State-Space to Transfer Function Conversion

Hello! Welcome back to our course on multivariable control.

In the last lesson, we saw how to transform a state-space model into a standard canonical form. A key step in that process involved finding the system's transfer function, which we took for granted. Today, we're going to master that conversion and its reverse, solidifying the bridge between the two most important ways of representing a dynamic system.

This lesson directly addresses the final outcome of our first module: converting between state-space representations and transfer function representations for a given system. Understanding this connection is vital, as it allows you to leverage tools from both the time domain (state-space) and the frequency domain (transfer function), a concept that should be familiar from your signal processing work.

From State-Space to Transfer Function

Let's start with the more straightforward direction: deriving a transfer function from a given state-space model $(A, B, C, D)$ .

Recall the standard LTI state-space equations:
$x ˙ (t) = A x (t) + B u (t)$
$y (t) = C x (t) + D u (t)$

To find the transfer function, which relates the output to the input in the frequency domain, we take the Laplace transform of both equations. A core assumption when deriving a transfer function is that all initial conditions are zero, so $x (0) = 0$ .

Applying the Laplace transform gives:
$s X (s) = A X (s) + B U (s)$
$Y (s) = C X (s) + D U (s)$

Our goal is to find the relationship $Y (s) = G (s) U (s)$ . To do this, we need to solve for $X (s)$ in the first equation and substitute it into the second.

Multivariable system representation 2019-04-24

The video 'Multivariable system representation' provides a clear, step-by-step algebraic derivation of this conversion. The instructor works through it as if calculating by hand, which makes the process very easy to follow.

Watch the section from 12:31 to 15:47. Pay close attention to the algebraic manipulation used to isolate X(s). Note the importance of the identity matrix 'I' and the proper handling of matrix multiplication (e.g., pre-multiplying by the inverse).

As the video shows, the derivation leads to this fundamental result:

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

This single formula is the bridge from any linear state-space representation to its unique transfer function.

The Transfer Function Matrix (MIMO Systems)

For Single-Input-Single-Output (SISO) systems, $G (s)$ is a scalar transfer function. For the Multiple-Input-Multiple-Output (MIMO) systems that are our main focus, $G (s)$ becomes a Transfer Function Matrix.

Transfer Function Matrix for MIMO Systems — This diagram shows how a MIMO system can be viewed as a matrix of individual transfer functions, where each element $G_{ij}(s)$ relates the i-th output to the j-th input. The overall output vector $\mathbf{Y}(s)$ is the result of the transfer function matrix $G(s)$ acting on the input vector $\mathbf{U}(s)$.

The Characteristic Equation

Look closely at the term $(s I - A)^{- 1}$ . The inverse of a matrix $M$ is given by $\frac{adj ( M )}{d e t ( M )}$ , where $adj (M)$ is the adjugate matrix and $det (M)$ is the determinant.

This means the denominator of our transfer function is determined by $det (s I - A)$ . This polynomial is called the characteristic equation of the system.

The roots of the characteristic equation, $det (s I - A) = 0$ , are the poles of the system. They are also, by definition, the eigenvalues of the matrix A.

This is a critical insight. It connects the system's poles directly to the eigenvalues of the state matrix $A$ . We will use this extensively in the next module to analyze system stability.

A Practical Example

Let's see this formula in action.

LCS - 52a - State-space to transfer function

The video 'LCS - 52a - State-space to transfer function' provides an excellent worked example for a 3rd-order system.

Watch from 03:48 to 07:51. The first part (until 07:20) walks through the manual calculation of the transfer function, including the matrix inverse. The second part briefly introduces the MATLAB command ss2tf which is the practical way to perform this conversion.

Test your understanding!

Given the following state-space system:
$A = [0 - 2 1 - 3], B = [01], C = [10], D = [0]$
Calculate the transfer function $G (s)$ using the formula $G (s) = C (s I - A)^{- 1} B$ .

