Sliding mode control in the presence of input delay: A singular perturbation approach

Sliding Mode Control (SMC) in the presence of small, unavoidable input delay as may be present in controller implementation is studied. Linear systems with bounded matched disturbances and uncertain system matrices are considered, where input delay in the SMC will produce oscillations or potentially even unbounded solutions. Without a priori knowledge of the bounds on the state-dependent terms as required by existing methods  , the design objective is to achieve ultimate boundedness of the closed-loop system with a bound proportional to the delay and disturbance bounds. This is a non-trivial problem because the relay gain depends on the state bound, whereas the latter bound depends on the relay gain. A controller with linear gain proportional to the scalar 1μ is proposed, which for small enough μ>0μ>0 produces a closed-loop singularly perturbed system and yields the desired ultimate bound. A constructive Linear Matrix Inequality (LMI)-based solution for evaluation of both the design parameters and the ultimate bound is derived. The superiority of the proposed control over existing methodologies that ignore input delay within the design is demonstrated through an example.