Sequential synthesis of the time-optimal control in real time

The paper considers the sequential synthesis method for time-optimal control of linear systems. The method is based on piecewise constant finite controls that ensure approximate solutions for time-optimal control problems. A sequence of finite controls is thereafter transformed into the optimal control. The appropriate computations are reduced to a sequence of linear algebraic equation problems and the integration of a matrix differential equation over the intervals of control switching points and final point change. It is proven that the sequence of finite controls converge to the optimal control. The sliding mode conditions are obtained, as well as the control structure modifications for the motions on switching manifolds. The initial approximations reducing considerably computational complexity are considered. The computational algorithm, together with the modelling and numerical results, are presented.     

On the relationship between the problem of anisotropy-based controller synthesis and classical optimal control problems

Consideration is given to the interrelation between different problems of \(\mathcal{H}_2 \)-and \(\mathcal{H}_\infty \)-optimization and their relationship with the theory of anisotropy-based control. It was shown that the anisotropy-based controller of the full order for the completely defined linear system minimizes the mean amount of the transmitted information between the input and output of the closed system.     

Parametric synthesis method of integral systems on the basis of root locus curves of Kharitonov polynomials

The method is developed for the synthesis of a stable set of characteristic polynomials of an interval dynamic system on the basis of the initial unstable set with the use of the system model in the form of a free root portrait. The synthesis is carried out by changing the boundaries (adjustment) of the interval of uncertainty of a free polynomial term, which enables affording the stability without a change of the configuration of the root portrait of the system. As a criterion of the adjustment, the distance is assigned that is measured along trajectories of the root portrait; in particular, a new polynomial can be defined by the nearest polynomial to the prescribed one with due regard for requirements of the quality of the system.     

Decomposition of problems of optimal control and estimation for discrete systems with fast and slow variables

Consideration was given to the linear-quadratic problem of optimal control for the discrete linear system with fast and slow variables under incomplete information about system state. Decomposition of the discrete matrix Riccati equations was carried out. The proposed decomposition algorithm relies on a geometrical approach using the properties of the invariant manifolds of slow and fast motions of the nonlinear multirate discrete systems as basis. The splitting transformation was constructed in the form of asymptotic decomposition in the degrees of a small parameter.     

Systems with a switch and estimation of absolute stability region

Two methods are proposed for estimation of absolute stability region in the space of parameters. The first method uses nontrivial necessary conditions, which follow from stability of switching systems. The second method is a modification of the harmonic balance approach; it is used for approximate determination of existence of periodic modes that arise when stability is lost.     

On instability of control systems for a certain class of objects and methods of its elimination

In practice of design of control systems, the cases occur when some of the roots of the transfer function of a controllable object are disposed on the imaginary axis of the complex plane. The optimal controller constructed for such objects, despite its realizability, will not afford the robustness properties in the system. The methods of removal of this phenomenon are given. The comparative estimate of the solution of this problem is provided both in the space of states and in the input-output relations (in the space of operators).     

Approximate solution of the control problem of supplies with many intervals and concave cost functions

The problem of searching for a cost-minimum plan of supplies of uniform products to one consumer is considered. The set of admissible intervals of the supply volume and concave cost functions of supplies within each interval are preassigned for each supplier. The totally polynomial ?-approximate algorithm for the given problem and the pseudopolynomial exact algorithm for its partial case are suggested.     

Joint detection and estimation of the Yule-Furry processes

An algorithm of joint detection of the Yule-Furry birth process and estimation of its start instant was designed. Results of the numerical studies were presented. Features of the solution of the problems of joint detection and estimation of this class were revealed.     

Exponential dissipativeness of the random-structure diffusion processes and problems of robust stabilization

Consideration was given to the class of systems described by a finite set of the controllable control-affine diffusion Ito processes with stepwise transitions defined by the evolution of the uniform Markov chain (Markov switchings). For these systems, the notion of exponential dissipativity was introduced, and its theory was developed and used to estimate the possible variations of the output feedback law under which the system retains its robust stability. For the set of linear systems with uncertain parameters, proposed was a two-step procedure for determination of the output feedback control based on comparison with the stochastic model and providing their simultaneous robust stabilization. At the first step, the robust stabilizing control is established by means of an iterative algorithm. Then, the possible variations of the feedback law for which the robust stability is retained are estimated by solving a system of matrix linear inequalities. An example was presented.     

