In this work, a novel method to design optimal linear surfaces for the sliding model control of non-square, nonlinear systems is developed by incorporating principles from nonlinear model predictive control which have been widely applied to nonlinear systems. The time-invariant parameters of the sliding mode control surfaces are optimally selected to achieve some performance objective using a predictive control framework. The optimization of these surface parameters is subject to constraints that arise from the stability conditions imposed by the sliding mode control law and the physical limits on the system such as control input saturation. The procedure is demonstrated experimentally using an underactuated, open-loop unstable, fast dynamic system.

Model predictive control allows for optimal performance based on a specified objective. However, it also requires on-line optimization resulting in long computation times that can limit its applicability in many fast dynamic systems. Sliding mode control is a stabilizing nonlinear controller that computes quickly, so it can be used for systems with fast dynamics, but its performance is a complex function of the system and surface parameters. We propose the use of predictive control to optimally redefine the surface parameters of the sliding mode controller in a multi-rate, cascade control configuration. The time-invariant parameters of the linear sliding mode control surfaces are determined by minimizing a specified performance objective under sliding mode control. Constraints in this minimization include satisfying the controller stability criteria on the surface parameters, the maximum and minimum limits on the control, and any dynamic restrictions on the state trajectory. The result is a nonlinear optimization problem similar to that posed in nonlinear predictive control except that the decision variables are the linear surface parameters for the sliding mode controller as opposed to the actual control and state trajectory.

There are a number of advantages to this approach. Almost any performance objective can easily be incorporated into the sliding mode control framework for nonsquare systems. The resulting nonlinear dynamic optimization problem is of low dimension leading to a more reliable solution within a small sample period. Undesirable state trajectories can be eliminated through the use of dynamic state constraints in the optimization problem. Finally, a stabilizing sliding mode control is always present to control the system as the surface parameters are continually re-optimized to achieve the desired performance objective.