System for controlling a position of a mass

A lithographic apparatus comprising:

• a radiation system for providing a beam of radiation;
• a support structure for supporting a patterning device, said patterning device serving to pattern said beam of radiation according to a desired pattern;
• a substrate holder for holding a substrate;
• a projection system for projecting said patterned beam onto a target portion of said substrate;
• a control system to control a position of a movable object associated with said lithographic apparatus by applying a control force to said movable object in accordance with a desired mass acceleration, said movable object comprising at least one of said support structure and said substrate holder, said control system comprising:
• a first input that receives a signal based on a desired position of said movable object;
• a second input that receives a feedback signal indicative of an actual position of said movable object;
• a comparator unit configured to determine a signal indicative of a difference between said desired movable object position and said actual movable object position;
• a control unit configured to produce a signal indicative of said control force based on said difference between said desired movable object position and said actual movable object position;
• an estimator unit configured to calculate an estimated relation between said control force and status information of said movable object, said movable object status information comprising an indication of at least one of a position of said movable object, a velocity of said movable object, and an acceleration of said movable object;
• and
• a third input that receives a feed-forward signal indicative of said desired movable object acceleration,
• wherein said control system determines said control force based on said estimated relation and said desired movable object acceleration.

The lithographic apparatus of 15, wherein said estimator unit calculates said estimated mass by:

• 10 m ^ = ∑ i = 1 n ⁢ ⁢ λ n - i ⁡ ( a i · f i ) ∑ i = 1 n ⁢ ⁢ λ n - i ⁡ ( a i ) 2
• where:
• {circumflex over (m)}=estimated mass
• ai (i=1, 2, 3, 4, . . . , n) is an acceleration sample.
• fi (i=1, 2, 3, 4, . . . , n) is a control force sample.
• λ is a forgetting factor.

A method of controlling a position of a mass by applying a control force to said mass in accordance with a desired mass acceleration, said method comprising:

• determining information indicative of a difference between a desired mass position and an actual mass position;
• producing information indicative of said control force based on said difference between said desired mass position and said actual mass position;
• obtaining status information of said mass indicative of at least one of said position of said mass, a velocity of said mass, and an acceleration of said mass;
• calculating an estimated relation between said control force and said mass status information;
• and
• determining said control force based on said estimated relation and said desired mass acceleration.

A device manufacturing method comprising:

• providing a substrate that is at least partially covered by a layer of radiation-sensitive material, said substrate being held by a substrate holder;
• providing a beam of radiation;
• configuring said beam of radiation with a desired pattern along its cross-section, said desired pattern provided by a patterning device supported by a support structure;
• projecting said patterned beam of radiation onto a target portion of said layer of radiation-sensitive material;
• and
• controlling a position of at least one of said substrate holder and said support structure by applying a control force to said at least one of said substrate holder and said support structure in accordance with a desired acceleration, wherein said position control comprises:
• determining information indicative of a difference between a desired position of said at least one of said substrate holder and said support structure and an actual position of said at least one of said substrate holder and said support structure;
• producing information indicative of said control force based on said difference between said desired position of said at least one of said substrate holder and said support structure and said actual position of said at least one of said substrate holder and said support structure;
• obtaining status information of said at least one of said substrate holder and said support structure indicative of at least one of said position of said at least one of said substrate holder and said support structure, a velocity of said at least one of said substrate holder and said support structure, and an acceleration of said at least one of said substrate holder and said support structure;
• calculating an estimated relation between said control force and said status information;
• and
• determining said control force based on said estimated relation and said desired acceleration.