Quasi-invariant control: Design and functional capabilities

The questions of physical realizability, stability, and control errors were considered by the example of a simplest mathematical model of the linear minimum phase system of quasi-invariant control. The problem of designing the quasi-invariant control was solved on this basis, thus providing the opportunity to extend control by designing a nonlinear system combining the quasi-invariant and classical control strategies.     

Invariance and astatism in systems not measuring perturbations

Control systems not being able to measure perturbations are considered. A necessary and sufficient condition for the astatism with respect to external perturbations to be possible is formulated basing on the comparison of absolute invariance and astatism. An unconventional method to ensure astatism implementing jet pulse equipment is analysed.     

Optimality conditions of sliding modes and the maximum principle for control problems with the scalar argument

Various types of optimality criteria and conditions, which define the set of admissible solution, are brought to the canonical form. For the problem set in this form, the optimality conditions of sliding modes are stated. It is shown that the optimality conditions emerge from these conditions in the form of the maximum principle for problems with a scalar argument and an arbitrary combination of the optimality criterion and constraints.     

Features of motion of dynamic systems with disturbances in the neighborhood of manifolds of switchings

The method of computation of control in real time of a linear system with disturbance is suggested. The system of linear algebraic equations is obtained, which links the deviations of phase coordinates to the deviations of initial conditions of the normalized conjugate system and to the deviation of the finite moment. The calculations reduce to the sequence of the solutions of systems of linear algebraic equations and the integration of a matrix differential equation over transfer intervals of the control switching moments and the finite moment of time. The correction of switching moments and the finite moment of control in the accompaniment of the phase trajectory of motion of a controllable object is considered. Simple constructive conditions of the origin of the sliding mode, motions of the representative point over manifolds of switchings, and changes of the control structure in accompanying the phase trajectory of the system motion are obtained. The convergence of the computational method is proved.     

Identification of frequency of a shifted sinusoidal signal

The article deals with the issue of identification of an unknown frequency of a shifted sinusoidal signal y(t) = σ ₀ + σ sin(ωt + ?). A new approach taken to estimate the frequency of a shifted sinusoidal signal is suggested, which is a robust approach relative to undetermined disturbances present in the measurement of a useful signal. In contrast to known analogs, the given approach permits controlling the time of estimation of the unknown frequency ω. The dimension of the identification algorithm suggested is less than that of known analogs.     

On approximation of the subdifferential of the nonsmooth penalty functional in the problems of optimal control

For the problem of optimal program control with terminal equality constraints which is considered as an “elementary operation” within the framework of the algorithm to calculate the optimal positional control, a smooth approximation the Frechét-nondifferentiable penalty functional was proposed. Examples of numerical experiments were presented.     

Design of suboptimal robust controllers under unknown weights of perturbations

In the classical formulations of the problems of design of the robust optimal controllers, the equations of the nominal plant and the weights of the permissible perturbations are assumed to be known. In the present paper, consideration was given to a nonclassical formulation of the design problem where the weights of perturbations are assumed to be unknown and subject to estimation from the measurement data. The multivariable discrete-time controlled plant was described by a given transfer matrix with perturbations in the irreducible factors and a bounded external perturbation. The problem of determination with a predefined accuracy of the perturbation weights that are best coordinated with the measurements and of their corresponding suboptimal controller was solved.     

Method of state space expansion in the design of autonomous control

The results of constructive analysis and block design of the problems of autonomous control in the nonlinear dynamic systems were presented. As compared with the well-known results of the control theory, the method of state space expansion enabled augmentation of the class of systems that are controlled autonomously in the output variables. The state observers on sliding modes were used for the dataware of the basic algorithms. The procedures developed feature multilevel decomposition of the problem of high-dimensionality design into smaller independent subproblems.     

Control of motion of the autonomous underwater vehicle at trajectory survey of the oceanic physical fields

Consideration was given to the control of motion of the autonomous underwater vehicle performing automatic measurements and mapping of the oceanic physical fields. In the general case, it is required to set up a control maintaining motion along the field isolines relying on the trajectory measurements of the level and field gradient components and identification and contouring of the anomalies by the characteristic points and generalized reference points. A practical example based on the experience of bathymetrical and hydrophysical measurements in the Arctic region using an autonomous underwater vehicle was discussed.