Show answer

Calculate $s I - A$ :
$s I - A = s [1001] - [0 - 2 1 - 3] = [s 2 - 1 s + 3]$
Calculate the inverse $(s I - A)^{- 1}$ :
The determinant is $det (s I - A) = s (s + 3) - (- 1) (2) = s^{2} + 3 s + 2$ .
$(s I - A)^{- 1} = \frac{1}{s ^{2} + 3 s + 2} [s + 3 - 2 1 s]$
Calculate $G (s) = C (s I - A)^{- 1} B$ :
$G (s) = [10] (\frac{1}{s ^{2} + 3 s + 2} [s + 3 - 2 1 s]) [01] = \frac{1}{s ^{2} + 3 s + 2} [10] [s + 3 - 2 1 s] [01] = \frac{1}{s ^{2} + 3 s + 2} [s + 3 1] [01] = \frac{1}{s ^{2} + 3 s + 2}$
So, the transfer function is $G (s) = \frac{1}{s ^{2} + 3 s + 2}$ .

From Transfer Function to State-Space

Now for the other direction. This conversion is more complex because, as we saw in the last lesson, the state-space representation for a given transfer function is not unique.

Because there are infinite possibilities, we don't just find any state-space model; we aim to construct one in a specific, useful structure—a canonical form. The previous lesson introduced you to the Controllable and Observable Canonical Forms. Now, we will use them as a systematic method for conversion.

Transformation: Transfer Function ↔ State Space

The webpage 'Transformation: Transfer Function ↔ State Space' from Swarthmore College provides a clear, formulaic method for this conversion. We will focus on the sections for Controllable and Observable canonical forms.

Read the sections 'Controllable Canonical Form (CCF)' and 'Observable Canonical Form (OCF)'. For each form, focus on the final 'Key Concept' box, which gives the general matrix formulas. These allow you to write down the state-space matrices by simple inspection of the transfer function coefficients.

Let's summarize the key procedure from the reading. For a general nth-order transfer function (assuming a strictly proper system, where the numerator order is less than the denominator):

$G (s) = \frac{b _{n - 1} s ^{n - 1} + \dots + b _{1} s + b _{0}}{s ^{n} + a _{n - 1} s ^{n - 1} + \dots + a _{1} s + a _{0}}$

Controllable Canonical Form (CCF)

You can directly construct the CCF matrices as follows:

A_{cc f} = 00 ⋮ 0 - a_{0} 10 ⋮ 0 - a_{1} 01 ⋮ 0 - a_{2} \dots \dots ⋱ \dots \dots 00 ⋮ 1 - a_{n - 1}, B_{cc f} = 00 ⋮ 01

C_{cc f} = [b_{0} b_{1} b_{2} \dots b_{n - 1}], D_{cc f} = [0]

Notice the denominator coefficients form the last row of $A$ , and the numerator coefficients form the $C$ matrix.

Observable Canonical Form (OCF)

Similarly, you can construct the OCF by applying the principle of duality ( $A_{oc f} = A_{cc f}^{T}$ , $B_{oc f} = C_{cc f}^{T}$ , $C_{oc f} = B_{cc f}^{T}$ ):

A_{oc f} = 010 ⋮ 0 001 ⋮ 0 \dots \dots \dots ⋱ \dots 000 ⋮ 1 - a_{0} - a_{1} - a_{2} ⋮ - a_{n - 1}, B_{oc f} = b_{0} b_{1} b_{2} ⋮ b_{n - 1}

C_{oc f} = [00 \dots 01], D_{oc f} = [0]

This provides a mechanical and reliable way to go from any transfer function to a valid state-space model. In practice, you would often use a tool like MATLAB's tf2ss command, which also generates a state-space representation (typically a variant of the controllable canonical form).

Conclusion

Today we've closed the loop on the two primary representations of LTI systems. You are now equipped to move fluidly between the state-space and transfer function domains.

Key Takeaways:

State-Space to Transfer Function: The conversion is unique and is given by the formula $G (s) = C (s I - A)^{- 1} B + D$ . This applies to both SISO and MIMO systems.
Characteristic Equation: The denominator of the transfer function is determined by $det (s I - A) = 0$ . The roots of this equation are the system poles and the eigenvalues of the state matrix $A$ .
Transfer Function to State-Space: This conversion is not unique. We can systematically generate valid state-space models in canonical forms (like CCF or OCF) by inspecting the transfer function coefficients.

Preview of the Next Lesson:

We've established that the eigenvalues of the $A$ matrix are the poles of the system. This connection is profoundly important for stability analysis. In our next lesson, we will begin Module 2 by formally exploring how to determine the stability of a system by analyzing the eigenvalues of the state matrix. This will be our first step in understanding the fundamental properties of multivariable systems.

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