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This paper proposes an uncertainty and disturbance estimator (UDE)-based controller for nonlinear systems with mismatched uncertainties and disturbances, integrating the UDE-based control and the conventional backstepping scheme. The adoption of the backstepping scheme helps to relax the structural constraint of the UDE-based control. Moreover, the reference model design in the UDE-based control offers a solution to address the “complexity explosion” problem of the backstepping approach. Furthermore, the strict-feedback form condition in the conventional backstepping approach is also relaxed by using the UDE-based control to estimate and compensate “disturbance-like” terms including nonstrict-feedback terms and intermediate system errors. The uniformly ultimate boundedness of the closed-loop system is analyzed. Both numerical and experimental studies are provided.

Using Laplace transforms we extend the Economic Production Quantity (EPQ) model by analysing cash flows from a Net Present Value (NPV) viewpoint. We obtain an exact expression for the present value of the cash flows in the EPQ problem. From this we are able to derive the optimal batch size. We obtain insights into the monotonicity and convexity of the present value of each of the cash flows, and show that there is a unique minimum in the present value of the sum of the cash flows in the extended EPQ model. We also obtain exact point solutions at several values in the parameter space. We compare the exact solution to a Maclaurin series expansion and show that serious errors exist with the first order approximation when the production rate is close to the demand rate. Finally we consider an alternative formulation of the EPQ model when the opportunity cost of the inventory investment is made explicit.