State Variable Models

Syllabus: State variable model and solution of state equation of LTI systems.

5 hand-picked visual resources on this topic, 1 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

State model and transfer function

The eigenvalues of $A$ are the poles of the system, so stability can be read from $A$ alone without ever forming the transfer function. The transfer function is unique; the state model is not — any similarity transform gives another valid one with the same eigenvalues.

$\dot x=Ax+Bu,\qquad y=Cx+Du$
$G(s)=C(sI-A)^{-1}B+D$
$\text{poles}=\text{eigenvalues of }A;\ \ \text{stable}\Leftrightarrow\Re\lambda_i<0$
$\bar A=P^{-1}AP\ \text{has the same eigenvalues}$
similarity transformation

See it

State transition matrix and the solution

$\phi(t)=e^{At}$ is found either by inverse-Laplace of $(sI-A)^{-1}$

Controllability and observability

Both are rank tests. For an $n$ -state system, full rank $n$ means controllable (or observable); a rank deficiency means some mode cannot be driven by the input, or cannot be seen at the output.

$Q_c=[\,B\ \ AB\ \ A^2B\ \cdots\ A^{n-1}B\,],\quad \text{rank}=n$

← Bode, Nyquist, Margins and Compensation

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