Controllability and Observability

Since the output $y(n)$ in Fig.E.1 is a linear combination of the input and states $x_i(n)$, one or more poles can be canceled by the zeros induced by this linear combination. When that happens, the canceled modes are said to be unobservable. Of course, since we started with a transfer function, any pole-zero cancellations should be dealt with at that point, so that the state space realization will always be controllable and observable. If a mode is uncontrollable, the input cannot affect it; if it is unobservable, it has no effect on the output. Therefore, there is usually no reason to include unobservable or uncontrollable modes in a state-space model.^E.6

A physical example of uncontrollable and unobservable modes is provided by the plucked vibrating string of an electric guitar with one (very thin) magnetic pick-up. In a vibrating string, considering only one plane of vibration, each quasi-harmonic^E.7 overtone corresponds to a mode of vibration [86] which may be modeled by a pair of complex-conjugate poles in a digital filter which models a particular point-to-point transfer function of the string. All modes of vibration having a node at the plucking point are uncontrollable at that point, and all modes having a node at the pick-up are unobservable at that point. If an ideal string is plucked at its midpoint, for example, all even numbered harmonics will not be excited, because they all have vibrational nodes at the string midpoint. Similarly, if the pick-up is located one-fourth of the string length from the bridge, every fourth string harmonic will be ``nulled'' in the output. This is why plucked and struck strings are generally excited near one end, and why magnetic pick-ups are located near the end of the string.

A basic result in control theory is that a system in state-space form is controllable from a scalar input signal $u(n)$ if and only if the matrix

$$\left[B,\, AB,\, A^2 B,\, \dots,\, A^{N-1}B\right]$$

has full rank (i.e., is invertible). Here, $B$ is $N\times 1$. For the general $N\times q$ case, this test can be applied to each of the $q$ columns of $B$, thereby testing controllability from each input in turn. Similarly, a state-space system is observable from a given output if and only if

$$\left[ \begin{array}{l} C\\ CA\\ CA^2\\ \dots\\ CA^{N-1} \end{array}\right]$$

is nonsingular (i.e., invertible), where $C$ is $1\times N$. In the $p$-output case, $C$ can be considered the row corresponding to the output for which observability is being checked.