State-Space to Transfer Function Conversion

Mastering the conversion between state-space and transfer function representations is a critical skill in control systems engineering. While the transfer function offers a compact, input-output view in the frequency domain, the state-space model provides an internal, time-domain description of a system's dynamics. Being able to move fluidly between these two frameworks allows you to leverage the best tools from classical and modern control theory, whether you're designing a filter, stabilizing a drone, or analyzing a chemical process.

The Fundamental Conversion Formula

The core mathematical link from a state-space description to a single-input, single-output (SISO) transfer function is elegant and direct. A linear time-invariant (LTI) system in state-space form is given by:

$$\dot{x}=Ax+Bu$$ $$y=Cx+Du$$

Here, $x$ is the state vector, $u$ is the input, and $y$ is the output. The matrices $A$, $B$, $C$, and $D$ define the system's dynamics, input coupling, output relationship, and direct feedthrough, respectively.

To find the transfer function $G(s)=Y(s)/U(s)$, we take the Laplace transform of these equations, assuming zero initial conditions. This yields $sX(s)=AX(s)+BU(s)$ and $Y(s)=CX(s)+DU(s)$. Solving the state equation for $X(s)$ gives $X(s)=(sI-A{)}^{-1}BU(s)$, where $I$ is the identity matrix. Substituting into the output equation produces the definitive formula:

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

The term $(sI-A{)}^{-1}$ is the resolvent matrix, and its determinant, $|sI-A|$, gives the system's characteristic equation. This formula is your primary tool for extracting the frequency-domain behavior from any state-space model.

Step-by-Step Derivation and a Worked Example

Let's walk through the application of the formula with a concrete system. Consider a system with matrices: $$A=\left[\begin{array}{cc}0& 1\\ -2& -3\end{array}\right],B=\left[\begin{array}{c}0\\ 1\end{array}\right],C=\left[\begin{array}{cc}1& 0\end{array}\right],D=[0]$$

Step 1: Form $(sI-A)$. $$sI-A=s\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]-\left[\begin{array}{cc}0& 1\\ -2& -3\end{array}\right]=\left[\begin{array}{cc}s& -1\\ 2& s+3\end{array}\right]$$

Step 2: Find the inverse $(sI-A{)}^{-1}$. The inverse of a 2x2 matrix $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is $\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$. Here, the determinant is $s(s+3)-(-1)(2)=s^2+3s+2$. Thus, $$(sI-A{)}^{-1}=\frac{1}{s^2+3s+2}\left[\begin{array}{cc}s+3& 1\\ -2& s\end{array}\right]$$

Step 3: Multiply $C(sI-A{)}^{-1}B$. First, compute $(sI-A{)}^{-1}B$: $$\frac{1}{s^2+3s+2}\left[\begin{array}{cc}s+3& 1\\ -2& s\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]=\frac{1}{s^2+3s+2}\left[\begin{array}{c}1\\ s\end{array}\right]$$ Now, multiply by $C$: $$C(sI-A{)}^{-1}B=\left[\begin{array}{cc}1& 0\end{array}\right]\cdot \frac{1}{s^2+3s+2}\left[\begin{array}{c}1\\ s\end{array}\right]=\frac{1}{s^2+3s+2}$$

Step 4: Add $D$ (which is zero). Therefore, the transfer function is: $$G(s)=\frac{1}{s^2+3s+2}$$

This process, though algebraically intensive for larger systems, is systematic and universally applicable.

From Transfer Function to State-Space: Controllable Canonical Form

The reverse journey—creating a state-space model from a given transfer function—is not unique. One state-space realization can produce a given $G(s)$, but infinitely many realizations exist. However, certain forms are standardized and particularly useful. The controllable canonical form is one such systematic realization. It is directly constructed from the coefficients of the transfer function's denominator and numerator.

