scholarly journals SYNTHESIS OF AN ADAPTIVE AUTOMATIC SYSTEM OF AIRCRAFT CONTROLS WITH MULTIDIMENSIONAL PI-REGULATOR

2019 ◽  
Vol 27 (4) ◽  
pp. 79-85
Author(s):  
Aramis Viktorovich Tishchenko ◽  
Anatoly Mikhailovich Kulabukhov ◽  
Victor Alexandrovich Masalskiy
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Minimum Mean Square Error ◽  
Center Of Mass ◽  
Control Object ◽  
The State ◽  
Quadratic Functional ◽  
State Variables ◽  
Control Actions ◽  
Conditional Minimization

The article presents the synthesis of a functional diagram of an adaptive automatic control system (ACS) for controlling an aircraft with an automatically reconfigurable multidimensional PI controller, which provides the minimum static and minimum mean square error of control with minimal energy consumption for the formation of the control exposure. The synthesis of ACS algorithms is performed as a result of solving the problem of conditionally minimizing the quadratic functional of the generalized work (taking into account restrictions on state variables and control actions given by differential equations of the control object (CO) and inequalities). The mathematical description of the multidimensional CO is carried out using the CO model in the state space, which automatically takes into account the mutual influence of individual control loops on each other. As the state variables of the aircraft, linear displacements, speeds and accelerations of the center of mass of the aircraft, and angular displacements, speeds and accelerations of the rotational movement of the aircraft relative to the center of mass are used. The matrix equation of dynamics of the aircraft is formed by a system of nonlinear differential equations of the first order of forces and moments of forces acting on the aircraft. To ensure the minimum static control error, integrators are included in the ACS (for each control action). The algorithm for the formation of control actions of the extended CO, providing the declared properties of the ACS, is obtained as a result of solving the problem of conditional minimization of the generalized work functional. The task of conditional minimization of a functional with constraints is performed by the maximum principle. The resulting two-point boundary value problem is transformed by the invariant immersion method into a Cauchy problem for optimal values of state variables. The evaluation of the characteristics of a specific adaptive ACS for the spacecraft is expected to be obtained as a result of further research by mathematical modeling.

Filippov solutions of vector Dirichlet problems

2020 ◽  
Vol 70 (2) ◽  
pp. 401-416
Author(s):  
Hana Machů
Keyword(s):  
Differential Equations ◽  
The State ◽  
State Variables ◽  
Dirichlet Problems ◽  
Filippov Solutions ◽  
Natural Notion ◽  
Growth Restrictions ◽  
The One ◽  
The Right ◽  
Effective Growth

Abstract If in the right-hand sides of given differential equations occur discontinuities in the state variables, then the natural notion of a solution is the one in the sense of Filippov. In our paper, we will consider this type of solutions for vector Dirichlet problems. The obtained theorems deal with the existence and localization of Filippov solutions, under effective growth restrictions. Two illustrative examples are supplied.

MODELS OF WWER-1000 NUCLEAR REACTOR WITH DIVISION INTO ZONES ON VERTICAL AXIS FOR INFORMATION TECHNOLOGY OF CONTROL

2021 ◽  
Vol 4 ◽  
pp. 105-116
Author(s):  
Valeriy Severyn ◽  
◽  
Elena Nikulina ◽  
◽  
Keyword(s):  
Information Technology ◽  
Differential Equations ◽  
Mathematical Models ◽  
Nuclear Power ◽  
Nuclear Reactor ◽  
Control Object ◽  
Vertical Axis ◽  
State Variables ◽  
Relative State ◽  
Systems Of Differential Equations

Mathematical models of the WWER-1000 nuclear power reactor have been developed with division into zones along the vertical axis in the form of nonlinear systems of differential equations with dimensionless relative state variables. Models in a given number of zones along the vertical axis represent neutron kinetics, gradual heat release, thermal processes in fuel, cladding and coolant, changes in the concentration of iodine, xenon and boron. The parameters of mathematical models have been calculated based on the design and technological parameters of the V-320 series nuclear reactor. A general model of the reactor as a control object with division into zones along the vertical axis, as well as models with control of absorbing rods and boric acid, are obtained. Integration of the obtained systems of differential equations for given initial conditions allows one to obtain changes in all state variables in the reactor zones along the vertical axis. In particular, from the change in power in the zones along the vertical axis, the axial offset is calculated as the relative value of the difference between the powers of the upper and lower halves of the reactor core. The developed reactor models with dimensionless relative state variables use a minimum number of calculations, allow calculating the change in the axial offset, and are included in the information technology for controlling the power units of nuclear power plants to optimize the maneuvering modes of the WWER-1000 V-320 series reactor.

