0.4 Controllability and observability grammians

Controllability grammian

The finite controllability grammian at time $t$∞ is defined as follows.

This grammian has two important properties. First, $P(t)=P^{*}(t)\ge 0$ . Secondly, the columns of $P(t)$ span the controllable space, i.e. $\mathrm{im}(P(t))=\mathrm{im}(C(A, B))$ It can be shown that the state defined by $A$ and $B$ is controllable if, and only if, $P(t)$ is positive definite for some $t> 0$ .

Using the controllability grammian, we can determine how to most efficiently take a system from the zero state to a certain state $\stackrel{}{x}$ . Given that $\stackrel{}{x}$ is in the controllable space, there exists $$ such that

In general, this minimal energy is exactly equal to $\stackrel{}{}^{*}P(\stackrel{}{T})\stackrel{}{}$ . If the system is controllable, then this formula becomes

Observability grammian

The finite observability grammian at time $t$∞ is defined as

Using this grammian, we can find an expression for the energy of the output $y$ at time $T$ caused by the system's initial state $x$ :

Infinite grammians

Consider a continuous-time linear system defined, as per normal, by the matrices $A$ , $B$ , $C$ , and $D$ . Assuming that this system is stable (i.e. all of its eigenvalues have negative real parts), both the controllability and observability grammians are defined for $t$∞ .

In the case of infinite grammians, the equations for minimal energy state transfer and observation energy drop their dependence on time. Assuming stability and complete controllability, the minimal energy required to transfer from zero to state $x_c$ is