A new sliding control strategy for nonlinear system solved by the Lie-group differential algebraic equation method

For a control problem of nonlinear system x ̇ = f ( x , u , t ) , the optimal control by minimizing a performance index is reformulated to be a set of differential algebraic equations (DAEs) with the Lagrangian being partially replaced by an exponentially time-decaying constraint: L 1 ( x , t ) = A 0 e - α t , and meanwhile the control force is bounded by | u | ⩽ u max . Then, we develop an implicit GL ( n , R ) Lie-group DAE (LGDAE) method to find u ( t ) by solving the DAEs: x ̇ = f ( x , u , t ) and L 1 ( x , t ) - A 0 e - α t = 0 . Similarly, we propose a new sliding mode control (SMC) strategy by using the LGDAE to solve the control force, where in addition to the equivalent control force we add a compensated control force which is used to quickly steer and continuously enforce the state trajectory on the sliding surface. This novel SMC is robust and is chattering-free for regulator problem and finite-time tracking problem of nonlinear systems. Furthermore, we combine the above two methods as being a two-stage controller for the forced nonlinear Duffing oscillator by stabilizing it to an equilibrium point. The present SMC together with the LGDAE is also used to stabilize the state trajectory of some uncertain chaotic systems to a desired state point. Its robustness against uncertainty is obvious.