Statistical characteristics of attainability set of controllable systems with random coefficients

For controllable systems with random coefficients we study a property of statistical invariance, satisfied with given probability. We obtain sufficient conditions for invariance of a set with respect to controllable system expressed in terms of Lyapunov functions and shift dynamic system. We study the statistical characteristics of attainability set of a controllable system which is parameterized by metric dynamic system.     

Approximate construction of attainability sets of control systems with integral constraints on the controls

The approximate construction of attainability sets of control systems with quadratic integral constraints on the controls is considered. It is assumed that a control system is non-linear with respect to the phase variable and linear with respect to the variable which describes the controlling action. The approximation of the attainability sets of a control system is accomplished in several stages. The latter class of controls generates a finite number of trajectories of the system. The trajectories of the system are then replaced by Euler broken lines. An estimate of the accuracy of the Hausdorff distance between the attainability set and the set which has been approximately constructed is obtained.     

The optimal choice of control constraints

The problem of the optimal choice of the limits of a set of possible values of the control during motion for the purpose of obtaining the required form of the attainability set of a linear dynamical system in a specified time interval is considered. Using the method, in which these sets are approximated by ellipsoids, the problem of controlling the parameters of the ellipsoid containing the control vector is solved. Then a functional, which depends on the matrix of the ellipsoid, containing the phase vector, reaches its maximum. The order in which the corresponding formulae are used is illustrated using the example of a simple mechanical system. The results obtained are suitable for systems in which, instead of the control vector, there is an interference vector with controllable boundaries of possible changes and can be extended to stochastic systems.     

Statistical characteristics of continuous functions and statistically weakly invariant sets of controllable system

We continue the investigation of expansion of a concept of invariance for sets which consists in studying statistically invariant sets with respect to control systems and differential inclusions. We consider the statistical characteristics of continuous functions: Upper and lower relative frequency of containing for graph of a function in a given set. We obtain conditions under which statistical characteristics of two various asymptotical equivalent functions coincide; then by the value of one of them it is possible to calculate the value of another one. We adduce the equality for finding relative frequencies of hitting functions the given set in the case when the distance from the graph of one of functions to the given set is a periodic function. A consequence of these statements are conditions of statistically weak invariance of a set with respect to controlled system. For some almost periodic functions we obtain the formulas by which we can calculate the mean values and the statistical characteristics. We also consider the following problem. Let the number λ₀ ∈ [0, 1] be given. It is necessary to find the value c(λ₀) such that the upper solution z(t) of the Cauchy problem does not exceed c(λ₀) with the relative frequency being equal λ₀. Depending on statement of the problem, a value z(t) can be interpreted as the size of population, energy of a particle, concentration of substance, size of manufacture or the price of production.     

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.     

Internal ellipsoidal estimates of attainability sets for linear systems under conditions of uncertainty

The problem of constructing internal ellipsoidal estimates of the geometric difference between two ellipsoids and applying the estimated results for the attainability sets of linear systems with a disturbance is considered. An addition to the existing method of constructing the difference between two ellipsoids is presented, and the previous constraints are removed. In the process of validating the addition, some relationships between certain properties of constructed ellipsoidal estimations and set convexity are given, being the data for the problem. A method for estimating the attainability sets for linear systems with a disturbance, equivalent to the existing approach to systems without disturbances, are given. The disturbances are considered using the obtained results.     

Linear stationary control systems over the Boolean semiring and their graphs of modules

We consider linear stationary dynamical systems over the Boolean semiring. We analyze the properties of complete observability, identifiability, attainability, and controllability of a system. We define the notion of the “graph of modules” of totally controllable totally attainable Boolean linear stationary systems by analogy with spaces of modules in the case of systems over fields. The above-mentioned graph is described in the simplest case of one-dimensional inputs and outputs. We prove the weak connectedness of this oriented graph.     

Geometry and asymptotics of wavefronts

Methods of convex analysis and differential geometry are applied to the study of properties of nonconvex sets in the plane. Constructions of the theory of α-sets are used as a tool for investigation of problems of the control theory and the theory of differential games. The notions of the bisector and of a pseudovertex of a set introduced in the paper, which allow ones to study the geometry of sets and compute their measure of nonconvexity, are of independent interest. These notions are also useful in studies of evolution of sets of attainability of controllable systems and in constructing of wavefronts. In this paper, we develop a numerically-analytical approach to finding pseudovertices of a curve, computation of the measure of nonconvexity of a plane set, and constructing front sets on the basis these data.In the paper, we give the results of numerical constructing of bisectors and wavefronts for plane sets. We demonstrate the relation between nonsmoothness of wavefronts and singularity of the geometry of the initial set. We also single out a class of sets whose bisectors have an asymptote.     

