Controllability and observability

This paper introduces the concepts of controllability and observability.

In a control system, sometimes we need to discuss about the controllability and observability, basically the controllability and observability is about the actuator and sensor, it doesn’t mind what kinds of control strategy you are applying.

Controllability

Definition

Controllability concerns whether the state of a state-space equation can be controlled form the input.

Definition 1 A state $\dot{\mathbf{x}}(t)=A(t) \mathbf{x}(t)+B(t) \mathbf{u}(t)$ is controllable if for any initial state $x_0$ and any final state $x_1$, there exists an input that transfer $x_0$ to $x_1$ in a finite time.

Proof

Every student in control engineering knows the famous proof of controllability that if the $n\times np$ controllability matrix

$$\begin{equation} \mathcal{C}=\left[\mathbf{B}\ \mathbf{A B}\ \mathbf{A}^{2}\ \mathbf{B}\ \cdots \ \mathbf{A}^{n-1} \mathbf{B}\right] \end{equation}$$

has rank $n$ (full row rank), then the system is controllable. A detailed proof can be found from reference¹.

Reachability

In a linear system, the controllability and reachability are the same concepts.

Observability

Definition

Observability concerns whether the initial state can be observed form the output.

Proof

Like the controllability, a system is observable if the $nq \times n$ observability matrix

$$\begin{equation} \mathcal{O}=\left[\begin{array}{c}{\mathbf{C}} \\ {\mathbf{C A}} \\ {\vdots} \\ {\mathbf{C A}^{n-1}}\end{array}\right] \end{equation}$$

is full col rank.

reference

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1. 1.Chen C T. Linear system theory and design[M]. Oxford University Press, Inc., 1998. ↩