Linear control system controllability

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Discussion Overview

The discussion revolves around the concept of controllability in linear control systems, specifically focusing on the implications of controllability when using output feedback instead of state feedback. Participants explore the relationship between the controllability of the system defined by matrices (A,B) and the ability to control the system through output feedback.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions how controllability theory applies to output feedback, suggesting that if the system (A,B) is not controllable, then no output feedback law can control the system from any initial condition.
  • Another participant agrees, stating that if the states cannot be controlled, then the outputs also cannot be controlled, although they note that partial controllability and system-level stability might still be possible.
  • A third participant reflects on their own problem, indicating a lack of system-level stability due to unstable uncontrolled modes.
  • There is a suggestion to modify the system to address issues of uncontrollability, although it is unclear how this would be achieved.
  • One participant clarifies that the problem being discussed originates from an old exam, indicating a theoretical context rather than a practical application.

Areas of Agreement / Disagreement

Participants generally agree that if the system (A,B) is not controllable, then output feedback cannot control the system from any initial condition. However, there is some discussion about the possibility of partial controllability and stability, indicating that the conversation remains somewhat unresolved regarding the nuances of these concepts.

Contextual Notes

The discussion does not resolve the implications of partial controllability or how to modify the system to achieve controllability. There are also assumptions about the definitions of controllability and stability that are not explicitly stated.

Who May Find This Useful

Readers interested in control theory, particularly those studying linear control systems and the implications of feedback methods on system controllability.

FOIWATER
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Hello,

In a linear system of the form $$\dot{x}=Ax+Bu$$ and $$y=Cx$$ we can use state feedback control $$u=Kx$$ assuming we know the state, or can observe it, to control the system if (A,B) is controllable (all states are reachable).

How does the theory on controllability apply to output feedback? So that the feedback is not taken from the state of the system using an observer, but from the output.

If the original system (A,B) is not controllable, is it also true that no OUTPUT feedback law exists to control the system from any initial condition?

My initial reaction is that the system is controllable from neither the state feedback or output feedback methods. My basis for this assumption is that the theory of controllability is fundamentally based on the system (A,B), and is not developed from the starting point of any feedback. But I am unsure.

Sorry if my question is not well posed - any information appreciated.
 
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FOIWATER said:
If the original system (A,B) is not controllable, is it also true that no OUTPUT feedback law exists to control the system from any initial condition?

yes.

Think about it this way. the outputs are simply the C matrix multiplied by the states. If you cannot control what the states are, you cannot control what the output is. That means with feedback there is still no way you can control the output of all of the states.

note... you can still have partial control-ability and system level stability.
 
Yes that makes sense, thankyou donpacino.

Yeah, in my problem I am noticing that I do not have system level stability, since the uncontrolled modes are not stable.

Thanks
 
no problemo

is this a real life problem, or a schoolwork/textbook problem. Is it purely theoretical or is there a real system?
If you 'need' to solve this problem, you can look into modifying the system such that it is no longer uncontrolable
 
It is a problem I found on an old exam from a course that I am not taking.
 

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