<Emphasis Type="Italic">D</Emphasis>-decomposition technique state-of-the-art

It is a survey of recent extensions and new applications for the classical D-decomposition technique. We investigate the structure of the parameter space decomposition into root invariant regions for single-input single-output systems linear depending on the parameters. The D-decomposition for uncertain polynomials is considered as well as the problem of describing all stabilizing controllers of the certain structure (for instance, PID-controllers) that satisfy given H _∞-criterion. It is shown that the D-decomposition technique can be naturally linked with M-Δ framework (a general scheme for analysis of uncertain systems) and it is applicable for describing feasible sets for linear matrix inequalities. The problem of robust synthesis for linear systems can be also treated via D-decomposition technique.     

The <Emphasis Type="Italic">MAP</Emphasis> + <Emphasis Type="Italic">MAP/PH</Emphasis>/1/<Emphasis Type="Italic">N</Emphasis> queuing system with single and batch arrivals of customers

The single-server queuing system with finite buffer was considered. The customers may arrive one-by-one or in batches. Arrivals of single customers and their batches obey the Markov input processes. The customers from a batch taken for servicing come one at a time at the exponentially distributed time intervals. The numbers of customers in batches are distributed geometrically. The time of customer servicing has a phase-type distribution. The numbers of batches and single customers that may be simultaneously accepted by the system are controllable parameters. The joint distribution of the number of batches and the number of customers in system, loss probabilities, distribution of the time of batch sojourn, and problems of optimization were analyzed.     

H<Subscript>2</Subscript>-optimal tuning algorithms for fixed structure controllers

An H₂-method of optimal tuning is proposed for a fixed order controller. The SISO plant model is considered in state space. The H₂-method of tuning parameter design is based on the minimization of a transient process closeness criterion for appropriate open-loop and closed-loop control systems and their reference models. The controller tuning algorithms use the plant parameter estimations obtained during the plant parameter identification. The analytical expressions are obtained for the square of H₂-norm of a stable dynamic system. The following theorem is proven: the minimum necessary conditions for the functionals of transfer function H₂-norm of open-loop and closed-loop systems are the same as the minimum necessary conditions for the Frobenius norm of the controller parameter tuning polynomial.     

Output control of linear systems with unmodeled dynamics

Recently, an output control law for linear parametrically uncertain plants was proposed by one of the authors of this paper. Here, the feasibility of this approach is substantiated in the presence of unmodeled asymptotically stable dynamics. A theorem is proved which formulates the feasibility conditions for the control law for systems with unmodeled dynamics.     

Nonlocal improvement of controls in state-linear systems with terminal constraints

The method is suggested for nonlocal improvement in state-linear problems of optimal control with terminal constraints. The improvement of controls is performed with the fulfillment of all terminal constraints. The method has the possibility to improve controls satisfying the maximum principle.     

Linear systems with multiple singular values

Properties of linear dynamical systems are studied in the case of Hankel singular values of high multiplicity for monosingular and bisingular systems. Singular values of the Hankel operator take only one or two values for such systems. Properties of transfer functions are analysed. Algorithms of transfer function analysis and design are developed. Relations between frequency characteristics and Hankel singular values are established.     

Control of incompletely controllable differential-difference systems with delay

A problem of control of linear autonomous differential-difference systems for violation of the condition of complete controllability and uncontrolled initial state is studied. A problem of system trajectory nullification and its delay there within any specified time period is posed. A new condition for solvability of this problem is proved and a constructive solution is proposed.     

Polynomial modal control for sampled data systems with delay

The problem of modal sampled-data control for continuous-time linear time-invariant plant with delay is considered. The characteristic matrix of the system is constructed. An algorithm is given for generating the set of causal discrete-time controllers that place eigenvalues of the characteristic matrix at specified points of the complex plane.     

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.     

Constructive methods of control optimization in nonlinear systems

The paper investigates methods of optimal program and point-to-point control in nonlinear differential systems. The proposed methods are based on the idea of discretizing the continuous-time problem. In case of program control, the method of projected Lagrangian is used, which involves solution of an auxiliary problem with linearized constraints by the reduced gradient method; in case of point-to-point control, Bellman’s optimality principle is employed for a “grid” over an approximating solvability tube of the system for a specified goal set with approximation of the point-to-point control by families of controlling function or parameters. An example is given, where the optimal point-to-point control is calculated for a model of maneuvering aircraft.     

Tracking in linear MIMO systems under external perturbations

Consideration was given to tracking the given trajectories of the output variables of the linear multiple-input multiple-output dynamic systems under uncontrollable external perturbations. A procedure to rearrange the mathematical model of a control plant in a consistent block form of controllability and observability of the output (measurable) variables with regard for the external perturbations was developed within the framework of the block approach. This form underlies the decomposition procedures of feedback design enabling one to track the given trajectories invariantly to the external perturbations.     

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.     

A two-sector model of developing systems

A two-sector model of developing systems of V.M. Glushkov is studied; it is described by a system of nonclassical equations of the Volterra type that includes equations with variable lower limits of both first and second kinds. Numerical methods for solving systems of nonclassical Volterra equations are constructed.     

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.     

Robust stabilization of continuous indeterminate systems

The system of an arbitrary order with one scalar control is considered. It is assumed that coefficients of the system and vectors of distribution of control are bounded and are functionals of an arbitrary character. With the aid of the Lyapunov function in a quadratic form with the constant Jacobian matrix of coefficients of a special form, the conditions for bounds of a change of the coefficients are obtained, in the fulfillment of which the linear stationary control by the state is synthesized. In this control, the system becomes exponentially stable on the whole.     

On absolute stability of discrete systems

Modernization is considered of the criterion of absolute stability of discrete systems [1, 2], which uses the method of quadratic transformation of the state vector. The modernization aim is to obtain the criterion form that does not explicitly depend on the notions of this method. The new formulation permits performing all calculations in the space of the initial dimension, which reduces the complexity of the problem solution. This work represents the development of the approach suggested in [3] for continuous systems.     

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.     

Optimal digital filtering of continuous signals in time lag systems

The problem of the optimal estimation of continuous processes by discrete measurements in the presence of time lag (delay) is considered. On the basis of the theory of parametric transfer functions, an optimal, periodically nonstationary filter is developed, which affords a minimum of the estimation error variance at any instant of time. The comparison is performed of the obtained solution with the optimal stationary filter, which ensures a minimum of the mean (by continuous time) error variance. It is shown that in the problem of estimation of the Markov process of the first order, a simpler stationary filter with the fixer of order zero is insignificantly inferior to the optimal filter.     

Invariance for impulsive control systems

The conventional concepts of invariance are extended in this article to include impulsive control systems represented by measure driven differential inclusions. Invariance conditions and some of their main features are derived. The solution concept plays a critical role in the extension of the conditions for conventional problems to the impulsive control context.     

Functional diagnosis of the nonlinear dynamic delay systems

Consideration was given to the problem of functional diagnosis of the nonlinear dynamic delay systems. Its solution implies design of redundancy relations whose verification underlies conclusions about system operability and the kind of system defect. For the systems obeying differential equations with a polynomial right-hand side, a method was proposed for designing the redundancy relations. It admits that some or all constant coefficients of the polynomials are unknown.