Methodology of Mathematical Error-Based Tuning Sliding Mode Controller

Design a nonlinear controller for second order nonlinear uncertain dynamical systems is one ofthe most important challenging works. This paper focuses on the design of a chattering freemathematical error-based tuning sliding mode controller (MTSMC) for highly nonlinear dynamicrobot manipulator, in presence of uncertainties. In order to provide high performance nonlinearmethodology, sliding mode controller is selected. Pure sliding mode controller can be used tocontrol of partly known nonlinear dynamic parameters of robot manipulator. Conversely, puresliding mode controller is used in many applications; it has an important drawback namely;chattering phenomenon which it can causes some problems such as saturation and heat themechanical parts of robot manipulators or drivers.In order to reduce the chattering this research is used the switching function in presence ofmathematical error-based method instead of switching function method in pure sliding modecontroller. The results demonstrate that the sliding mode controller with switching function is amodel-based controllers which works well in certain and partly uncertain system. Pure slidingmode controller has difficulty in handling unstructured model uncertainties. To solve this problemapplied mathematical model-free tuning method to sliding mode controller for adjusting the slidingsurface gain (λ ). Since the sliding surface gain (λ) is adjusted by mathematical model free-basedtuning method, it is nonlinear and continuous. In this research new λ is obtained by the previous λmultiple sliding surface slopes updating factor α. Chattering free mathematical error-basedtuning sliding mode controller is stable controller which eliminates the chattering phenomenonwithout to use the boundary layer saturation function. Lyapunov stability is proved inmathematical error-based tuning sliding mode controller with switching (sign) function. This controller has acceptable performance in presence of uncertainty (e.g., overshoot=0%, risetime=0.8 second, steady state error = 1e-9 and RMS error=1.8e-12).