Control system, method to increase a bandwidth of a control system, and lithographic apparatus

A method to increase a bandwidth of a control system providing an output signal on the basis of an input signal, the control system comprising a linear integrator and a linear gain parallel to the linear integrator, wherein the linear integrator is arranged to provide a first control signal and wherein the linear gain is arranged to provide a second control signal, the method comprising:

• switching to an integrator mode when the value of the second control signal passes zero;
• and
• switching to the gain mode when:
• eu<kh−1u2,
• wherein e is the input signal of the controller, u is the output signal of the controller in the integrator mode, and kh is a gain of the linear gain,
• wherein in the integrator mode the first control signal is used as the output signal, and
• wherein in the gain mode the second control signal is used as the output signal, and
• wherein the output signal of the controller controls a position or speed variable of one or more of:
• a stage,
• a lens,
• a handler,
• a stage position system, or
• a radiation source system.

A lithographic apparatus comprising:

• a support constructed to support a patterning device, the patterning device being capable of imparting a radiation beam with a pattern in its cross-section to form a patterned radiation beam;
• a substrate table constructed to hold a substrate;
• and
• a projection system configured to project the patterned radiation beam onto a target portion of the substrate,
• wherein the lithographic apparatus comprises a control system to control a variable of the lithographic apparatus, the control system comprising:
• a controller arranged to provide an output signal on the basis of an input signal, the controller comprising:
• a linear integrator arranged to provide a first control signal;
• a linear gain in parallel to the linear integrator and arranged to provide a second control signal;
• and
• a selector arranged to switch between an integrator mode in which the first control signal is used as the output signal of the controller and a gain mode in which the second control signal is used as the output signal of the controller,
• wherein the selector is arranged to:
• switch to the integrator mode when the value of the second control signal passes zero;
• and
• switch to the gain mode when:
• eu<kh−1u2,
• wherein e is the input signal of the controller, u is the output signal of the controller, and kh is the gain of the linear gain, and
• wherein the output signal of the controller controls a position or speed variable of one or more of:
• a stage of the lithographic apparatus,
• a lens of the lithographic apparatus,
• a substrate handler,
• the substrate table;
• a stage position system, or
• a radiation source system.

A control system, comprising:

• a controller arranged to provide an output signal on the basis of an input signal, the controller comprising:
• a linear integrator arranged to provide a first control signal;
• a linear gain in parallel to the linear integrator and arranged to provide a second control signal;
• and
• a selector arranged to select the first control signal or the second control signal as the output signal of the controller,
• wherein the selector is arranged to:
• switch to the first control signal as the output signal when the value of the second control signal passes zero;
• and
• switch to the second control signal as the output signal when an absolute value of the first control signal is the same or larger than an absolute value of the second control signal, and
• wherein the output signal of the controller controls a position or speed variable of one or more of:
• a stage,
• a lens,
• a handler,
• a stage position system, or
• a radiation source system.

A method to increase a bandwidth of a controller providing an output signal on the basis of an input signal, the controller comprising a linear integrator and a linear gain parallel to the linear integrator, wherein the linear integrator is arranged to provide a first control signal, and wherein the linear gain is arranged to provide a second control signal, the method comprising:

• selecting the first control signal as the output signal of the controller when the value of the second control signal passes zero;
• and
• switching to the second control signal as the output signal of the controller when an absolute value of the first control signal is the same or larger than an absolute value of the second control signal, and
• wherein the output signal of the controller controls a position or speed variable of one or more of:
• a stage,
• a lens,
• a handler,
• a stage position system, or
• a radiation source system.