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Many nonlinear physical phenomena, including various power systems, can be modeled, analyzed, and controlled using the framework of port-Hamiltonian systems. Moreover, this framework can offer more advanced methods to cope with modern challenges in the energy sector. This paper presents a comprehensive and systematic survey of recent studies on the application of the port-Hamiltonian approach to power systems. Over a hundred of relevant research works are reviewed to show the vast capabilities of this approach and point out its possible gaps. The works are classified according to the type of power systems under study. The analysis of the articles shows that the vast majority of them are dedicated to controller design, a much smaller part of the works deals with the modeling and stability issues, and only a few consider the problem of optimal control. Moreover, the paper discusses current challenges and future trends in this direction.

Input-output linearization by state feedback is applied to a flux-controlled active magnetic bearing (AMB) system, operated in the zero-bias mode. Two models of the AMB system are employed. The first one is described by the third-order dynamics with a flux-dependent voltage switching scheme, whereas the second one is the fourth-order system, called self-sensing AMB, since it does not require the measurement of the rotor position. In the case of that system we had to find the flat outputs to guarantee its stability. The proposed control schemes are verified by means of numerical simulations performed within the Matlab environment.

The paper describes a symbolic computation software package NLControl, dedicated to the study of various modelling, analysis, synthesis problems for nonlinear control systems. Most of the developed functions are based on algebraic approach of differential forms, and on the theory of non-commutative polynomial rings. For a few control problems an alternative solution is implemented in terms of functions algebra. The package addresses both continuous-and discrete-time systems as well as systems, defined on homogeneous time scales. Comparison of NLControl and Mathematica built-in nonlinear control functions is given.

The paper addresses a state space realization problem of a set of higher order delta-differential input–output equations, defined on a homogeneous time scale. The algebraic framework of differential one-forms is applied to formulate necessary and sufficient solvability conditions. This approach applies the total differential operator to analytic system equations to obtain the infinitesimal system description in terms of one-forms. This representation can be converted into polynomial system description by interpreting the polynomial indeterminate as the delta derivative acting on one-forms. The system description in terms of two matrices over skew polynomial ring is then used to derive explicit formulas for the differentials of state coordinates that significantly simplify the calculations. The formulas are found from the left quotients computed by the left Euclidean division algorithm.

For a nonlinear input-output equation defined on homogeneous time scale, an explicit formula is given allowing to compute recursively the differentials of the state coordinates. The recursive formula is based on the Euclidean division algorithm.

The purpose of this paper is to present necessary and sufficient condition for irreducibility of discrete-time nonlinear multi-input multi-output system which extends directly the corresponding condition for the linear case. The condition is presented in terms of the greatest common left divisor of two polynomial matrices related to the input-output equations of the system. The basic difference is that unlike the linear case the elements of the polynomial matrices belong to a non-commutative polynomial ring. This condition provides a bases for finding the equivalent minimal irreducible representation of the i/o equations which is a suitable starting point for constructing an observable and accessible state space realization.