Bilinear transformation of continuous time state space system

Question

I'm trying to understand the derivation of the bilinear transform for a set of continuous time state-space matrices. I've found plenty of websites which list steps to perform the conversion (here 1 or here 2 or here 3) - but haven't found any derivation or insight into how they came about. When I attempt to do it myself, I keep getting terms in z which I can't seem to get rid of.

To make the problem clear, if I define the bilinear transform as:

$$ s=\frac{\alpha\left(z-1\right)}{z+1} $$

Where alpha is usually given as 2/T.

The transfer function of a continuous-time state-space system can be given as:

$$ H(s) = \mathbf{C}(s\mathbf{I}-\mathbf{A})^{-1}\mathbf{B} + \mathbf{D} $$

How do I derive the expressions for the matrices of the discrete time version bilinear transformed model:

$$ H(z) = \mathbf{C_d}(z\mathbf{I}-\mathbf{A_d})^{-1}\mathbf{B_d} + \mathbf{D_d} $$

When I make the variable change, I keep getting extra terms in z which I don't know what to do with. I'm kinda hoping this is just me failing to have a strong grasp of matrix operations. Any insight (including if I am going about this completely the wrong way) would be really appreciated!

--- Edit with answer.

Thanks to Klaz for posting an answer to this first. I managed to work through the problem today and thought I would post my own alternate working here.

Given the typical state-space equations:

$$ s\mathbf{Q}(s) = \mathbf{A}\mathbf{Q}(s) + \mathbf{B}\mathbf{X}(s) $$ $$ \mathbf{Y}(s) = \mathbf{C}\mathbf{Q}(s) + \mathbf{D}\mathbf{X}(s) $$

Focus on the state update equation first and make the bilinear substitution:

$$ \frac{\alpha\left(z-1\right)}{z+1} \mathbf{Q}(z) = \mathbf{A}\mathbf{Q}(z) + \mathbf{B}\mathbf{X}(z) $$

We manipulate this expression to make the LHS $z\mathbf{Q}(z)$:

$$ \alpha\left(z-1\right) \mathbf{Q}(z) = \mathbf{A}\left(z+1\right)\mathbf{Q}(z) + \mathbf{B}\left(z+1\right)\mathbf{X}(z) $$

$$ z\left(\alpha\mathbf{I}-A\right)\mathbf{Q}(z) = \left(\alpha\mathbf{I}+A\right)\mathbf{Q}(z) + \mathbf{B}\left(z+1\right)\mathbf{X}(z) $$

$$ z\mathbf{Q}(z) = \left(\alpha\mathbf{I}-A\right)^{-1}\left(\alpha\mathbf{I}+A\right)\mathbf{Q}(z) + \left(\alpha\mathbf{I}-A\right)^{-1}\mathbf{B}\left(z+1\right)\mathbf{X}(z) $$

This instantly gives us $\mathbf{A_d}$ as:

$$ \mathbf{A_d} = \left(\alpha\mathbf{I}-A\right)^{-1}\left(\alpha\mathbf{I}+A\right) $$

We know that the linear $(z+1)$ term of $\mathbf{X}(z)$ can be applied wherever $\mathbf{Q}(z)$ is used. We apply this in the output expression to give the new output discrete equation as:

$$ z\mathbf{Q}(z) = \left(\alpha\mathbf{I}-A\right)^{-1}\left(\alpha\mathbf{I}+A\right)\mathbf{Q}(z) + \left(\alpha\mathbf{I}-A\right)^{-1}\mathbf{B}\mathbf{X}(z) $$ $$ \mathbf{Y}(z) = \mathbf{C}\left(z+1\right)\mathbf{Q}(z) + \mathbf{D}\mathbf{X}(z)$$ $$ \mathbf{Y}(z) = \mathbf{C}z\mathbf{Q}(z) + \mathbf{C}\mathbf{Q}(z) + \mathbf{D}\mathbf{X}(z)$$

We can now pull out $\mathbf{B_d}$ as:

$$ \mathbf{B_d} = \left(\alpha\mathbf{I}-A\right)^{-1}\mathbf{B} $$

Because we know an expression for $z\mathbf{Q}(z)$, we can formulate $\mathbf{Y}(z)$ as:

$$ \mathbf{Y}(z) = \mathbf{C}\left(\mathbf{A_d}\mathbf{Q}(z) + \mathbf{B_d} \mathbf{X}(z) \right) + \mathbf{C}\mathbf{Q}(z) + \mathbf{D}\mathbf{X}(z)$$

$$ \mathbf{Y}(z) = \mathbf{C}\left(\mathbf{A_d}+\mathbf{I}\right)\mathbf{Q}(z) + \left(\mathbf{C}\mathbf{B_d} + \mathbf{D}\right)\mathbf{X}(z) $$

Giving:

$$ \mathbf{C_d} = \mathbf{C}\left(\mathbf{A_d}+\mathbf{I}\right) $$ $$ \mathbf{D_d} = \mathbf{C}\mathbf{B_d} + \mathbf{D} $$

Answer 1

I've had the same question last week, but I've managed to find how to derive it (getting rid of those $z$ terms is indeed tricky). I will give here detailed demonstration of how to arrive to the result given in 1 (with, in your notation, $\alpha = 2 \lambda$).

