On a discrete game problem with non-convex control vectograms

In a normed space of finite dimension, a discrete game problem with fixed duration is considered. The terminal set is determined by the condition that the norm of the phase vector belongs to a segment with positive ends. In this paper, a set defined by this condition is called a ring. At each moment, the vectogram of the first player's controls is a certain ring. The controls of the second player at each moment are taken from balls with given radii. The goal of the first player is to lead a phase vector to the terminal set at a fixed time. The goal of the second player is the opposite. In this paper, necessary and sufficient termination conditions are found, and optimal controls of the players are constructed.

Partial stability of linear systems with respect to a given component of the phase vector

A new geometric approach to the study of the partial stability of linear systems is proposed, which is based on the application of the geometric theory of linear operators. Using the theory of conjugate spaces and conjugate linear operators, bases are constructed in which the system under study takes the canonical form. A cyclic subspace with respect to the conjugate linear operator is considered. A basis is constructed for the dual space of a linear operator, in which its matrix takes the canonical form. This basis corresponds to the dual basis of the original linear space. Then, in a pair of bases of dual spaces the system under study takes the simplest form. The geometric properties of the system are realized using a non-singular linear transformation in the space of a part of the components of the system’s phase vector. This allows us to decompose the system under study in order to obtain necessary and sufficient conditions for the partial stability of the linear system. In an equivalent system, an independent subsystem is distinguished, whose nature of stability determines the behavior of the investigated component of the original system’s phase vector. The relationship between the partial stability of the system and the existence of an invariant subspace of a linear operator characterizing the dynamics of the system is established. The canonical form of the resulting subsystem makes it easy to exclude auxiliary variables and write an equation equivalent to this system. The application of the obtained results to the solution of the problem of partial stability for linear systems with constant coefficients belonging to the classes of ordinary differential equations, discrete systems and systems with deviating argument is shown. An example of a linear system of differential equations is given to illustrate the result obtained.

Discrete game problem with ring-shaped terminal set

In a normed space of finite dimension a discrete game problem with fixed duration is considered. The terminal set is determined by the condition that the norm of the phase vector belongs to a segment with positive ends. In this paper, a set defined by this condition is called a ring. The aim of the first player is to lead a phase vector to the terminal set at fixed time. The aim of the second player is the opposite. In this paper, optimal controls of the players are constructed. Computer simulation of the game process is performed. A modification of the original problem, in which at an unknown time there is a change in the dynamics of the first player, is considered.

Synchronization of oscillations for coupled Van der Pol oscillators by output

A number of automatic control tasks, in particular, the synchronization of trajectories, the tracking task, control by a reference system are associated with the synthesis of control algorithms for dynamic cascade systems, which are a set of interconnected active subsystems. In this paper, the oscillation synchronization problem is considered for two Van der Pol coupled oscillators. It is assumed that the driven subsystem depends on the external control action, in addition, the phase vector is not fully known. On the first step the solution of the problem of synchronization in the form of state feedback is written. The aim of the work is to find the synchronizing control in the form of feedback on the state estimation. Such a formulation is relevant, since for many practical applications of control theory, a typical situation is when the complete state vector of the system is unknown and only some of the functions of the state variables - the outputs of the system are accessible to measurement. One can try to use the control law obtained from feedback by replacing the state with its estimate obtained by observer - a special dynamical system whose state eventually approaches (asymptotically or exponentially) to the state of the original system. In this case a question arises whether such control will be solving the synchronization problem. In mathematical control theory, in particular for the stabilization problem of dynamical systems, similar questions constitute the content of the known principle of separation. For the observation problem solving the apparatus of the method of synthesis of auxiliary invariant relations for constructing a nonlinear observer was used. In accordance with this approach a nonlinear observer is constructed for the system under consideration, which ensures the exponential estimates of the phase vector. It is further shown that the use in the control law instead of the state of the system of its evaluation under simultaneously solving the problems of observation and synchronization leads to the local solution of the problem under consideration.

Side Information Mitigation with new Phase Sequence for Hybrid PTS Scheme

This paper focus on presenting a new phase sequence based hybrid PTS approach for OFDM systems to mitigate the problem of PAPR without use of side information. The proposed approach is the exponential constant multiplied for each phase vector so that it can accommodate more number of sub carriers. The modified phase vectors are then multiplied to the original phase vectors so as to obtain a modified version of phase vectors that includes an offset. Experimental results shows that the PAPR is reduced about 04~0.8 db when compared against the traditional PTS scheme deployed for Hybrid systems.

Linear Pursuit Differential Game under Phase Constraint on the State of Evader

We consider a linear pursuit differential game of one pursuer and one evader. Controls of the pursuer and evader are subjected to integral and geometric constraints, respectively. In addition, phase constraint is imposed on the state of evader, whereas pursuer moves throughout the space. We say that pursuit is completed, if inclusiony(t1)-x(t1)∈Mis satisfied at somet1>0, wherex(t)andy(t)are states of pursuer and evader, respectively, andMis terminal set. Conditions of completion of pursuit in the game from all initial points of players are obtained. Strategy of the pursuer is constructed so that the phase vector of the pursuer first is brought to a given set, and then pursuit is completed.