Geometric control of hybrid systems

In this paper, we present a geometric approach for computing controlled invariant sets for hybrid control systems. While the problem is well studied in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support functions.     

Estimation of the attainability set for a linear system based on a linear matrix inequality

The problem under consideration is the construction of an attainability set’s internal approximation for a full controllable linear time-invariant system. This approximation is obtained as an intersection of two domains given by quadratic forms. One of these forms is based on parameters of the original system. The other form is produced by the solution to some linear matrix inequality. The method proposed here is illustrated by a numerical example.     

A dynamic programming method for controlled linear systems with delays

A linear system with permanent delay is considered. A method of dynamic programming for constructing attainability sets and solving the problem of target control for the systems is used. The expressions for value functionals described by solutions to the corresponding Hamilton-Jacobi-Bellman equation are obtained. It is proved that these value functionals calculated by means of convex analysis satisfy the above equations. Strategies for synthesized control for the problem of hitting on the target set are given.     

Sets invariant under an integral constraint on controls

In this paper we study the invariance of given sets with respect to a system with distributed parameters. The considered system is described by a heat conductivity equation whose right-hand side written in the additive form contains a control. For the initial data we obtain sufficient conditions for the strong and weak invariance of the set that represents the graph of a given multivalued mapping.     

Controlling solutions to a linear differential equation

Both global attainability and global reducibility problems for a controllable system equivalent to a linear differential equation are positively solved. Furthermore, the existence of a linear equation having a given Cauchy matrix on a given segment and coinciding with given equations from the left and right of that segment is proved. The results obtained allow one to construct a linear equation with the fundamental solutions system possessing preassigned properties.     

Axiomatic systems for rough set-valued homomorphisms of associative rings

Axiomatic approaches to study approximation operators are one of the primary directions for the investigation of rough set theory. In this paper, we provide some axiomatic systems of lower and upper approximation operators in rough set theory. We also apply the axiomatic systems of generalized rough sets for definitions of generalized lower and upper approximations with respect to an ideal of a ring and discuss some of their significant properties.     

Hypercubes are minimal controlled invariants for discrete-time linear systems with quantized scalar input

Quantized linear systems are a widely studied class of nonlinear dynamics resulting from the control of a linear system through finite inputs. The stabilization problem for these models shall be studied in terms of the so-called practical stability notion that essentially consists in confining the trajectories into sufficiently small neighborhoods of the equilibrium (ultimate boundedness).We study the problem of describing the smallest sets into which any feedback can ultimately confine the state, for a given linear single-input system with an assigned finite set of admissible input values (quantization). We show that the family of hypercubes in canonical controller form contains a controlled invariant set of minimal size. A comparison is presented which quantifies the improvement in tightness of the analysis technique based on hypercubes with respect to classical results using quadratic Lyapunov functions.     

Reachability realization and stabilizability of switched linear discrete-time systems

In this paper, the reachability realization of a switched linear discrete-time system, which is a collection of linear time-invariant discrete-time systems along with some maps for “switching” among them, is addressed. The main contribution of this paper is to prove that for a switched linear discrete-time system, there exists a basic switching sequence such that the reachable (controllable) state set of this basic switching sequence is equal to the reachable (controllable) state set of the system. Hence, the reachability (controllability) can be realized by using only one switching sequence. We also discuss the stabilizability of switched systems, and obtain a sufficient condition for stabilizability. Two numeric examples are given to illustrate the results.     

Controllability properties of constrained linear systems

In this paper, we present results on constrained controllability for linear control systems. The controls are constrained to take values in a compact set containing the origin. We use the results on reachability properties discussed in Ref. 1.We prove that controllability of an arbitrary pointp inR ⁿ is equivalent to an inclusion property of the reachable sets at certain positive times. We also develop geometric properties ofG, the set of all nonnegative times at whichp is controllable, and ofC, the set of all controllable points. We characterize the setC for the given system and provide additional spectrum-dependent structure.We show that, for the given linear system, several notions of constrained controllability of the pointp are the same, and thus the setC is open. We also provide a necessary condition for small-time (differential or local) constrained controllability ofp.This work was supported in part by NSF Grant ECS-86-09586.     

Normal form of real differential equations

In a neighborhood of a fixed point we consider an autonomous analytic system of ordinary differential equations. We establish the existence of a normalizing transformation for which the normal form retains the properties of the original system such as reality and invariance with respect to a linear change of variables. For real systems we consider the problem of existence of an analytic transformation into normal form and the problem of existence of a finitely smooth transformation into a linear system.