Passivity of nonlinearity incremental systems: application to PI stabilization of nonlinear RLC circuits
In the paper, the authors investigate a class of nonlinear systems described by the set of vector nonlinear equations \(dx/dt=f(x) + g\cdot u\), \(y=h(x)\). Here, \(x\), \(y\) and \(u\) are column vectors containing functions of time \(t\), \(f\) and \(h\) are nonlinear vector mappings and \(g\) is a constant matrix of full rank. All matrices have adequate sizes. If \(x_0\) is a fixed non-zero equilibrium point, one can associate to the initial system a new system where the evolution is described in terms of the deviations from the equilibrium point, i.e. \(x-x_0\). This last system is called the incremental system (IS). It is worth pointing out that the behavior of the IS is of importance in nonlinear circuit theory. A standard result states that in the class of linear time-invariant systems (LTIS), if the initial system is passive, then its associated IS has the same property. Then the following nontrivial question can be asked: Under what conditions is the implication which is true for the class LTIS, also true for nonlinear systems? Some important results pertaining to this question are given in the present paper for the class described by the aforementioned system. The main quantity which here is under attention is precisely the storage function, denoted by \(H(x)\). For the class of LTIS such a function exists and is non-negative. The results obtained in the paper are expressed in three propositions. In Proposition 1, the canonic field taken into account is assumed to be passive, to have a non-negative storage function and to satisfy a certain inequality involving the variables of the associated IS. Then its IS is also passive and the form of its \(H(x)\) is given. Proposition 2 gives conditions under which the storage function has a unique global minimum. It is then shown that, under the conditions contained in Proposition 1, the system can be globally stabilized with a linear PI control. In order to avoid some difficult calculations of \(H(x)\) for an RLC network, it is assumed that current controlled resistors are in series with inductors and voltage-controlled resistors are in parallel with capacitors. Finally, Proposition 3 gives the main result stating that an RLC network containing a) passive inductors and capacitors whose energy functions are twice continuously differentiable and strictly convex, b) passive resistors whose characteristic functions are monotone non-decreasing, has the following properties: its IS is also passive and the whole system can be globally stabilized with a PI controller. We think the paper gives a significant contribution in the areas of incremental systems and global stabilization. It could open new ways in nonlinear circuit theory.
scientific article; zbMATH DE number 5504340