Problems In Control Theory Research Articles

SEE Transform Technique in Control Theory

As in the Laplace transform technique, a new integral technique called SEE (Sadiq-Emad-Eman) transform is submitted. Showed propositions and important properties of SEE integral technique for differentiate of functions and showed shifting property for SEE technique. Got transfer function of dynamic system in control theory applying SEE integral technique. Solved some important problems in control theory via SEE technique. Showed that the Laplace integral technique and every integral transform technique like the Laplace transform available in the literature are particular cases of the SEE technique.

Stages and main problems of the century-long control theory and system identification development. Part 5. Principles and problems in control and identification of complex systems of various nature based on cognitive maps impulse processes models

Stages and main problems of the century-long control theory and system identification development. Part 5. Principles and problems in control and identification of complex systems of various nature based on cognitive maps impulse processes models

A new method for the approximation of certain non-linear systems

Abstract A common problem in systems and control theory is to provide an approximation to non-linear systems. We provide a novel approach as a general solution to this problem originally conceived by Gamkrelidze. We consider and solve a general approximation problem which provides the fundamentals for various switching-type systems thus encompassing a wide range of systems theory problems.

Planar Walking of a Five-Link Biped Robot over a Stepped Surface with Obstacles of Different Heights and Lengths

A problem of modeling biped locomotion is an actual and complex problem of modern control theory. In the present paper, we deal with a special case when a five-link biped robot with knees moves along a given stepped surface. We describe the robot motion by a hybrid system which consists of a system of ordinary differential equations at the single support phase and an algebraic relation at the double support phase. The main contribution of this paper is the construction of a periodic motion that satisfies a set of conditions inherent in human walking. The results of the paper will be useful to those who are interested in control theory applications.

Equivalent Differential Equations in Problems of Control Theory and the Theory of Hamiltonian Systems

Equivalent Differential Equations in Problems of Control Theory and the Theory of Hamiltonian Systems

THE MEDIAN TREE

Mathematical models in which objects have a finite discrete structure are used to solve a wide range of practical problems. Such are, for example, many problems in auto-matic design, operation research, control theory, network planning, and computational geometry. Often, oriented or non-oriented graphs are considered as such models, which can adequately describe the structure of the problem of interest to us. Among these tasks, a certain place is occupied by tasks related to network issues of optimal placement of ware-houses, certain service points, emergency, medical services, etc. When studying such issues, it becomes necessary to identify, in a certain sense, the central objects in the networks un-der consideration, and in order to find them, the concept of the shortest path between these two given elements plays an essential role. A definition of the medial vertex of a graph is provided in the article – the vertex whose sum of distances to the others is minimal. The median of a non-oriented tree con-sisting of medial vertices is considered, and it is proved that the median of each such tree has either one vertex or two connected by an edge. This result is a definite generalization of a similar property of the center of a tree consisting of central vertices. The question of the relationship between the center and median of the same tree is also studied. The examples of trees with ten vertices show that such a direct connection does not exist, and there can be all four logically possible cases: the same tree can contain one or two central and one or two medial vertices.

Stages and main tasks of the century-long control theory and system identification development

The article provides a review of the mathematical description of the dynamics of continuous and discrete linear stationary systems and objects, used at the development stage of the classical theory of automatic control in the form of mathematical models of the «input-output» type. The time and frequency characteristics of continuous and discrete control systems are described, typical links of stationary systems are considered, parametric discrete models of objects as part of typical digital control loops are presented. Stochastic discrete autoregressive models of stationary time series used to describe the dynamic objects in the synthesis of digital control systems are considered. A review of standard control laws for the implementation of continuous and discrete controllers has been completed. A method for synthesizing discrete controllers for multidimensional controlled objects with different, unknown and changing delays is considered, through which variable delays are compensated in the characteristic equation of a closed-loop control system. A common technique for synthesizing one-dimensional and multidimensional controllers for stochastic objects with delays based on ARMAX models is considered. An analysis of approaches to identifying delays in controlled objects is carried out and a method for identifying delays when using input-output models is considered, based on the calculation and comparison of impulse responses for extended and non-extended models of the controlled object. An analysis of the advantages and disadvantages of «input-output» type models is given, as well as the possibilities of their application for solving various classes of control theory problems.

