A General Approach to the Solution of the Bounded Control Synthesis Problem in a Controllability Problem

Feedback Control Design for Systems with x-Discontinuous Right-Hand Side

“…with the constraints u ∈ ⊂ R. Describe the main idea of the controllability function method from [8]. Assume that the ancillary function = (x) (controllability function) exists.…”

“…The controllability function method (based on the idea of Lyapunov function) was suggested in 1979 for solving the admissible feedback synthesis problem for systems with smooth right-hand sides [8][9][10].…”

Stepwise synthesis of constrained controls for single input nonlinear systems of special form

“…, nm, one by one from left to right and take into account the following remark: if the column q i (x) = ad k a b j (x) is deleted then all columns of the form q i+ms (x) = ad k+s a b j (x) for all s ≥ 1 such that i + ms ≤ nm should be deleted as well. This algorithm is the same as the algorithm for linear controllable systems with a multidimensional control given in the paper [18] and is analogous to the algorithm given in [4,5,6] for linearization of affine systems with multidimensional control.…”

“…Our main tool is the controllability function method proposed in [18,19] for solving the synthesis problem of admissible positional constrained control. Later it was developed for different classes of systems and different statements of the synthesis problem, for example, for infinite systems [20], for systems in a finite-dimensional space with constraint on a control [21] and its derivatives [22] which called inertial control in [23, p.292] and so on.…”

“…In this section we construct a control which transfers any initial point to a subspace. This is done by use of the controllability function method [18,19].…”