Controllability, Observability and The Kalman Decomposition

Lecture slides: controllability-observability-and-the-kalman-decomposition.pdf Download controllability-observability-and-the-kalman-decomposition.pdf

Reading suggestion: KJA_pp151_162.pdf Download KJA_pp151_162.pdf

We can build our intuition for what can be achieved using feedback control through the concept of controllability. Here we review the basic definition.

   

We can test for controllability by checking the rank of the controllability matrix. We now use the Cayley-Hamilton theorem to understand why this is the case, and also introduce the concept of the controllability Gramian.

   

Observability is used to understand how information about the state can be deduced from the output measurements. We review the basic definitions and tests for observability.

   

It is possible to understand controllability and observability in terms of subspaces. We begin by reviewing the key subspaces related to the matrix equation $y=Ax$ (column space, row space, null space and left null space). We then relate these to the unreachable and/or unobservable parts of the state space of an uncontrollable and/or unobservable part of a state-space model.

  

A state transformation can be used to represent a state-space model in a way that reveals its controllable and observable parts. This special decomposition is called the Kalman decomposition, and also has strong connections to pole zero cancellations in the transfer function associated with a state-space model.

   

Something that I forgot to discuss in the video - it is possible for the sub-matrices in the Kalman decomposition to be empty matrices (ie matrices with one or both of their dimensions equal to zero). The happens when a state-space model is controllable and/or observable (the matrices in the uncontrollable or/and unobservable parts are empty).