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# Observability

Module by: Thanos Antoulas, JP Slavinsky. E-mail the authors

Summary: (Blank Abstract)

Observability is the tool we use to investigate the internal workings of a system. It lets us use what we know about the input utut and the output ytyt to observe the state of the system xtxt.

To understand this concept let's start off with the basic state-space equations describing a system: x=Ax+Bu x A x B u y=Cx+Du y C x D u If we plug the general solution of the state variable, xtxt, into the equation for ytyt, we'd find the following familiar time-domain equation:

yt=CeAtx0+0tCeA(tτ)Buτdτ+Dut y t C A t x 0 τ 0 t C A t τ B u τ D u t
(1)

Without loss of generality, we can assume zero input; this will significantly clarify the following discussion. This assumption can be easily justified. Based on our initial assumption above, the last two terms on the right-hand side of time-domain equation are known (because we know utut). We could simply replace these two terms with some function of tt. We'll group them together into the variable y0ty0t. By moving y0ty0t to the left-hand side, we see that we can again group yty0t yt y0t into another replacement function of tt, y_ty_t. This result has the same effect as assuming zero input. y_t=yty0t=CeAtx0 y_t yt y0t C A t x 0 Given the discussion in the above paragraph, we can now start our examination of observability based on the following formula:

yt=CeAtx0 y t C A t x 0
(2)

The idea behind observability is to find the state of the system based upon its output. We will accomplish this by first finding the initial conditions of the state based upon the system's output. The state equation solution can then use this information to determine the state variable xtxt.

base formula seems to tell us that as long as we known enough about ytyt we should be able to find x0x0. The first question to answer is how much is enough? Since the initial condition of the state x0x0 is actually a vector of nn elements, we have nn unknowns and therefore need nn equations to solve the set. Remember that we have complete knowledge of the output ytyt. So, to generate these nn equations, we can simply take n1n1 derivatives of base formula. Taking these derivatives is relatively straightforward. On the right-hand side, the derivative operator will only act on the matrix exponential term. Each derivative of it will produce a multiplicative term of AA. Then, as we're dealing with these derivatives of ytyt at t=0t0, all of the exponential terms will go to unity ( eA0=1 A 0 1 ). y0=Cx0 y 0 C x 0 ddty0=CAx0 t1 y 0 C A x 0 d2dt2y0=CA2x0 t2 y 0 C A 2 x 0 dn1dtn1y0=CAn1x0 tn1 y 0 C A n 1 x 0 This can be re-expressed in matrix notation. ( y0 d1y0dt1 d2y0dt2 dn1y0dtn1 )=( C CA CA2 CAn1 )x0 y 0 t1 y 0 t2 y 0 tn1 y 0 C CA CA2 C A n 1 x 0

The first term on the right-hand side is known as the observability matrix, σCAσCA:

σCA=( C CA CA2 CAn1 ) σCA C CA CA2 C A n 1
(3)

We call the system completely observable if the rank of the observability matrix equals nn. This guarantees that we'll have enough independent equations to solve for the nn components of the state xtxt.

Whereas for controllability we talked about the system's controllable space, for observability we will talk about a system's unobservable space, XunobsXunobs. The unobservable space is found by taking the kernel of the observability matrix. This makes sense because when you multiply a vector in the kernel of the observability matrix by the observability matrix, the result will be 00. The problem is that when we get a zero result for ytyt, we cannot say with certainty whether the zero result was caused by xtxt itself being zero or by xtxt being a vector in the nullspace. As we cannot give a definite answer in this case, all of these vectors are said to be unobservable.

One cool thing to note is that the observability and controllability matrices are intimately related:

σCAT=CATCT σ C A C A C
(4)

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#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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