Numerical Optimization of Regulators for Automatic Control System

V.A. ZHMUD, L.V. DIMITROV, J. NOSEK



NUMERICAL OPTIMIZATION OF REGULATORS FOR AUTOMATIC CONTROL SYSTEM









NOVOSIBIRSK
2019

UDC 621.375.087.9:681.515/516 (075.8) Z 62




Reviewers:
Dr. of Technical Sciences, Prof. G.A. Frantsuzova, Novosibirsk State Technical University, Novosibirsk, Russia
Dr. of Physics and Mathematical Sciences, Corresponding member of RAN, Prof. A.V. Taichenachev, Institute of Laser Physics, Novosibirsk, Russia




       Zhmud Vadim A.
Z 62 Numerical Optimization of Regulators for Automatic Control System: Textbook for higher education / Vadim A. Zhmud, Lubomir V. Dimitrov, Jaroslav Nosek. - Novosibirsk: NSTU Publisher, 2019. - 296 p.
          ISBN 978-5-7782-3802-2
          The book is intended for students and PhD students in the field of automatics, mechatronics, robotics, digital or analog feedback control. Par-ticullary, in NSTU (Novosibirsk, Russia) it is intended for the students enrolled in the educational field 27.03.04 “Control in Technical Systems” in the course “Computer-aided design tools and control systems” (for Bachelors in their 4th year). It includes training materials and guidelines for selfassessment (questions for self-testing).
          The basic knowledge of previously studied mathematical disciplines is required for successful study of this course.
          The book is prepared due to the financial support from the European Foundation Erasmus+, projects KA-107A and SmartCity.






UDC 621.375.087.9:681.515/516 (075.8)


ISBN 978-5-7782-3802-2

               © Vadim A. Zhmud, Lubomir V. Dimitrov,

                 Jaroslav Nosek, 2019
© Novosibirsk State Technical University, 2019

Content


Introductory words...........................................................7
1.  Introduction .............................................................8
   1.1. The subject of study: locked dynamical control systems ...............8
   1.2. Feature of ACS: negative feedback....................................10
   1.3. The task of regulator calculation....................................15
   1.4. Questions for self-control...........................................18
2.  Terminology and mathematical tools of TAC ...............................20
   2.1. Basic requirements and mathematical tools for the system.............20
   2.2. Requirements for the possibility of the physical realization of the model......................................................................22
   2.3. The choice of the regulator structure ...............................24
   2.4. Regulator with fixed coefficients ...................................25
   2.5. Regulators with a non-integer integration and differentiation .......27
   2.6. Classification of developing direction of methods of the designing of ACS....................................................................28
   2.7. Adaptive and self-tuning regulators .................................30
   2.8. Adaptive regulators and prospects of this approach...................31
   2.9. Self-tuning regulators...............................................34
   2.10.  Robust regulators .................................................35
   2.11.  Advantages of digital regulators ..................................40
   2.12.  The two kind of tasks of regulator numerical optimization .........42
   2.13.  Objectives and tasks for the further research .....................43
3.  Visual simulation of unlocked stuctutres ................................45
   3.1. The window of the software VisSim....................................45
   3.2. Start of the working in the software ................................49
   3.3. Tuning of the parameters of the simulation and optimization..........51
   3.4. Choice of the time sampling step ....................................54

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   3.5. Choice of the integrating method......................................56
   3.6. Tuning of the parameters of simulation and optimization...............60
   3.7. Simulation of the response of the linear link.........................61
   3.8. Simulation of the response of non-linear links........................64
4.  Visual simulation of the locked structures ...............................66
   4.1. Simulation of the locked linear system ................................66
   4.2. The differences between theoretical analysis, simulation and practical results ...................................................................73
5.  Statement of the task of optimization of locked structures and tools for its resolving ...........................................................76
   5.1. Demands to the locked loops of the model for simulation ...............76
   5.1.1. Demands for the possibility of simulation ..........................76
      5.1.2.  Demands to the feedback loop from the point of view of the adequacy of the regulators to their tasks ...............................77
      5.1.3.  Demands to the cost function ...................................81
      5.1.3.  Additional demands to the locked systems........................83
   5.2. Structure for the regulator optimization ..............................84
   5.3. Tools of the cost functions ...........................................86
   5.4. An example of the system quality analysis based on the cost function...91
   5.5. Grounds for choosing of weight coefficients in complex cost functions .93
6.  Numerical optimization of the locked structures ..........................96
   6.1. Procedure for the automatic optimization of regulators ................96
   6.2. Fixing of the coefficients ............................................104
   6.3. Optimization of the ensemble of systems for the robust control.........105
   6.4. The validity of the model for optimizing of the regulator..............108
   6.5. Forced limitation of the regulator coefficients .....................111
   6.6. Forced limitation of the frequency range of the model used in optimizing of the regulator ....................................................114
7.  Dividing of the motion: the use of multiple drives (MISO) ...............117
   7.1. Justification for an excessive number of the influencing actions on the object ...................................................................117
   7.2. Combining of the advantages of the different drives .................117
8.  Dividing of motions: using of multiple sensors (SIMO) ...................126
   8.1. Combining of the advantages of the different sensors ................126
   8.2. Simultaneous combing of the advantages of the different sensors and different drives .........................................................133