Given a general SISO transfer function: $$G(s)=\frac{b_n{s}^n+b_{n-1}{s}^{n-1}+...+b_{1}s+b_0}{s^n+a_{n-1}{s}^{n-1}+...+a_{1}s+a_0}+D$$ (where the $b_i$ and $a_i$ coefficients are scalar constants), the controllable canonical form matrices are:

$$A=\left[\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & 1\\ -a_0& -a_1& -a_2& \cdots & -a_{n-1}\end{array}\right],B=\left[\begin{array}{c}0\\ 0\\ ⋮\\ 0\\ 1\end{array}\right]$$ $$C=\left[\begin{array}{ccccc}{b}_0& b_1& b_2& \cdots & b_{n-1}\end{array}\right],D=D$$

This structure places the denominator coefficients in the bottom row of the $A$ matrix and packs the numerator coefficients into the $C$ matrix. Its key property is that it is guaranteed to be controllable, meaning all system modes can be influenced by the input $u$.

From Transfer Function to State-Space: Observable Canonical Form

Dual to the controllable form is the observable canonical form. For the same general transfer function $G(s)$, the matrices take on a transposed structure:

$$A=\left[\begin{array}{ccccc}0& 0& \cdots & 0& -a_0\\ 1& 0& \cdots & 0& -a_1\\ 0& 1& \cdots & 0& -a_2\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & 1& -a_{n-1}\end{array}\right],B=\left[\begin{array}{c}{b}_0\\ b_1\\ b_2\\ ⋮\\ b_{n-1}\end{array}\right]$$ $$C=\left[\begin{array}{ccccc}0& 0& \cdots & 0& 1\end{array}\right],D=D$$

Here, the denominator coefficients appear in the last column of $A$, and the numerator coefficients populate the $B$ matrix. This realization is guaranteed to be observable, meaning all system modes can be inferred by measuring the output $y$.

These two canonical forms illustrate a deep principle in linear systems theory: the duality between controllability and observability. The matrices of the observable form are essentially the transpose of those in the controllable form, with the roles of $B$ and $C$ swapped.

Common Pitfalls

1. Assuming $(sI-A)$ is Always Invertible: The conversion formula $G(s)=C(sI-A{)}^{-1}B+D$ is only valid where $(sI-A{)}^{-1}$ exists—that is, where $s$ is not an eigenvalue of $A$. These eigenvalues (poles) are precisely where $G(s)$ blows up to infinity. The transfer function describes the system's behavior for all complex $s$ except at these pole locations.

1. Ignoring Pole-Zero Cancellations: When converting from state-space to transfer function, if the state-space model is not both controllable and observable, pole-zero cancellation will occur in the derived $G(s)$. The cancelled pole, while absent from the transfer function, is still a dynamic mode present in the internal state-space description. This leads to "hidden" dynamics that can cause instability, even if the transfer function appears stable. Always check the minimality of your realization.

1. Misapplying Canonical Forms to Improper Transfer Functions: The canonical forms described assume the transfer function is proper (numerator degree $\le$ denominator degree). If you have an improper transfer function (numerator degree > denominator degree), you must first perform long division to extract a direct feedthrough term $D$, leaving a proper fractional part to be realized in the $A,B,C$ matrices.

Summary

• The universal formula to convert a state-space model $(A,B,C,D)$ to a transfer function is $G(s)=C(sI-A{)}^{-1}B+D$. This process involves matrix algebra centered on the resolvent $(sI-A{)}^{-1}$.
• The reverse conversion is not unique. The controllable canonical form provides a systematic state-space realization guaranteed to be controllable, with the $A$ matrix in companion form and the denominator coefficients in its bottom row.
• The observable canonical form provides a dual realization guaranteed to be observable, featuring a transposed structure with denominator coefficients in the last column of $A$.
• These conversions are the essential bridge between modern state-space analysis (focused on internal state control and observation) and classical frequency-domain techniques (focused on input-output stability and response), allowing for a comprehensive approach to linear system analysis and design.