A dynamical system with state , input , and output is observable on the interval if given output and input , we can uniquely determine initial state and time . Here, we'll specifically study controllability for linear systems, in 🇺🇸 State Space Model form, as they have closed-form expressions.

For the LTV system

the closed-form solution is

Since we know , we can compute the last two terms. The question of observability comes down to finding from

and thus is specifically determined by whether the operator has null space . Multiplying with , which is invertible and doesn't change the null space, we have

Using the property of 🍱 Matrix > Adjoints, the null space of is equivalent to that of the matrix

We call the observability Grammian. In summary, a linear system is observable if this observability Grammian has null space , or equivalently has full rank.

Observability Matrix#

For something more concrete, we focus on LTI systems, where the state transition matrix

Plugging this into the observability Grammian, we have

Following the same intuition as that of 🕹️ Controllability > Controllability Matrix, we define the observability matrix as

This has the same range as , because expanding gives us terms, and integrating simply makes combinations of these terms. Thus, we conclude that the system is observable if this has full rank.

Output Feedback#

In many applications, we cannot directly measure the states of the state space model and thus cannot use 🕹️ Controllability > State Feedback. However, if the system is observable, we're able to estimate the state using known quantities: specifically, the observer computes estimate given the input and output .

If we simulate the system using , we have

Between this simulated system and the real system, we can measure the error by plugging in the transition definitions,

From this equation, we can see that if is Hurwitz, the error will converge to zero naturally.

However, if is not Hurwitz or takes too long to converge, we can use a matrix to correct the estimate, specifically by comparing the measured output and predicted output :

Using this transition, our updated error follows

Similar to the strategy in 🕹️ Controllability, if we choose such that is Hurwitz, then we can make the error converge to zero. If the system is observable, such a is possible. However, even if some eigenvalues of can't be altered by , but already have negative real parts, can still be Hurwitz; in this case, the system is called detectable.

Finally, combining this observer with a controller, we have the equations

Plugging back into the first equation gives us

Separation Principle#

In the form above, it's not immediately clear whether output feedback obeys the same stability properties as state feedback, as detailed in 🕹️ Controllability. However, the Separation Principle allows us to guarantee stability: for the controller above, the closed loop system can be written as

and the eigenvalues of

are that of and combined. This is important, since it means that we can separately design state-feedback gain and observer gain , such that as long as and are both stable, the resulting output feedback controller is also stable.

Output Sensitivity#

Finally, we can determine how sensitive the output is to each component of the initial state. With initial state and no input for , our output is simply

with total energy

Thus, we see that the output can be characterized by the observability Grammian described above. If is aligned with an eigenvector associated with a large eigenvalue of , then it will produce a larger output.