Integral Sliding Mode Control of Multi-input Nonlinear Uncertain Non-affine Systems

This chapter is devoted to examine the control of multi-input uncertain non affine systems. Despite the great importance of this subject the literature has relatively few contributions, often not fully satisfactory. The authors attempt to identify the reasons for this in the difficulties of dealing with uncertain and time and state dependent matrices multiplying the control in the sliding output dynamics called High Frequency Gain matrices. For constant nonsingular matrices it would seem sufficient to know a set of matrices, the so-called unmixing set, and use enumerative techniques to identify on line the element which serves to make effective the control algorithm. The problem up to now is the cardinality of this set which is the product of factorial and exponential of the dimension of the matrix making impractical this approach even for low scale systems. The issue becomes more complicated if the matrix HFG is state-dependent and time. In this case the best that, in the current state of the research, can be done is to assume the above matrix with eigenvalues in the positive half-plane for any value of its arguments, and adopt a particular technique called integral sliding mode. Generalization to nonsingular matrices seems to be a very hard task since everything must be time and state varying.