A Computer Package for the Synthesis of Optimal Control Systems with Arbitrary Performance Index

This work is proposed to continue the discussion of the problems, theoretical foundations and practical features of the construction and synthesis of robust control systems with high gain, allowing us to control multidimensional nonlinear dynamic objects of high dimensional with functional uncertainties. If problems could not be solved at the level where they appeared, it is necessary to rise the level of understanding of the laws of nature, or in the words of master Lui-Shi Chun Qiu (China, 3rd century BC): "The boy of five chi growth leads the bull by the bridle and the bull obeys him in everything. This is because the person in this case follows naturalness" (the laws of nature). The judo philosophy ("soft way") is based on the principles of using the power and energy of the opponent to achieve victory. The purpose of this work is to demonstrate the theoretical aspects and practical features of the methods of synthesis of optimal control systems by the criterion of maximum reproduction accuracy using the example of robust systems, which allow to control dynamic objects with functional uncertainties, including unstable objects, no minimal-phase objects, neutral objects and objects with differentiation properties. The simplicity (at the level of the engineer) and universality, mathematical rigor and physical validity of this approach is based on the judo philosophy: suppressing the dynamics of a functionally uncertain object and external disturbances by the infinitely large gain with the finite control signal and at the same time maintaining sustainability. Theoretically exhaustive solution of the problem of robust control is given by the idea of constructing systems that are stable with an unlimited increase of the gain coefficient. The sustainability properties are valid for optimal systems that were synthesized using quadratic quality functionals that do not explicitly depend on the control signal, and using a restriction on the control signal. It is significant that in contrast to continuous systems with un-measurable disturbances and not well known control object (in which the conditions of invariance imply the use of infinitely large gain), in relay (discontinuous) systems the equivalent effect is achieved with the help of finite control signal. A nice bonus is the highest accuracy which leads to mathematically zero error of regulation, thus all error coefficients (of position, speed, acceleration, acceleration derivative, etc.) is also equal to zero in the presence of external and internal interferences. In fact, the optimal accuracy control system is equivalent to a system with astatism of the n-th order: the regulator contains n serial connected integrators.

Анотація . Розроблено метод оптимального налаштування ПІ-регуляторів при врахуванні обмежень. Метод ґрунтується на зведенні задачі налаштування до задачі безумовної оптимізації. Для цього було запропоновано спеціальний комплексний критерій, який характеризується бажаними топологічними властивостями. Останні отримані за допомогою вагових коефіцієнтів, що враховують вимоги щодо стійкості системи, обмежень та комплексу критеріїв, які відображають небажані показники регулювання. Наперед задані топологічні властивості критерію дали змогу знайти його глобальний. У роботі для цієї мети була використана модифікація методу рою часточок ME - PSO , яка характеризується переініціалізацією рою при умові його стагнації. Для оцінки ефективності використання методу були виконані розрахунки коефіцієнтів ПІ-регулятора при обмеженні за перерегулюванням (прийнято, що воно повинно бути нульовим). Критерієм оптимізації була швидкодія регулювання. У результаті отримано, що для чотирьох тестових передаточних функцій (другого та третього порядків; одна передаточна функція відповідає об'єкту регулювання із часовою затримкою) швидкодія регулювання значно збільшується при забезпеченні обмеження за перерегулюванням. Отримані результати порівнювались із тими, що відповідають регуляторам, що налаштовані із використанням інших (відомих у інженерній практиці) методів налаштування ПІ-регуляторів (порівнювані методи дають змогу отримати нульове перерегулювання). Крім того, аналіз якості автоматичного регулювання був проведений із використанням показників: середньоінтегральної похибки та середньоінтегрального регулювання. Розроблений підхід доволі універсальний. Він може використовуватись для синтезу оптимальних систем регулювання у тому числі і нелінійних. Ключові слова: регулятор, оптимізація, критерій, обмеження, налаштування, метод рою часточок

Synthesis of optimal stochastic control systems by the method of successive approximations: PMM vol. 43, no. 1. 1979, pp. 7–16

This paper presents a unified canonical transformationand generating function approach, including associated numerical algorithms,for linear time-varying optimal control problems with various terminalconstraints. Generating functions are employed to find the optimal controllaw by solving Hamiltonian two-point-boundary-value problems. Thetime-varying optimal control laws constructed by the second type generatingfunction do not have infinite feedback gain at terminal time, which isdifferent from other existing solutions. Motivated by practical design oftime-varying optimal control systems, a structure-preserving matrixrecursive algorithm is proposed to solve coupled time-varying matrixdifferential equations of the generating function; derivation of therecursive algorithm is based on symplectic formulation of canonicaltransformation. To preserve symplectic structure of matrices in therecursive computation, state transition matrices of the Hamiltonian systemare calculated by Magnus series. In fact, the canonical transformation andgenerating function method leads to a geometric perspective to thedesign and computation of optimal control systems.%control systems synthesis and computation.

The optimal performance of linear pulse-frequency-modulated (PFM) control systems is investigated. A performance index consisting of a weighted sum of the quadratic terminal state error and fuel consumption is used. The direct synthesis of optimal PFM control systems is difficult, but upper and lower bounds on the optimal values of the performance index can be established without difficulty. These bounds are formed by the solutions of two auxiliary optimal control problems. When the upper and lower bounds are close to one another, PFM control may be replaced by a more easily implemented mode of control. Two additional modes-pulse-amplitude modulation and pulse-width modulation-also fall within these bounds.