So we define our new discrete-time function transfer as

$$ \begin{array}{rcl} H_d(z) &=& H(\frac{\alpha\left(z-1\right)}{z+1}) \\ &=& \mathbf{C} \left[ \frac{\alpha\left(z-1\right)}{z+1}\mathbf{I}-\mathbf{A} \right]^{-1}\mathbf{B} + \mathbf{D} \\ &=& \frac{z+1}{\alpha} \mathbf{C} \left[ \left(z-1\right)\mathbf{I}- \frac{z+1}{\alpha}\mathbf{A} \right]^{-1}\mathbf{B} + \mathbf{D} \\ &=& \frac{z+1}{\alpha} \mathbf{C} \left[ z \left(\mathbf{I} - \frac{1}{\alpha} \mathbf{A} \right) - \left(\mathbf{I} + \frac{1}{\alpha} \mathbf{A}\right) \right]^{-1}\mathbf{B} + \mathbf{D} \\ \end{array} $$

For sake of notation, let $\mathbf{P} = \mathbf{I} - \frac{1}{\alpha} \mathbf{A}$ and $\mathbf{Q} = \mathbf{I} + \frac{1}{\alpha} \mathbf{A}$ (important remark useful later on: $\mathbf{P} + \mathbf{Q} = 2 \mathbf{I}$). Then

$$ \begin{array}{rcl} H_d(z) &=& \frac{z+1}{\alpha} \mathbf{C} \left[ z \mathbf{P} - \mathbf{Q} \right]^{-1}\mathbf{B} + \mathbf{D} \\ &=& \frac{z+1}{\alpha} \mathbf{C} \left[ z \mathbf{I} - \mathbf{P}^{-1} \mathbf{Q} \right]^{-1} \mathbf{P}^{-1} \mathbf{B} + \mathbf{D} \\ &=& \frac{z+1}{\sqrt{2 \alpha}} \mathbf{C} \left[ z \mathbf{I} - \mathbf{P}^{-1} \mathbf{Q} \right]^{-1} \left( \sqrt{\frac{2}{\alpha}} \mathbf{P}^{-1} \mathbf{B} \right) + \mathbf{D} \\ &=& \frac{z+1}{\sqrt{2 \alpha}} \mathbf{C} \left[ z \mathbf{I} - \mathbf{A_d} \right]^{-1} \mathbf{B_d} + \mathbf{D} \end{array} $$

where we defined

$ \mathbf{A_d} = \mathbf{P}^{-1} \mathbf{Q} $ and $ \mathbf{B_d} = \sqrt{\frac{2}{\alpha}} \mathbf{P}^{-1} \mathbf{B} $. Furthermore, we have

$$ \begin{array}{rcl} \frac{z+1}{\sqrt{2 \alpha}} \mathbf{C} &=& \frac{1}{\sqrt{2 \alpha}} \mathbf{C} \left( z + 1 \right) \\ &=& \frac{1}{\sqrt{2 \alpha}} \mathbf{C} \left( z \mathbf{I} + \mathbf{I} + \mathbf{A_d} - \mathbf{A_d} \right) \\ &=& \frac{1}{\sqrt{2 \alpha}} \mathbf{C} \left[ \left( z \mathbf{I} - \mathbf{A_d} \right) + \left( \mathbf{I} + \mathbf{A_d} \right) \right] \\ &=& \frac{1}{\sqrt{2 \alpha}} \mathbf{C} \left[ \left( z \mathbf{I} - \mathbf{A_d} \right) + \mathbf{P}^{-1} \left( \mathbf{P} + \mathbf{Q} \right) \right] \\ &=& \frac{1}{\sqrt{2 \alpha}} \mathbf{C} \left[ \left( z \mathbf{I} - \mathbf{A_d} \right) + 2 \mathbf{P}^{-1} \right] \end{array} $$

which gives us

$$ \begin{array}{rcl} H_d(z) &=& \frac{1}{\sqrt{2 \alpha}} \mathbf{C} \left[ \left( z \mathbf{I} - \mathbf{A_d} \right) + 2 \mathbf{P}^{-1} \right] \left[ z \mathbf{I} - \mathbf{A_d} \right]^{-1} \mathbf{B_d} + \mathbf{D} \\ &=& \frac{1}{\sqrt{2 \alpha}} \mathbf{C} \mathbf{B_d} + \sqrt{\frac{2}{\alpha}} \mathbf{C} \mathbf{P}^{-1} \left[ z \mathbf{I} - \mathbf{A_d} \right]^{-1} \mathbf{B_d} + \mathbf{D} \\ &=& \mathbf{C_d}(z\mathbf{I}-\mathbf{A_d})^{-1}\mathbf{B_d} + \mathbf{D_d} \end{array} $$ with $ \mathbf{C_d} = \sqrt{\frac{2}{\alpha}} \mathbf{C} \mathbf{P}^{-1} $ and $ \mathbf{D_d} = \mathbf{D} + \frac{1}{\sqrt{2 \alpha}} \mathbf{C} \mathbf{B_d} $. Replacing $\mathbf{P}$ and $\alpha = 2 \lambda$ gives the results in 1.

PS: the same idea can be used to prove the discrete-time state-space representation found using Generalized Bilinear Transform ($s \leftarrow \alpha \frac{z - 1}{\beta z + \left(1-\beta\right)}$) or First-order Holder methods.