Developments in nonlinear control theory have made it possible to design controllers for systems having non-smooth nonlinearities in their dynamics. Hydraulic systems that use inexpensive proportional valves are examples of such systems, where non-smooth nonlinearities arise due to valve geometry and spool imperfections. Without a proper valve model, however, nonlinear analysis and control of these hydraulic systems is not possible. We have developed nonlinear equations for a generic proportional valve model and have used them to obtain simplified flow rate expressions under generally accepted assumptions. These equations related a set of geometric spool properties and physical model variables to the flow rate through the valve ports. The development focuses on obtaining a single set of flow rate equations for the cases of critical center, overlapped, and underlapped proportional valves. The proportional valve model equations are useful for system simulation and for nonlinear controller design.

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