Multi-input sliding mode control of nonlinear systems with uncertain affine dependence on the control

In the extension to multi-input non linear uncertain systems of the sliding mode methodology a crucial role is played by the matrix pre multiplying the control in the dynamic equation of the sliding output. If this matrix is perfectly known and invertible it is possible to transform a multi-input sliding mode control problem in an almost decoupled set of single input problems. If this matrix is uncertain nothing can be done in general, and the investigation is oriented to find condition ensuring the feasibility of control strategies in a progressively increasing set of uncertain matrices. In the case of uncertain but constant matrices it is possible, in principle, to manage the case in which the matrix in question is invertible. The corresponding adaptive or switching strategy suffer of the curse of dimensionality of the so called Unmixing Set. In this paper the case of time and state varying uncertain matrix is dealt with. The more general class of such a matrices for which there is, at least locally, a solution of the problem is found. The introduction of artificial integrators in the output channel (the integral sliding mode control methodology) allows the practical implementation of the control law without requiring the a priori knowledge of parameters featured by the solution of a relevant nonlinear Lyapunov equation.