Show that the pair (C, A) is observable if and only if the matrix 𝒪^H 𝒪 is invertible.

Name: Problem 11, Chapter 19: Linear Algebra in Action
Uploaded: 2020-05-23T05:05:15-08:00
Duration: 6 min 28 s
Channel: Jimmy Yao
Description: Show that the pair (C, A) is observable if and only if the matrix 𝒪^H 𝒪 is invertible.

step 1 So the problem here, we want to argue are a is a linear independent. So the column of the a is l

Show that the pair (C, A) is observable if and only if the...

Key Concepts

Invertibility of the Observability Gramian

The matrix formed by the product of the conjugate transpose of the observability matrix and the observability matrix itself (often referred to as the observability Gramian) is invertible if and only if the observability matrix has full column rank. Invertibility here implies that there is no non-zero state that can result in zero measurements, thereby confirming the system's observability.

Observability Matrix

The observability matrix is a matrix built from the output matrix and successive products with the state matrix. Its full column rank is a necessary and sufficient condition for system observability. This matrix encapsulates how the state information is spread into the outputs over time.

Observability

Observability is a fundamental property in control theory that determines whether the internal state of a system can be inferred from its external outputs over time. If a system is observable, then one can design observers or estimators to reconstruct the state from output measurements.

Transcript

00:03 So the problem here, we want to argue are a is a linear independent.

00:09 So the column of the a is linear independent is equivalent to say a is invertible.

00:19 So it is actually a is invertible.

00:24 If and only if a transpose a is also invertible.

00:32 So this is because if we see this direction.

00:37 So here for this direction, if a the colon is the columns of a is linear independent, then we know a is invertible.

01:09 So if a is invertible, and so we also know a transpose is invertible.

01:18 So this is be a because the rank of a is equal to rank of a transpose.

01:25 So yeah, so they have the inverse.

01:28 So say we denotes the inverse of a transpose b, a transpose inverse, and the inverse of a will be a inverse.

01:39 So we say a transpose inverse and a inverse will be the inverse of a transpose a transpose a.

01:49 This is because if they time together, it's equal to a inverse and a transpose inverse a transpose and a transpose and equal to a transpose i, a is equal to a inverse a is equal to i.

02:11 So this means it is the inverse matrix of a transpose a.

02:19 So a transpose a transpose a is invertible.

02:26 So for 2, this direction.

02:29 So we know a transpose a is invertible.

02:36 Due to the rank nullity theorem, we know that the rank of a transpose a plus the nullity of a inverse transpose a is equal to a.

02:56 So here a transpose a is invertible, so we know this rank is equal to n.

03:02 So the nullity should equal to zero...

Please give Ace some feedback