Correction of the optimal regulator based on solving the inverse problem of optimal stabilization with vector control

Objective. The problem of designing controllers that implement a given programmed movement of a controlled object and the problem of determining the movement of a dynamic system are two main problems in classical control theory. This article discusses the solution of direct and inverse optimal stabilization problems. The state vector is assumed to be completely available for measurement.Method. Based on the optimality ratio linking the weight coefficients of the quadratic quality functional and the optimal gain matrix, which closes the control object, it is proposed to use a numerical method for determining the functional matrices. Mathematical models of autonomous fully controlled objects were used for the study, the formation of which was carried out randomly, in particular, according to the normal distribution law.Result. The initial stage of the solution is associated with modal synthesis, the result of which is a proportional regulator that provides stabilization of the control object by the location of the poles of the synthesized system. The next step is to determine the weighting coefficients of the functional by numerically solving the optimality ratio. The final stage is the solution of the direct optimal stabilization problem, which is based on the Lagrange variational problem. As a result, the optimal regulator is calculated, which, when switched on in a closed system instead of a modal one, reduces the duration of the transient process.Conclusion. The proposed approach of the authors allows minimizing to a certain extent the transients of the adjusted control system.

Визначення програми керування кутом тангажа твердопаливного ракетно-го об’єкта

This paper analyzes the trends in the improvement of the performance characteristics of guided missiles with solid-propellant sustainer engines and identifies the features and requirements for flight trajectories, design parameters, and control programs. Within the framework of the optimal control theory, the comprehensive problem of simultaneous optimization of a missile’s design parameters and control systems is formulated. An approach to the formation of missile flight control programs in the form of polynomials is developed further, thus making it possible to reduce the optimal control theory problem to a simpler problem of nonlinear mathematical simulation. The proposed approach to control program development is used at the initial design stage to form a wide range of guided missile trajectories. Use is made of a methodology for the ballistic and aeroballistic flight range optimization of the design parameters and flight control programs of a canard missile. The missile flight range depends essentially on the values of the design and trajectory parameters and control programs chosen for optimization. Because of this, the optimization of the chosen parameters (maybe, other parameters too) in the solution of specific target problems seems to be the indispensable initial stage of missile design. For the considered missile trajectories with a vertical launch where the Mach number takes different values, optimal programs of pitch time variation that maximize the flight range are determined. The analysis of the optimization results for different trajectories shows that the optimal program in active flight with a vertical launch is the linear time dependence of the pitch angle. The application package developed allows one to determine flight control programs optimal in a given class of functions and advisable design parameters and basic performance characteristics of guided missiles for various aerodynamic designs and flight schemes as early as at the initial design stage to an accuracy required for design studies. This makes it possible to analyze design alternatives, thus improving the quality of solution of problems arising at the initial design stage and reducing the time and the cost of design work on new missiles.

Novel Kinds of Fractional λ–Kinetic Equations Involving the Generalized Degenerate Hypergeometric Functions and Their Solutions Using the Pathway-Type Integral

In recent years, fractional kinetic equations (FKEs) involving various special functions have been widely used to describe and solve significant problems in control theory, biology, physics, image processing, engineering, astrophysics, and many others. This current work proposes a new solution to fractional λ−kinetic equations based on generalized degenerate hypergeometric functions (GDHFs), which has the potential to be applied to calculate changes in the chemical composition of stars such as the sun. Furthermore, this expanded form can also help to solve various problems with phenomena in physics, such as fractional statistical mechanics, anomalous diffusion, and fractional quantum mechanics. Moreover, some of the well-known outcomes are just special cases of this class of pathway-type solutions involving GDHFs, with greater accuracy, while providing an easily calculable solution. Additionally, numerical graphs of these analytical solutions, using MATLAB Software (latest version 2023b), are also considered.

