Hybrid discrete-continuous systems: I. stability and stabilizability

For hybrid discrete-continuous linear stationary systems, we consider the basic problems of qualitative control theory, namely, stability and stabilization. For such systems, we obtain parametric criteria for asymptotic and exponential stability ensuring a prescribed stability exponent and a rank criterion for stabilizability. We consider the problem of finding the minimum number of inputs for which the considered system is stabilizable. We suggest an effective algorithm for constructing the matrix describing the structure of the input device of a minimum-input stabilizable system. An example illustrating the results is given.     

Spectrum control in linear systems with incomplete feedback

For a linear stationary control system closed by a linear incomplete feedback, we obtain necessary and sufficient conditions for the solvability of the spectrum control problem in the case of coefficients of special form.     

Necessary and sufficient conditions in a spectrum control problem

For a linear stationary control system closed by a linear output feedback and for a bilinear stationary control system, we obtain new necessary and sufficient conditions for the solvability of the eigenvalue spectrum control problem in the case of a special form of the coefficients.     

Spectrum control problem for strongly irregular periodic vibrations of linear systems with zero mean of the coefficient matrix

We consider a linear periodic control system with zero mean of the coefficient matrix and with linear state feedback control periodic with the same period. We obtain necessary and sufficient conditions for the solvability of the frequency spectrum control problem with a given goal set for strongly irregular periodic vibrations. In this problem, one should find a feedback coefficient such that the closed system has a strongly irregular periodic solution with the desired frequencies.     

Numerical methods of stabilizer construction via guidance control

This work concerns guidance stabilization of non‐autonomous control systems. Global stabilization problem is usually quite complex and difficult to solve. To overcome this difficulty, guidance control is used. A guidance stabilizer uses a trajectory of a globally asymptotically stable auxiliary system as a guide. A local stabilizer keeps the trajectory of the system in a neighborhood of a solution of the auxiliary system. In this way, the trajectory of the system tends to the equilibrium position. The main idea of this method is to solve the global stabilization problem by applying local stabilization methods. The presented procedure also yields additional possibilities for the design of a stabilizer that eliminates the peak effect, that is, the large deviation of the solutions from the equilibrium position at the beginning of the stabilization process. This effect represents a serious obstacle to the design of cascade control systems and to guidance stabilization. The optimal control problem used in this paper eliminates this effect that we have when we apply known construction methods of local stabilizers to obtain a high speed of damping of the control systems trajectories. According to this approach and using ε‐strategies introduced by Pontryagin in the frame of differential games theory, the stabilizing control is constructed as a function of time defined in a small time interval and not as a feedback. An application to a mechanical stabilization problem is provided here. Copyright © 2012 John Wiley & Sons, Ltd.     

Modelling and system identification of active magnetic bearing systems

Active magnetic bearing (AMB) systems have recently attracted much attention in the rotating machinery industry due to their advantages over traditional bearings such as fluid film and rolling element bearings. The AMB control system must provide robust performance over a wide range of machine operating conditions and over the machine lifetime in order to make this technology commercially viable. An accurate plant model for AMB systems is essential for the aggressive design of control systems. In this paper, we propose two approaches to obtain accurate AMB plant models for the purpose of control design: physical modelling and system identification. The former derives a model based upon the underlying physical principles. The latter uses input – output data without explicitly resorting to physical principles. For each problem, a brief summary of the theoretical derivation and assumptions is given. Experimental results based on data collected from an AMB test facility at the United Technologies Research Center provide a vehicle for a comparison of the two approaches.     

Minimal stabilization of vector (MISO and SIMO) systems

We consider the problem of constructing a stabilizer described by a system of linear differential equations and such that a given dynamical system becomes stable after being closed by the feedback produced by the stabilizer. Moreover, we require that the dimension of the stabilizer, that is, the dimension of its state vector, be minimal. We assume that the given system has either a single input and multiple outputs (a SIMO system) or, on the opposite, multiple inputs and a single output (a MISO system).     