The algorithm of structural and parametric synthesis of automatic control systems for limitedly uncertain non-linear objects with unstable balance status and right own values in their exit matrix is proposed. The pecu-liarity of the structural synthesis algorithm is that synthesized automatic control systems use differential com-pensating connection with two components. The signal of one of them is proportional to the signal of disa-greement between the required and measured values of the state vector of the system. A signal of another com-ponent is formed by the sign-operated integral feedback on the module of the control influence. The algorithm of parametric synthesis is reduced to solving the problem at a conditional minimum of integral regulatory error when limiting the characteristic frequency multi-member module to an algorithmically linearized system.

The synthesis of control systems for overall processes remains a challenge. Whilst a number of automated approaches have been explored academically, in practice systems are mainly synthesised using experience and intuition in conjunction with dynamic simulation software. An approach is presented in this paper for the systematic synthesis of overall process control systems which is simple, practical and appropriate for use at the undergraduate teaching level. Many potential problems can be identified and avoided in the synthesis of process control systems by ensuring that the control system can maintain a steady-state control of the overall mass, the mass of the individual components and, where necessary, multiple phases. Processes with recycles are vulnerable to process control problems when the overall mass balance of the system containing the recycle is not correctly managed. This can be overcome by self-regulation (perhaps involving process design changes) or by changing the structure of the control system. Achieving and maintaining the material balance are the main purposes of the majority of controllers in most chemical processes. Without robust delivery of material balance control, higher level control objectives (such as quality controls) are unlikely to be met reliably. This paper presents a step-wise approach to the synthesis of overall process control systems that avoids many potential problems by identifying which process parameters can be manipulated to achieve the intended process mass balance and designing the control structure accordingly.

Computer-aided control system design (CACSD) encompasses a broad range of methods, tools, and technologies for system modelling, control system synthesis and tuning, dynamic system analysis and simulation, as well as validation and verification. The domain of CACSD enlarged progressively over decades from simple collections of algorithms and programs for control system analysis and synthesis to comprehensive tool sets and user-friendly environments supporting all aspects of developing and deploying advanced control systems in various application fields. This entry gives a brief introduction to CACSD and reviews the evolution of key concepts and technologies underlying the CACSD domain. Several cornerstone achievements in developing reliable numerical algorithms; implementing robust numerical software; and developing sophisticated integrated modelling, simulation, and design environments are highlighted.

MODELLING AND DESIGN OF CONTROL SYSTEMS BASED ON PETRI NETS MODELS

Main concepts and definitions.- Formalization of dynamic system investigation problems.- A priority information obtainment and analysis on input actions and its derivatives.- Robust Wiener filtering.- Analysis of control error dispersion on numerical characteristics of actions.- Use of canonical presentations for sequence of spectral density moments.- Analysis of maximal control error.- Synthesis of control systems with preset ensured operation accuracy.- Optimization of control systems on criterion of highest ensured accuracy.- Digital robust systems.

A new two-step approach for solving a control system synthesis problem by symbolic regression methods

A two-time scale discrete control system is considered. The closed-loop optimal linear quadratic (LQ) regulator for the system requires the solution of a full-order algebraic matrix Riccati equation. Alternatively, the original system is decomposed into reduced-order slow and fast subsystems. The closed-loop optimal control of the subsystems requires the solution of two algebraic matrix Riccati equations of order lower than that required for the full-order system. A composite, closed-loop suboptimal control is created from the sum of the slow and fast feedback optimal controls. Numerical results obtained for an aircraft model show a very close agreement between the exact (optimal) solutions and computationally simpler composite (suboptimal) solutions. The main advantage of the method is the considerable reduction in the overall computational requirements for the closed-loop optimal control of digital flight systems.

On synthesis of optimal bang-bang feedback control systems with quadratic performance index

Collision avoidance is very important problem in the domain of multi-robot interaction. In this paper we propose a new approach of collision avoidance in the context of the optimal control system synthesis problem definition with minimal information available. It is assumed that robots have a certain scope within which they can interact with static and dynamic phase constraints. A group of robots is considered to be homogeneous, and control system unit for reaching terminal states already available to robots. The control system which is responsible for collision avoidance is only activated when the nearest neighbor is located in the scope of the considered robot. The first important feature of this work is the fact that the collision avoidance between two robots is reciprocal with joint control system, without assigning priorities. Another key feature of this work is the complete absence of information about the environment and the current state of other robots at given time. Robots only share information with nearest neighbors if they locate in the scope of each other. We also present a computational experiment with mobile robots as control objects. A multilayer perceptron was used to approximate the control function. Weights of the perceptron were optimized in unsupervised paradigm by an algorithm belonging to the evolutionary strategies class. At the beginning of each epoch we generate a sample of collision scenarios for optimization, while the quality criterion of the achieved weights at the end of epoch is evaluated on a fixed test sample. Experimental results demonstrate strong ability of the optimized multilayer perceptron to map the relative state of two mobile robots to controls in order to avoid collisions.

Synthesis of an optimal discrete control system in the presence of control delay