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This notebook shows some techniques for dealing with discrete systems analytically using the (z) transform
51.1. Definition¶
The (z) transform of a sampled signal ((f^*(t))) is defined as follows:
- S, z = sympy.symbols('s, z') k = sympy.Symbol('k', integer=True) Dt = sympy.Symbol('Delta t', positive=True) 51.1. The z transform of a sampled signal ( f ∗ ( t)) is defined as follows: Z f ∗ ( t) = ∑ k = 0 ∞ f ( k Δ t) z − k.
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Note The notation is often abused, so you may also encounter * (mathcal{Z}[f(t)]), which should be interpreted as having the sampling implied * (mathcal{Z}[F(s)]), which implies that you should first calculate the inverse Laplace and then sample, so something like (mathcal{Z}[F(s)]=mathcal{Z}[mathcal{L}^{-1}[F(s)]]=mathcal{Z}[f(t)]) * Seborg et al use (mathcal{Z}[{F(s)}]) to mean the transfer function of (F(s)) with a sample and zero order hold infront of it, which in these notebooks will be expressed as (mathcal{Z}[{H(s)F(s)}]).
51.2. Direct calculation in SymPy¶
For a unit step, (f(t)=1) and we can obtain the (z) transform as an infinte series as follows:
Sympy can recognise this infinite series as a geometric series, and under certain conditions for convergence, it can find a finite representation:
To extract the first case solution, we use args:
Notice what has happened here: we have taken the infinite series and written it in a compact form. You should always keep in mind that these two forms are equivalent.
51.3. Transfer functions from difference equations¶
For a first order difference equation (the discrete equivalent of a first order differential equation):
If we interpret (z^{-n}) as an (n) time step delay, can write
This transforms our difference equation to
Leading to a discrete transfer function:
Unfortunately, sympy simplifies this expression using positive powers of (z)
Since I have not found an easy way to get sympy to report negative powers of (z), I find it convenient to define
51.4. Responses and inversion¶
To find the response of this system to the unit input, we can multiply the input and the transfer function. Note that this is equivalent to convolution of the polinomial coefficients.
Let’s evaluate that with numeric values for the coefficients:
Remember that the (z) transform was defined using the values of the sampled signal at the sampling points.
To obtain the values of the response at the sampling points (also called inverting the (z) transform), we need to expand the polynomial. We can do this using Taylor series. Sympy has a Poly class which can extract all the coefficients of the polynomial easily.
Notice that the coefficients are returned in decreasing orders of (q), but we want them in increasing orders to plot them.
We’ll be using this operation quite a lot so there’s a nice function in tbcontrol.symbolic that does the same thing:
We’ll compare the values we obtained above with the step response of a continuous first order system with the same parameters:
51.5. Calculation using scipy¶
We can get the same values without going through the symbolic steps by using the scipy.signal library.
Note this uses the transfer function in terms of (z) (not (z^{-1})).
51.6. Calculation using the control libary¶
I have found it useful to define
This allows us to do calculations in a relatively straightforward way:
If we remember that the inversion of the signal is the same as an impulse response, we can also get the same result as follows:
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SymPy is an open source python library for symbolic computing. In the SymPy community we all believe in the merits and greatness of the open source approach but for the moment let us focus on the 'symbolic computing' part.
Symbolic computing systems, also called computer algebra systems (CASs) are used in research and engineering for solving a variety of mathematical problems symbolically. In modern labs you will rarely see a researcher solving mundane equations by hand. They usually give them to a computer algebra system to solve so they can spend their time on more interesting and intricate work.
What does it mean to solve problems symbolically? It means that instead of giving a numerical solution, as many systems do, a CAS will derive the solution using symbolic algebra, the same way that you would do it if you were to do it by hand on paper. So if you tell SymPy to solve the equation x2 = 2 for x, it will give you √2 and -√2, exactly.
You can see for examples Quick examples to start to get an idea of the rich set of abstract operations that a CAS supports. Some of the math may be unknown to you but do not be afraid: there is much that you can do for the project that does not involve hard math. You can look also at Wolfram Alpha to see another example of a CAS. This one is more feature complete than SymPy, but proprietary, so the source code is not available to the public. Actually Wolfram Alpha is just one service based on the Mathematica CAS, a popular proprietary CAS.
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