Practical aspects of model predictive control in linear systems and cognitive maps

The subject of this paper is peculiarities of model predictive control (MPC) application in linear discrete-time system stablization. While stabilization in linear systems is already a well-studied problem in control theory, the MPC approach gives opportunity to produce faster stabilization trajectories at cost of higher amount of computations required. Significant progress in capabilities of computers since emergence of the control theory makes the MPC approach feasible in modern times. The MPC approach gives an opportunity to achieve significantly better results, but its application requires great care. It is due to many unobvious and undesirable effects it leads to if used incorrectly. These effects are discussed, explained and demonstrated one by one on examples in this paper. Analysis of their causes reveals requirements for MPC-based stabilizing control algorithm which allow resulting controller to operate reliably. It also appears that in most cases an optimal stabilization trajectory is not unique, i.e. it is possible to choose between optimal trajectories to improve some kind of secondary objective. In addition, as an example which is valuable by itself, stabilization in linear cognitive maps is discussed separately. Being an example of discrete-time linear system, linear cognitive maps are susceptible of application of the same control strategies and algorithms to their impulses. But if nature of linear cognitive map is disregarded, their state starts to wander under pressure of external random perturbation (i.e. noise) even though stabilizing controller mitigates their influence on cognitive map’s impulses. Ability of the MPC approach to consider secondary objectives allowed to mitigate this effect at least partially. In particular, it is achieved here by seeking a particular objective cognitive map state as a secondary objective in search for a stabilization trajectory. It is also demonstrated here, that only a certain hyperplane in cognitive map’s state-space is reachable under assumption, that its impulse is zero at the end of trajectory.

Design of Hypersonic Vehicle Time-Delay Compensation Controller based on Dynamic Surface Control

vehicles with input delays is described in this paper. Due to high dynamic characteristics of hypersonic vehicle, the strong uncertainties, high nonlinearity, strong coupling and control input time delays are challenging problems in the design of its control system. Therefore, it is necessary to design a control method that can handle input time delays. Finally, in this paper, the mathematical expression of the altitude nonlinear control theory problem of an aircraft with input time delay is given. Secondly, based on the backstepping method, the time delay compensation controller is designed. Thirdly, the backstepping method and dynamic control surface are used to design altitude angle control law. The stability and time delay compensation effect are proved. Finally, simulation results show that the suggested method can improve the stability and disturbance rejection performance for the system effectively, which has certain actual application value.

System Identification Algorithm for Systems with Interval Coefficients

In this research a new system identification algorithm is presented for obtaining an optimal set of mathematical models for system with perturbed coefficients, then this algorithm is applied practically by an “On Line System Identification Circuit”, based on real time speed response data of a permanent magnet DC motor. Such set of mathematical models represents the physical plant against all variation which may exist in its parameters, and forms a strong mathematical foundation for stability and performance analysis in control theory problems.

A Scalable Distributed Dynamical Systems Approach to Learn the Strongly Connected Components and Diameter of Networks

Finding strongly connected components (SCCs) and the diameter of a directed network play a key role in a variety of discrete optimization problems, and subsequently, machine learning and control theory problems. On the one hand, SCCs are used in solving the 2-satisfiability problem, which has applications in clustering, scheduling, and visualization. On the other hand, the diameter has applications in network learning and discovery problems enabling efficient internet routing and searches, as well as identifying faults in the power grid. In this paper, we leverage consensus-based principles to find the SCCs in a scalable and distributed fashion with a computational complexity of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}(Dd_{\rm{in-degree}}^{\max })$</tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$D$</tex-math></inline-formula> is the (finite) diameter of the network and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d_{\rm{in-degree}}^{\max }$</tex-math></inline-formula> is the maximum in-degree of the network. Additionally, we prove that our algorithm terminates in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$D+1$</tex-math></inline-formula> iterations, which allows us to retrieve the diameter of the network. We illustrate the performance of our algorithm on several random networks, including Erdö-Rényi, Barabási-Albert, and Watts-Strogatz networks.

A coupled problem described by time-dependent subdifferential operator and non-convex perturbed sweeping process

The aim of the present paper is to state the well-posedness of a new dynamical system driven by a differential inclusion involving time-dependent subdifferential operators with integral perturbation and a non-convex perturbed sweeping process. The current study is motivated by recent works on integro-differential sweeping processes and dynamical systems coupled by differential inclusions (involving sweeping processes and maximal monotone operators). In our development, we proceed using a discretization method, combining tools of first-order evolution problems of subdifferential type and those governed by the normal cone of $ r $-prox regular moving sets. We establish a new existence result to the class of dynamical systems under consideration, in the context of infinite dimensional Hilbert setting. This allows us to deal with a Bolza-type problem in optimal control theory.