Output Control by Linear Time-Varying Systems using Parametric Identification Methods

2020 ◽  
Vol 21 (7) ◽  
pp. 387-393
Author(s):  
V. Q. Dat ◽  
A. A. Bobtsov
Keyword(s):  
Differential Equation ◽  
Initial Conditions ◽  
Control Object ◽  
The State ◽  
Time Varying ◽  
State Variables ◽  
Matrix Differential Equation ◽  
Matrix Differential ◽  
Uniform Observability ◽  
Riccati Matrix

In this paper the problem of control for time-varying linear systems by the output (i.e. without measuring the vector of state variables or derivatives of the output signal) was considered. For the control design, the well-known online procedure for solving the Riccati matrix differential equation is chosen. This procedure involves the synthesis of linear static feedbacks on state variables in the case of known parameters of the plant. If state variables are not measured, then for the observer design using the matrix Riccati differential equation, using the dual scheme, which provides for the transposition of the state matrix and the replacement of the input matrix by the output matrix. It is well known that an observer of state variables built on the basis of a solution of the Riccati matrix differential equation ensures the exponential stability of a closed loop system in the case of uniform observability. Despite the fact that this type of observer can be classified as universal, its have a number of significant drawbacks. The main problem of such observers is the need for accurate knowledge of the parameters and the requirement for uniform observability, which in practice cannot always be realized. Thus, the problem of the new methods design for constructing observers of state variables of linear non-stationary systems is still relevant. Some time ago, a number of methods for the adaptive observers design of state variables for nonlinear systems were proposed. The main idea of the synthesis of observers was based on the transformation of the original dynamic system to a linear regression model containing unknown parameters, which in turn were functions of the initial conditions of the state variables of the control object. This approach in the English language literature is called PEBO. This paper, based on the PEBO method, proposes a new approach for the observers design of state for non-stationary systems. This approach provides the possibility of obtaining monotonic convergence estimates with transient time tuning.

scholarly journals Specific Mathematical Aspects of Dynamics Generated by Coherence Functions

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Ezzat G. Bakhoum ◽  
Cristian Toma
Keyword(s):  
Differential Equations ◽  
Complex Systems ◽  
Integral Equations ◽  
Quantum Physics ◽  
Initial Conditions ◽  
Coherence Function ◽  
Multiple Integral ◽  
The State ◽  
Integrodifferential Equations ◽  
State Variables

This study presents specific aspects of dynamics generated by the coherence function (acting in an integral manner). It is considered that an oscillating system starting to work from initial nonzero conditions is commanded by the coherence function between the output of the system and an alternating function of a certain frequency. For different initial conditions, the evolution of the system is analyzed. The equivalence between integrodifferential equations and integral equations implying the same number of state variables is investigated; it is shown that integro-differential equations of second order are far more restrictive regarding the initial conditions for the state variables. Then, the analysis is extended to equations of evolution where the coherence function is acting under the form of a multiple integral. It is shown that for the coherence function represented under the form of annth integral, some specific aspects as multiscale behaviour suitable for modelling transitions in complex systems (e.g., quantum physics) could be noticed whennequals 4, 5, or 6.

Convergence of the identification algorithm applied to the mutual inductance of the induction motor

2012 ◽  
Vol 61 (3) ◽  
pp. 337-345
Author(s):  
Andrzej Jąderko ◽  
Jędrzej Pietryka
Keyword(s):  
Differential Equations ◽  
Induction Motor ◽  
Mutual Inductance ◽  
The State ◽  
Position Vector ◽  
Identification Algorithm ◽  
State Variables ◽  
Vector Length ◽  
Parameter Versus ◽  
Observer System

Convergence of the identification algorithm applied to the mutual inductance of the induction motor A new observer of induction motor state variables is proposed in the paper. A nonlinearity of the main magnetic path is expressed as a function of a properly chosen parameter versus the position vector length. The value of the mutual inductance received in the identification algorithm is calculated exploiting the estimated values of the state variables. The coefficients appearing in the differential equations of the observer system are modified in each step of the algorithm on the basis of the calculated mutual inductance. The analysis of convergence of the identification algorithm is shown in this paper.

scholarly journals PATH FOLLOWING PROBLEM FOR UAV CARRYING PENDULUM

2021 ◽  
Vol 55 (1 (254)) ◽  
pp. 56-63
Author(s):  
Arman S. Shahinyan
Keyword(s):  
Inverse Problems ◽  
Equilibrium Point ◽  
Center Of Mass ◽  
Path Following ◽  
The State ◽  
State Trajectory ◽  
Control Actions ◽  
Inverse Problems Of Dynamics ◽  
Yaw Angle

The linearized dynamics of a UAV is considered along with a pendulum hanging from it. The state trajectories of the center of mass of the UAV are given. Given the trajectory of the center of mass of the UAV and the state trajectory of its yaw angle, we have to find the control actions and conditions under which the UAV would follow the path while holding the pendulum stable around its lower equilibrium point. The problem is solved using the method for solving inverse problems of dynamics. All the state trajectories of the system and all the control actions are calculated. The condition is obtained under which a solution to the path following problem exists. A specified simple trajectory is chosen as an example for visualizing the results.

THE MARGINAL STABILITY OF AUTONOMOUS DIFFERENTIAL EQUATIONS

1993 ◽  
Vol 03 (03) ◽  
pp. 785-788
Author(s):  
N.N. GREENBAUN
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Steady State Solution ◽  
Usual Method ◽  
State Solution ◽  
Marginal Stability ◽  
State Variables ◽  
Autonomous Differential Equations

Solutions in the vicinity of a steady state solution to a system of autonomous nonlinear differential equations are of interest to modelers. The usual method for determining marginal stability of the steady state is the Routh-Hurwitz criterion. The method offered here is less complicated and more efficient when the number of state variables exceeds three.

