Lecture 4 slides

We now shift our focus to state-space models. Our first lecture on this topic treats coordinate changes, zeros, state feedback and observers.

Reading suggestion for further study: KJA_pp139_150.pdf

The covered material is treated in the hand-in exercises 2.1, 2.2a-c.

State-space Models

We first introduce the state-space models, and confirm that transfer function models and state-space models are really describing the same thing. State-space models just offer a different perspective, and unlock a range of powerful tools from linear algebra that we can now use for system analysis.

Next we translate a few concepts from transfer functions into the state-space world by showing that poles and zeros can be determined from a state-space model by solving eigenvalue problems.

A curious point here is that this shows something deeper than the observation that transfer functions and state-space give useful perspectives on the same object. It is really saying that linear algebra and complex-analysis give different viewpoints on the same mathematics. We will another example of this when we study coordinate changes.

Coordinate Changes

An important tool in the analysis of state-space models are coordinate changes. These can be used to change the coordinates of the state variable in a way that reveals more structural features about the A, B, C and D matrices. We've already seen one example of this - the controllable canonical form. We review another which can be used to perform partial fraction expansion with eigenvalue analysis.

State-Feedback

State feedback is a simple control system design method that can be used to place closed loop poles arbitrarily. It is not a very practical method since to be applied it requires that we are able to measure the system state. But it forms the backbone of the combined state feedback/observer design method. Note some extra videos on this topic are included in the extra stuff at the end of the video.

Observers

State feedback requires state measurements, which are typically not available in practice. The value of the state at every point in time can however be estimated using an observer. We introduce the key ideas.

Putting it all Together

By combining an observer with a state feedback controller we do get a practical and useful design method. We show how this can be done, and discuss some of the closed loop properties of this method.

Extra stuff

A while back I made some videos on state-feedback and observer design, where we go through some examples and design rules in a bit more detail. This might help refresh your memory on some of these topics.