Effect of Controllability and Observability On Stability

In this post, I would post the email conversation that I had with my friends over the topic "the effect of controllability and observability on stability". I am thankful to the student Asmini who actually raised this question.

My response :

In simple terms, controllability is the virtue of a system by which one can control the state of the system. In technical terms, controllability is the ability to be able to control each state variable or to be able to take it from one point in state space to another point. There is subtle difference between the terms "Controllability" and "Reachability". Don't worry about this second term at present. Both are more or less same. If a system is controllable, that means we can design a controller to control all states. That's good for us, is n't it?

On the other hand, observability is the virtue of system which enables us to observe all the states. I mean if a system is observable, we can use sensors to record all state variables. Note that this is not always the case. We may not have access to all states of a system. Again, it is good to have a completely observable system.

Now coming to your question, what is the effect of controllability and observability on stability?

Well, I think if a system is completely controllable or observable, then it is possible to design controllers which can stabilise the system. But if the system is either uncontrollable or unobservable, then it would be difficult to stabilise it. In some case, it would be impossible to design a controller to stabilize the system.

Indrani's Response:

I assume that we know the meaning of "stability of a system". If a system
dynamics is not stable we can make it stable by application of a proper
control input. On the other hand, controllability of a system implies
that it is possible to force the system to a particular state by
application of a control input. If a state is uncontrollable then no
input will be able to control that state. Thus if a system state is not
controllable as well as  the corresponding dynamics is not stable then it
will not be possible to make that state stable by applying a control
input.


Similarly if a state is not observable then the controller will not be
able to determine its behavior from the system output and hence not be
able to use that state to stabilize the system. Thus controllability and
observability of a system are two important properties which are to be
considered before designing a controller.

Prem's response:

Yes... Indrani's answer is correct... There is no correlation between stability and controllability. A controllable system can be either unstable or stable.

But controllability means that if the system is unstable then you can stabilize it using proper state feedback. Observability indicates whether we can deduct the states from the observed output so that states will be available for stabilization.

If some of the nodes are not observable and those nodes are already stable then also we can stabilize.

so we can classify into two cases.

i) An observable and controllable system can be stabilizable with state feedbacm.

ii) If some of the states are unobservable and those nodes are stable, then again we can design state feedback to design the stabilizing the controller with observable states.

I am still waiting for Awhan's response and I would update this post once I hear from him. But following thing can be concluded about this topic.

• Controllability and Observability properties are desirable for controller design. This is the first thing to be checked before designing a controller.
  
  
• There is no correlation between Controllability and Observability with Stability of the system. Stability is the property of the system (Plant + Controller). On the other hand controllability and Observability are the properties of Plant alone.