Application of methods of conditional multidimensional minimization to the ballistic trajectory calculation problem

2017 ◽  
pp. 59-62
Author(s):  
A. A. Dubrovina
Keyword(s):  
Differential Equations ◽  
Entire Range ◽  
Nonlinear Differential Equations ◽  
Operating Time ◽  
Physical Significance ◽  
Trajectory Calculation ◽  
Flight Path Angle ◽  
Conditional Minimization ◽  
Ballistic Trajectory ◽  
Range Rate

The paper focuses on the problem of calculating the approximate ballistic missile trajectory, the calculation ensuring that the missile travels from a given launch point to the finish point and covering the entire range rate for the missiles of the type considered. The missile trajectory is defined by a system of nonlinear differential equations. A different range is achieved by changing the initial values of the flight-path angle and the operating time of the missile stages. Due to the physical significance, these variables are constrained. The problem of multidimensional conditional minimization by the method of barrier functions with minimization of Nelder - Meed method

scholarly journals Benchmark: A Nonlinear Reachability Analysis Test Set from Numerical Analysis

10.29007/6dcf ◽  
2018 ◽  
Author(s):  
Hoang-Dung Tran ◽  
Luan Viet Nguyen ◽  
Taylor T Johnson
Keyword(s):  
Numerical Analysis ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Differential Algebraic Equations ◽  
Hybrid Automata ◽  
State Variables ◽  
Algebraic Equations ◽  
Test Set ◽  
Verification Methods ◽  
Highly Nonlinear

The field of numerical analysis has developed numerous benchmarks for evaluating differential and algebraic equation solvers. In this paper, we describe a set of benchmarks commonly used in numerical analysis that may also be effective for evaluating continuous and hybrid systems reachability and verification methods. Many of these examples are challenging and have highly nonlinear differential equations and upwards of tens of dimensions (state variables). Additionally, many examples in numerical analysis are originally encoded as differential algebraic equations (DAEs) with index greater than one or as implicit differential equations (IDEs), which are challenging to model as hybrid automata. We present executable models for ten benchmarks from a test set for initial value problems (IVPs) in the SpaceEx format (allowing for nonlinear equations instead of restricting to affine) and illustrate their conversion to several other formats (dReach, Flow*, and the MathWorks Simulink/Stateflow [SLSF]) using the HyST tool. For some instances, we present successful analysis results using dReach and SLSF.

Fuzzy Model of Situational Control of the Flight Parameters of an Autonomous Unmanned Aircraft under Uncertainty Conditions

2021 ◽  
Vol 22 (12) ◽  
pp. 650-659
Author(s):  
V. B. Melekhin ◽  
M. V. Khachumov
Keyword(s):  
Environmental Factors ◽  
Unmanned Aerial Vehicle ◽  
Fuzzy Model ◽  
Control Object ◽  
The State ◽  
Effective Control ◽  
Problem Situations ◽  
Control Actions ◽  
Aerial Vehicle ◽  
Control Principle

The article outlines the main problems of automatic planning of the behavior of an autonomous unmanned aerial vehicle in unstable air conditions. It is shown that the urgency of the problem is due to the fact that an autonomous unmanned aerial vehicle independently forms and implements its flight route without support from a ground control station. There is therefore a need to develop a method for automatic control of programmed movements associated with the implementation of the route constructed by the problem solver. To solve this problem we propose an approach to regulating the parameters of the state of dynamic objects based on the principle of situational control of the goal-directed behavior of complex systems in changing environmental conditions. The expediency of choosing this control principle is due to the fact that the state of an autonomous unmanned aerial vehicle during its flight is characterized by a large number of parameters and disturbing environmental factors. In order to effectively implement this control principle, we introduce the concept of a complete problematic situation, which consists of deviations of the state parameters of an autonomous unmanned aerial vehicle from the required values during flight and disturbing environmental factors. On this basis, a fuzzy model of situational control of the state parameters of an autonomous unmanned aerial vehicle functioning in an unstable environment is developed, in which linguistic variables and functions are used to provide a generalized presentation of reference problem situations, as well as to describe the deviations of the state parameters and disturbing environmental factors. The conditions are determined under which the reference indistinctly presented problem situations generalize the actual problem situations that arise at the control object. This makes it possible to significantly reduce the number of logical-transformational decision rules in the situational control model and to promptly automatically determine effective control actions in problematic situations that ensure the effective implementation of programmed movements of an autonomous unmanned aerial vehicle under conditions of uncertainty. In conclusion, it is shown that for the implementation of control actions which are selected on a situational basis with increased requirements for the accuracy of regulation of the time-varying parameters of the control object and a significant level of possible discrepancies between their actual and specified values in conditions of uncertainty, it is advisable to use indistinctly implemented proportional, integral and differential regulation laws.

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