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Predictive Control with Actuators Constraints using Lagrangian Multipliers

The Generalized Predictive Control (GPC) is based on a quadratic cost index. If it is minimized in order to get the minimum error within a predictive horizon, a linear controller is obtained. Problems appear because real processes present constraints (nonlinearities such as deadzone, saturation, etc). In general, this problem is transformed into a quadratic programming problem (QP), in which a quadratic cost index is minimized subject to linear constraints. It would be very interesting to use an analytical method for solving it instead of using a numerical method whose computational cost could be high. In this work, Lagragian multipliers will be used in order to solve the quadratic problem because it presents several advantages. So, when the control horizon is small, i.e. Nu = 2, the method is easy to implement. It is possible to use the method for greater control horizons, but the implementation results more difficult. Besides, an analytical solution can be obtained and with low computational cost. Finally, it is possible to obtain the global minimum instead of the local one numerical methods sometimes give.