Minimal stabilizers with arbitrary spectrum for SIMO and MISO systems

We consider two classes of linear dynamical systems, single-input-many-output (SIMO) systems and many-input-single-output (MISO) systems. For both classes, we consider the problem of synthesizing a stabilizer of minimum dimension providing pole assignment in the closed-loop system. We show that, for these classes, a minimal stabilizer providing pole assignment has dimension k _min = min{ν ? 1, µ ? 1}, where ν and µ are the controllability and observability indices of the original (SIMO or MISO) dynamical system.     

Hybrid discrete-continuous systems: II. Controllability and reachability

For hybrid discrete-continuous linear stationary systems, we consider basic problems of qualitative control theory such as controllability and reachability. For these systems, we obtain parametric criteria for relative and total controllability and relative and total reachability. We consider the problem of the evaluation of the minimum number of inputs for which the system has some controllability or reachability property. An example illustrating obtained results is considered.     

On some boundary value problems for a system of two second-order equations

For a two-point homogeneous boundary value problem for a system of two nonlinear second-order differential equations, we suggest sufficient solvability conditions (in particular, stated, like Bernstein conditions, in terms of the growth of the absolute values of the right-hand sides of the system with respect to the derivatives of the unknown functions). We obtain a priori estimates for solutions.     

Reconstruction of boundary controls in parabolic systems

In the paper, an inverse dynamic problem is considered. It consists in reconstructing a priori unknown boundary controls in dynamical systems described by boundary value problems for partial differential equations of parabolic type. The source information for solving the inverse problem is the results of approximate measurements of the states of the observed system’s motion. The problem is solved in the static case; i.e., to solve it, we use all the measurement data accumulated during some specified observation interval. The problem under consideration is ill-posed. To solve it, we propose the Tikhonov method with a stabilizer containing the sum of the mean-square norm and total time variation of the control. The use of such nondifferentiable stabilizer allows us to obtain more precise results than the approximation of the desired control in the Lebesgue spaces. In particular, this method provides the pointwise and piecewise uniform convergences of regularized approximations and makes possible the numerical reconstruction of the subtle structure of the desired control. In the paper, the subgradient projection method for obtaining a minimizing sequence for the Tikhonov functional is described and substantiated. Also, we demonstrate the two-stage finitedimensional approximation of the problem and present the results of numerical simulation.     

Control of spectrum in bilinear systems

For a bilinear stationary control system, we obtain necessary and sufficient conditions for the solvability of the spectrum control problem for the case in which the coefficients have a special form.     

Weak Invariance of a Cylindrical Set with Smooth Boundary with Respect to a Control System

We consider the problem of constructing resolving sets for a differential game or an optimal control problem based on information on the dynamics of the system, control resources, and boundary conditions. The construction of largest possible sets with such properties (the maximal stable bridge in a differential game or the controllability set in a control problem) is a nontrivial problem due to their complicated geometry; in particular, the boundaries may be nonconvex and nonsmooth. In practical engineering tasks, which permit some tolerance and deviations, it is often admissible to construct a resolving set that is not maximal. The constructed set may possess certain characteristics that would make the formation of control actions easier. For example, the set may have convex sections or a smooth boundary. In this context, we study the property of stability (weak invariance) for one class of sets in the space of positions of a differential game. Using the notion of stability defect of a set introduced by V.N. Ushakov, we derive a criterion of weak invariance with respect to a conflict control dynamic system for cylindrical sets. In a particular case of a linear control system, we obtain easily verified sufficient conditions of weak invariance for cylindrical sets with ellipsoidal sections. The proof of the conditions is based on constructions and facts of subdifferential calculation. An illustrating example is given.     

The reconstruction of controls by regularization methods with uneven stabilizers

The problem of reconstructing a previously unknown control (parameter) of a dynamical system using the results of approximate observations of the motion of this system is considered. It is proposed to use static and dynamic methods to solve this problem which, in their implementation, utilize the method of Tikhonov regularization with a stabilizer containing a variation of the simulating subsidiary control (parameter).The use of such a non-differentiable stabilizer enables one to obtain more refined results than the approximation of the required control in Lebesgue spaces. In particular, the piecewise-uniform convergence of the regularized approximations can be successfully substantiated by this route, which opens up the possibility of numerically reconstructing the fine structure of the required control.     

