Controllable system?











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So I have my equation of motion for an inverted pendulum on a cart: (linearised about the upright position and X denoting the position of the cart):



$$ ddot{theta}-theta = ddot{X} $$



Rather than dealing with a 4x4 system I’d like to set $ddot{X} = a theta +b dot{theta}$ but now I’m struggling to write my system in the form:



$$dot{x} = Ax + Bu$$



as I can’t determine what my control Bu is now that I’m not applying a direct force to the cart?






linear-algebra dynamical-systems control-theory classical-mechanics







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    So I have my equation of motion for an inverted pendulum on a cart: (linearised about the upright position and X denoting the position of the cart):



    $$ ddot{theta}-theta = ddot{X} $$



    Rather than dealing with a 4x4 system I’d like to set $ddot{X} = a theta +b dot{theta}$ but now I’m struggling to write my system in the form:



    $$dot{x} = Ax + Bu$$



    as I can’t determine what my control Bu is now that I’m not applying a direct force to the cart?






    linear-algebra dynamical-systems control-theory classical-mechanics







    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      So I have my equation of motion for an inverted pendulum on a cart: (linearised about the upright position and X denoting the position of the cart):



      $$ ddot{theta}-theta = ddot{X} $$



      Rather than dealing with a 4x4 system I’d like to set $ddot{X} = a theta +b dot{theta}$ but now I’m struggling to write my system in the form:



      $$dot{x} = Ax + Bu$$



      as I can’t determine what my control Bu is now that I’m not applying a direct force to the cart?






      linear-algebra dynamical-systems control-theory classical-mechanics







      share|cite|improve this question













      So I have my equation of motion for an inverted pendulum on a cart: (linearised about the upright position and X denoting the position of the cart):



      $$ ddot{theta}-theta = ddot{X} $$



      Rather than dealing with a 4x4 system I’d like to set $ddot{X} = a theta +b dot{theta}$ but now I’m struggling to write my system in the form:



      $$dot{x} = Ax + Bu$$



      as I can’t determine what my control Bu is now that I’m not applying a direct force to the cart?






      linear-algebra dynamical-systems control-theory classical-mechanics




      linear-algebra dynamical-systems control-theory classical-mechanics






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          You are only using one half of the equations of motion. The other half contains an external force, which will be your input $u$.






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            You are only using one half of the equations of motion. The other half contains an external force, which will be your input $u$.






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              You are only using one half of the equations of motion. The other half contains an external force, which will be your input $u$.






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                You are only using one half of the equations of motion. The other half contains an external force, which will be your input $u$.






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                You are only using one half of the equations of motion. The other half contains an external force, which will be your input $u$.







                share|cite|improve this answer












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                answered 8 hours ago









                Kwin van der Veen

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