inputs:
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”.
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