Vibrational stabilization and the Brockett problem

The present paper deals with the exposition of methods for solving the Brockett problem on the stabilization of linear control systems by a nonstationary feedback. The paper consists of two parts. We consider continuous linear control systems in the first part and discrete systems in the second part. In the first part, we consider two approaches to the solution of the Brockett problem. The first approach permits one to obtain low-frequency stabilization, and the second part deals with high-frequency stabilization. Both approaches permit one to derive necessary and sufficient stabilization conditions for two-dimensional (and three-dimensional, for the first approach) linear systems with scalar inputs and outputs. In the second part, we consider an analog of the Brockett problem for discrete linear control systems. Sufficient conditions for low-frequency stabilization of linear discrete systems are obtained with the use of a piecewise constant periodic feedback with sufficiently large period. We obtain necessary and sufficient conditions for the stabilization of two-dimensional discrete systems. In the second part, we also consider the control problem for the spectrum (the pole assignment problem) of the monodromy matrix for discrete systems with a periodic feedback.     

Control of the spectrum of irregular oscillations of linear systems with coinciding ranks of the matrix multiplying the control and the extended matrix

We consider a linear periodic control system such that the ranks of the matrix multiplying the control and the extended matrix consisting of the averaged coefficient matrix and the matrix multiplying the control are the same. We assume that the control has the form of feedback linear in the state variables and is periodic with the same period as the system itself. We pose the problem of control of the frequency spectrum of strongly irregular periodic oscillations with an objective set, that is, the problem of finding a feedback coefficient such that the closed system has a strongly irregular periodic solution with the desired frequencies. We obtain necessary and sufficient conditions for the solvability of this problem.     

On the hazard rate and reversed hazard rate orderings in two-component series systems with active redundancies

The lifetimes of two-component series systems with two active redundancies are compared using the hazard rate and the reversed hazard rate orders. We study the problem of where to allocate the spares in a system to obtain the best configuration. We compare redundancy at component level vs. system level using the likelihood ratio order. For this problem we find conditions under which there is no hazard rate ordering between the lifetimes of the systems.     

Sufficient conditions for the existence of a common stabilizer for a family of linear nonstationary plants

We consider the problem on the existence of a common stabilizer for a finite family of continuous linear scalar nonstationary plants. To solve the problem, we suggest to use a topological approach based on finding a common stabilizer in the form of a convex combination of stabilizers for the individual plants and on an application of well-known assertions about the properties of sets on simplices.     

Lower Bounds for the Rate of Convergence of Dynamic Reconstruction Algorithms for Distributed-Parameter Systems

We obtain lower bounds for the rate of convergence of reconstruction algorithms for distributed-parameter systems of parabolic type. In the case of a pointwise constraint on control for known reconstruction algorithms, we establish a lower bound on the rate of convergence, which shows that, given certain conditions, for each solution of the system one can choose such a collection of measurements so that the reconstruction error will not be less than a certain value. In the case of unbounded controls, we obtain lower bounds for a possible reconstruction error for each trajectory as well as for a given set of trajectories. For a system of special form, we construct an algorithm for which we obtain upper and lower bounds for accuracy having identical order for a specific choice of matching of the parameters.     

Difference method of increased order of accuracy for the Poisson equation with nonlocal conditions

In a rectangular domain, we consider the two-dimensional Poisson equation with nonlocal boundary conditions in one of the directions. For this problem, we construct a difference scheme of fourth-order approximation, study its solvability, and justify an iteration method for solving the corresponding system of difference equations. We give a detailed study of the spectrum of the matrix representing this system. In particular, we obtain a criterion for the nondegeneracy of this matrix and conditions for its eigenvalues to be positive.