Sliding mode control method having terminal convergence in finite time

A closed-loop sliding mode control method for a system of the form x=.function.(x)+u with convergence to equilibrium, robustness to parametric uncertainties, and stability upon the rate of change of uncertainties over a sliding surface, where x is the system state and u is the control for a closed-loop trajectory through a sliding surface s=e+.lambda.e=0 where e is the trajectory error (x-x.sub.d), and x.sub.d is the desired trajectory, and .lambda. is a positive constant, when information about said systems is not accurate, thus requiring the use of a derived estimate .function.(x) for .function.(x), by applying to said system a control in the form ##EQU87## where .alpha. is a positive constant and .beta..sub.n, .beta..sub.d are chosen parameters equal to (2i+1) where i belongs to a set of positive integers chosen for .beta..sub.n and .beta..sub.d so that both of said parameters will be odd integers and .beta..sub.d >.beta..sub.n, which, when substituted in said control method of the aforesaid form x=.function.(x)+u, yields a closed-loop sliding mode control method of the form ##EQU88## in which an initial state s.sub.i is to be always zero, and where .DELTA.=.function.(x)-.function.(x) and which yields a control equation ##EQU89## where .lambda. is a chosen parameter, .delta..sub.n, .delta..sub.d are as before chosen parameters equal to (2i+1) where i belongs to a set of positive integers chosen for .delta..sub.n and .delta..sub.d so that both of said parameters will be odd integers and .delta..sub.d >.delta..sub.n, whereby convergence to a steady state is achieved in finite time.