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

• A new sliding control strategy for nonlinear system is derived.
• Optimal control is reformulated to be a set of differential algebraic equation.
• The Lie-group GL(n, RR) is developed to find control force.
• The Lie group DAE method is effective to solve control problem.
• The controlled system is chattering-free with sliding surface being preserved very accurate.

For a control problem of nonlinear system ẋ=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: L1(x,t)=A0e-αtL1(x,t)=A0e-αt, and meanwhile the control force is bounded by |u|⩽umax|u|⩽umax. Then, we develop an implicit GL(n,R)GL(n,R) Lie-group DAE (LGDAE) method to find u(t)u(t) by solving the DAEs: ẋ=f(x,u,t) and L1(x,t)-A0e-αt=0L1(x,t)-A0e-αt=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.