PROSPECTS FOR THE JOINT USE OF UNMANNED AERIAL VEHICLES

In the article, the development of methods of combat use of unmanned aerial vehicles (UAVs) from the point of view of cause and effect processes in the system "UAVs - means of enemy air defense - purposes of combat mission" is considered by the authors. The main factors of development and changes in the methods of combat use of UAVs are analyzed. The basic patterns and connections of the process of development of methods of joint use of UAVs, the area of conditions for the expedient use of groups of UAVs are highlighted. The scientific problem of the stochastic synthesis of the automated control system for the group application of UAVs, as one of the problems of modern control theory and technical cybernetics, is substantiated and formalized. The question of the possibility of reducing the task of stochastic synthesis of the automated control system for the group application of UAVs to the set of control optimization tasks in linear (linearized) systems according to quadratic criteria is considered.

Limited-control optimal protocols arbitrarily far from equilibrium.

Recent studies have explored finite-time dissipation-minimizing protocols for stochastic thermodynamic systems driven arbitrarily far from equilibrium, when granted full external control to drive the system. However, in both simulation and experimental contexts, systems often may only be controlled with a limited set of degrees of freedom. Here, going beyond slow- and fast-driving approximations employed in previous studies, we obtain exact finite-time optimal protocols for this limited-control setting. By working with deterministic Fokker-Planck probability density time evolution, we can frame the work-minimizing protocol problem in the standard form of an optimal control theory problem. We demonstrate that finding the exact optimal protocol is equivalent to solving a system of Hamiltonian partial differential equations, which in many cases admit efficiently calculable numerical solutions. Within this framework, we reproduce analytical results for the optimal control of harmonic potentials and numerically devise optimal protocols for two anharmonic examples: varying the stiffness of a quartic potential and linearly biasing a double-well potential. We confirm that these optimal protocols outperform other protocols produced through previous methods, in some cases by a substantial amount. We find that for the linearly biased double-well problem, the mean position under the optimal protocol travels at a near-constant velocity. Surprisingly, for a certain timescale and barrier height regime, the optimal protocol is also nonmonotonic in time.

Robust Stability and Stable Member Problems for Multilinear Systems

In this paper, we consider robust stability and stable member problems for linear systems whose characteristic polynomials are nonmonic polynomials with multilinear uncertainty. For both problems, the results are given by using the reflection (box) coefficients and the extreme point property of multilinear functions defined on the box. Finding stable member in a polynomial family is one of the hard problems of linear control theory. This issue is considered by visualizing the cases n-l=2 and n-l=3. Necessary and sufficient conditions for robust stability and the existence of a stable member of the multilinear polynomial family using the reflection coefficients are obtained. Several examples are provided.

On solvability for a class of nonlinear systems of differential equations with the Caputo fractional derivative

We prove solvability and regularity of a solution to a class of fractional nonlinear systems of differential equations with the Caputo derivative by the use of the Leray-Schauder fixed point theorem. We show how the result can be applied in an optimal control theory problem governed by such nonlinear systems of fractional ODEs.

Structure of Optimal Control in Optimal Shaping of the Steel Arch

The paper presents the problem of optimal shaping of the H-bar cross-section of a steel arch that ensures minimal mass. Nineteen combinations of nine basic load states are considered simultaneously in the problem formulation. The optimal shaping task is formulated as a control theory problem within the formal structure of the maximum Pontriagin’s principle. Since the ranges of constraint activity defining the control structure are a priori unknown and must be determined numerically, assuming the proper control structure plays a key role in the task solution. The main achievement of the present work is the determination of a solution of the multi-decision and multi-constraint optimization problem of the arch constituting a primary structural system of the existing building assuring the reduction of the structure mass up to 42%. In addition, the impact of the assumed state constraint value on the solution structure is examined.