Linear control system controllability

In summary, the conversation discusses the application of controllability theory to output feedback and whether or not an output feedback law exists to control a system from any initial condition if the original system is not controllable. The conclusion is that if the original system is not controllable, then no output feedback law can control the system from any initial condition. This is because the outputs are dependent on the states, and if the states cannot be controlled, then the outputs cannot be controlled. It is also mentioned that system level stability may not be achieved in this case.
  • #1
FOIWATER
Gold Member
434
12
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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  • #2
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.
 
  • #3
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
 
  • #4
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
 
  • #5
It is a problem I found on an old exam from a course that I am not taking.
 

What is a linear control system?

A linear control system is a system where the output is directly proportional to the input, meaning that the response of the system is linear. This type of system is commonly used in engineering and science for controlling and regulating processes and devices.

What is controllability in a linear control system?

Controllability in a linear control system refers to the ability to manipulate the system's output by controlling the input. In other words, it is the degree to which a system can be controlled by its inputs.

How is controllability measured in a linear control system?

Controllability is measured using the controllability matrix, which is a mathematical tool used to determine the degree of controllability of a system. The matrix is calculated using the system's state-space representation and determines whether the system is fully controllable, partially controllable, or uncontrollable.

What factors affect the controllability of a linear control system?

The controllability of a linear control system is affected by several factors, including the number of control inputs, the dynamics of the system, and the placement of the control inputs. A system with more control inputs and faster dynamics is easier to control, while a system with poorly placed control inputs may be more difficult to control.

Why is controllability important in a linear control system?

Controllability is important in a linear control system because it determines the system's ability to achieve a desired output. A fully controllable system allows for precise and accurate control, while an uncontrollable system may lead to unstable or unpredictable behavior.

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