State-Space Realizations, Similarity & Canonical Forms

A realization is a particular set of state coordinates for a dynamic input/output behavior. Two realizations can have very different-looking matrices and state variables yet predict exactly the same external response. The distinction matters because controller and observer calculations act on coordinates, while actuators, sensors, limits, and initialization belong to the physical system.

Similarity changes coordinates, not dynamics #

Start with

$$\dot x=Ax+Bu,\qquad y=Cx+Du.$$

Choose an invertible matrix $T$ and define $z=Tx$. Since $x=T^{-1}z$,

$$\dot z=TAT^{-1}z+TBu,\qquad y=CT^{-1}z+Du.$$

Thus

$$A_z=TAT^{-1},\quad B_z=TB,\quad C_z=CT^{-1},\quad D_z=D.$$

Similarity preserves the characteristic polynomial and eigenvalues of $A$, the transfer matrix $C(sI-A)^{-1}B+D$, and whether the realization is controllable or observable. It does not preserve the numerical values, units, or direct physical interpretation of individual state coordinates.

A transfer function does not select unique state #

For zero initial state, an LTI realization produces

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

Converting a scalar $n$th-order differential equation to $n$ first-order equations creates one valid realization. Partial fractions may create a modal realization. A controllable companion form can place the denominator coefficients in a structured row. All can express the same $G(s)$.

The reverse conversion loses internal information. A transfer function cannot reveal an uncontrollable mode, because the input never excites it, or an unobservable mode, because the output never exposes it. Pole-zero cancellation may therefore hide internal unstable dynamics.

What common forms reveal #

For the monic denominator

$$p(s)=s^n+a_{n-1}s^{n-1}+\cdots+a_1s+a_0,$$

one common controllable phase-variable convention uses

$$A_c= \begin{bmatrix} 0&1&0&\cdots&0\\ 0&0&1&\cdots&0\\ \vdots&&&\ddots&\vdots\\ -a_0&-a_1&-a_2&\cdots&-a_{n-1} \end{bmatrix},\qquad B_c=\begin{bmatrix}0\\0\\\vdots\\1\end{bmatrix}.$$

Other texts and tools reverse the state order, so the ones and polynomial coefficients appear transposed or reflected. Always state the convention; a familiar label does not determine the matrix layout.

Minimality is structural #

A realization is minimal when no lower-order realization has the same transfer behavior. For finite-dimensional LTI systems, minimality is equivalent to being both controllable and observable. A nonminimal realization can be decomposed into four state classes: controllable/observable, controllable/unobservable, uncontrollable/observable, and uncontrollable/unobservable. Only the controllable/observable block is required by the external transfer behavior.

Minimal does not mean physically complete. A reduced input/output model may intentionally omit thermal, saturation, structural, or failure states that matter outside the identification experiment. Conversely, a physical model may retain an unactuated state because it is safety-relevant even though it cannot participate in one controller design.

Numerical conditioning is part of the representation #

The classical controllability matrix

$$\mathcal C=[B\;AB\;A^2B\;\cdots\;A^{n-1}B]$$

is central to theory and to companion-form derivations. Its powers of $A$ can, however, span wildly different magnitudes. Official current tooling therefore warns that companion transformations and raw rank tests can become numerically singular for moderate-order systems.

Practical rules:

• scale states, inputs, and outputs to comparable engineering magnitudes;
• prefer orthogonal staircase, Schur, or Gramian-based algorithms for computation;
• treat a reported numerical rank as tolerance- and scaling-dependent;
• verify the transformed model's input/output response and initial-state mapping;
• use canonical forms as explanatory coordinates, not as a mandatory implementation format.

The best realization is the one that makes the required inference reliable: physical coordinates for limits and diagnosis, modal coordinates for modes, balanced coordinates for reduction, and structurally stable numerical forms for computation.