import sympy
sympy.init_printing()

18. Valve equation

Let’s linearise the nasty nonlinear term in the equation percentage valve relationship in T4 Problem 4 (or T2 problem 4)

\[F = \underbrace{C_v \alpha^{x - 1}}_{\text{nonlinear}}\]

First we introduce the requisite symbols. Notice that we specify constraints on these variables, this will make simplifications better later on.

C_v, alpha, x = sympy.symbols('C_v, alpha, x', positive=True)

term = C_v*alpha**(x - 1)

We also introduce a barred versions of the variable. Sympy automatically constructs these to typesetting nicely.

xbar = sympy.symbols('xbar', positive=True)

For single variable expressions, we can use sympy.series to linearise for us. Note that he help for sympy.series references the help for sympy.Expr.series, which has a lot more detail about the operation of this function

sympy.series?

sympy.Expr.series?

Calling series by itself will result in an error term (the one with an \(\mathcal{O}\)). This is useful to estimate the error of the approximation.

sympy.series(term, x, xbar, 2)

$$\frac{C_{v} e^{\bar{x} \log{\left (\alpha \right )}}}{\alpha} + \frac{C_{v} \left(x - \bar{x}\right) e^{\bar{x} \log{\left (\alpha \right )}} \log{\left (\alpha \right )}}{\alpha} + O\left(\left(x - \bar{x}\right)^{2}; x\rightarrow \bar{x}\right)$$

But mostly we will be interested in the expression rather than the error, so we will remove that term with the removeO method:

lineq = sympy.series(term, x, xbar, 2).removeO()
lineq

$$\frac{C_{v} \left(x - \bar{x}\right) e^{\bar{x} \log{\left (\alpha \right )}} \log{\left (\alpha \right )}}{\alpha} + \frac{C_{v} e^{\bar{x} \log{\left (\alpha \right )}}}{\alpha}$$

18.1. Rewriting in terms of devation variables

While we are here, we can also rewrite in terms of deviation variables:

xprime = sympy.symbols("x'", positive=True)

lineq_deviation = lineq.subs({x: xprime + xbar})
lineq_deviation

$$\frac{C_{v} x' e^{\bar{x} \log{\left (\alpha \right )}} \log{\left (\alpha \right )}}{\alpha} + \frac{C_{v} e^{\bar{x} \log{\left (\alpha \right )}}}{\alpha}$$

19. A note about simplification

You will note that we specified positive=True for all our symbols when we created them. This is because the default assumptions about variables in SymPy are that they are complex. And for complex numbers, log is not a 1-to-1 function. See if you understand the following:

xbar, alpha = sympy.symbols('xbar, alpha')
sympy.exp(xbar*sympy.log(alpha)).simplify()

$$e^{\bar{x} \log{\left (\alpha \right )}}$$

xbar, alpha = sympy.symbols('xbar, alpha', positive=True)
sympy.exp(xbar*sympy.log(alpha)).simplify()

$$\alpha^{\bar{x}}$$

19.1. Multiple variables

Unfortunately, SymPy doesn’t have a built-in function for multivariate Taylor series, and consecutive application of the series function doesn’t do exactly what we want.

variables = x, y, z = sympy.symbols('x, y, z')
bars = xbar, ybar, zbar = sympy.symbols('xbar, ybar, zbar')

term = x*y*z

Note that the other variables are assumed to be constant here, so we don’t recover the answer we are looking for.

term.series(x, xbar, 2).removeO().series(y, ybar, 2).removeO()

$$x \bar{y} z + x z \left(y - \bar{y}\right)$$

The function tbcontrol.symbolic.linearise calculates a multivariable linearisation using the textbook formula. Note that it does not handle expressions which include derivatives or equalities, so don’t try to pass a full equation, just use it for the nonlinear terms.

import tbcontrol.symbolic

bars, linearexpression = tbcontrol.symbolic.linearise(term, variables)
linearexpression

$$\bar{x} \bar{y} \bar{z} + \bar{x} \bar{y} \left(z - \bar{z}\right) + \bar{x} \bar{z} \left(y - \bar{y}\right) + \bar{y} \bar{z} \left(x - \bar{x}\right)$$