37. State-space realizations#

State-Space RealizationsIn system theory, a realization is a state-space triplet \(\{A, B, C, D\}\) that represents a given transfer function \(H(s)\). For a strictly proper system, the relationship is defined by:

\[H(s) = C(sI - A)^{-1}B\]

Given a general \(n\)-th order transfer function:

\[H(s) = \frac{b_{n-1}s^{n-1} + \dots + b_1s + b_0}{s^n + a_{n-1}s^{n-1} + \dots + a_1s + a_0}\]

There are two primary canonical forms used for implementation and analysis.

37.1. Controller Canonical Form (CCF)#

The Controller Canonical Form is structured to make the system’s controllability obvious. It is often used in control design (e.g., pole placement) because the input \(u(t)\) enters the “chain” of integrators at the final stage.

State-Space Representation. The matrices for an \(n\)-th order system in CCF are:

\[\begin{split}A = \begin{bmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \\ -a_0 & -a_1 & -a_2 & \dots & -a_{n-1} \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{bmatrix}\end{split}\]
\[C = \begin{bmatrix} b_0 & b_1 & \dots & b_{n-1} \end{bmatrix}\]

Characteristics

  • Structure: The states are typically chosen as the output and its derivatives (\(x_1 = y, x_2 = \dot{y}, \dots\)) in the absence of zeros.

  • Controllability: A system in this form is always controllable, provided there are no pole-zero cancellations.

37.2. Observer Canonical Form (OCF)#

The Observer Canonical Form is the dual of the CCF. It is structured such that the output \(y(t)\) is directly influenced by all state variables, making the system’s observability easily verifiable.

State-Space Representation.

The matrices for an \(n\)-th order system in OCF are:

\[\begin{split}A = \begin{bmatrix} 0 & 0 & \dots & 0 & -a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ 0 & 1 & \dots & 0 & -a_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & -a_{n-1} \end{bmatrix}, \quad B = \begin{bmatrix} b_0 \\ b_1 \\ b_2 \\ \vdots \\ b_{n-1} \end{bmatrix}\end{split}\]
\[C = \begin{bmatrix} 0 & 0 & \dots & 0 & 1 \end{bmatrix}\]

Characteristics:

  • Duality: The \(A\) matrix of the OCF is the transpose of the \(A\) matrix of the CCF (\(A_{OCF} = A_{CCF}^T\)), and the \(B\) and \(C\) matrices are swapped and transposed (\(B_{OCF} = C_{CCF}^T\) and \(C_{OCF} = B_{CCF}^T\)).

  • Observability: A system in this form is always observable.

37.3. Example: realization of a second-order system#

From the specific case \(H(s) = \frac{s}{s^2 + a_1s + a_0}\):

Controller Realization

\[\begin{split}\dot{x} = \begin{bmatrix} 0 & 1 \\ -a_0 & -a_1 \end{bmatrix} x + \begin{bmatrix} 0 \\ 1 \end{bmatrix} u, \quad y = \begin{bmatrix} 0 & 1 \end{bmatrix} x\end{split}\]

Observer Realization

\[\begin{split}\dot{x} = \begin{bmatrix} 0 & -a_0 \\ 1 & -a_1 \end{bmatrix} x + \begin{bmatrix} 0 \\ 1 \end{bmatrix} u, \quad y = \begin{bmatrix} 0 & 1 \end{bmatrix} x\end{split}\]

37.4. Minimality and Similarity Transformations#

A realization is minimal if and only if it is both controllable and observable. The dimension of a minimal realization equals the degree of the denominator of the transfer function (the McMillan degree).Any two realizations \(\{A, B, C\}\) and \(\{\tilde{A}, \tilde{B}, \tilde{C}\}\) of the same transfer function are related by a similarity transformation \(T\) such that:

\[\tilde{A} = T A T^{-1}, \quad \tilde{B} = T B, \quad \tilde{C} = C T^{-1}\]