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9.  Modification of the cost functions ....................................133
   9.1. Providing of the power saving .....................................134
   9.2. The detector of the error growth ..................................139
   9.3. Optimization with the use of etalon transient response.............142
10.  New structures for single channel objects (SISO) .....................147
   10.1. Robust power saving two-channel regulator with single output .....148
   10.2. Robust power saving two-channel regulator with single output object, prone to oscillations ..........................................154
   10.3. Regulator with separation of “right” and “wrong” movements .......158
   10.4. The use of bypass channel for feedback control of an oscillating object..................................................................164
11.  Feedback systems with pseudo local loops .............................172
   11.1. About local and pseudo local loops ...............................172
   11.2. The control of objects with two integrators ......................173
   11.3. The control of objects with two integrators and non-linear positive feedback ...............................................................175
   11.4. The control of objects with three integrators and non-linear positive feedback ...............................................................178
12.  Design of a piecewise adaptive regulator .............................189
   12.1. Robust system as a prototype of the adaptive system ..............189
   12.2. An example of splitting a set of object parameters into subsets...191
   12.3. Identifying the belonging of the object model to a given subset...192
13.  Optimization of the regulator for multichannel objects (MIMO) ........194
   13.1. The task of control of multichannel object........................194
   13.2. The statement of the problem and the solvability conditions.......197
   13.3. Methods for solving the problem ..................................199
   13.4. The efficiency of the completeness of a PID-regulator when controlling a multi-channel object.............................................200
   13.5. The idea of Smith multichannel predictor..........................209
   13.6. Numerical optimization of the regulator for an object of dimension 3 х 3...................................................................232
14.  Investigation of the numerical optimization toolkit for control of an oscillatory unstable object .........................................252
   14.1. Statement of the problem .........................................253
   14.2. Methods for solving the problem, theory and practical results ....254
      14.2.1. Optimizing the PID-regulator.................................254

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      14.2.2. The optimization of PIDD-regulator .......................256
      14.2.3. Bypass channel application ...............................257
      14.2.4. Using pseudo-local links and an open object model.........259
      14.2.5. The danger of obtaining non-robust solution...............265
  14.3.  The proposed modification of the known  methods ...............267
  14.4.  Discussion ....................................................271
  14.5.  Conclusions for the chapter....................................272
15.  Practical recearch and laboratory works in the field of feedback control in technical universities of Liberec and Sofia ....................272
16.  Questions for control at exams ....................................284
Concluslion ............................................................287
Acknowledgments ........................................................288
References..............................................................289
Apendix.................................................................293

                INTRODUCTORY WORDS





    This fascinating manual introduces the reader to the complicated matter of automatic control systems in a pleasant and intriguing manner. I as a control theory specialist read in one breath. It is written in an accessible language and is illustrated with many examples. Moreover, it contains tasks taken from the practice. The textbook is written by authors from Russia, Bulgaria, and Czech Republic and is a result of a joint Tempus Project No. 517138 - JPCR that aims at double degree diplomas on Mechatronics issued by Novosibirsk Technical University and Sofia Technical University or Liberec University. My expertise in theory and practice of Automatic Control Systems makes me to believe that this textbook is useful both for Bachelors and Masters of the Technical University of Sofia who are trained in the fields "Mechatronics," "Mechatronic Systems", and "Industrial Engineering", and in particular in the following disciplines: Electronic control and control devices and systems, Mechatronics systems and management systems, Mechatronic systems and management systems, Control systems and control systems. In addition, this textbook could be of benefit to PhD students and researchers working in areas close to the automatic control system.


Assoc. prof. Pancho Tomov Faculty of Mechanical Engineering Technical University of Sofia, Specialist in Automation

                1. INTRODUCTION





            1.1.      The subject of study: locked dynamical control systems


    The subject of this work is locked automatic control systems (ACS) [1-18]. An example of such system can be any robotic arm assembly, as well as any part of robot manipulator. Such systems are needed not only in robotics but also in all areas of engineering, technology, industry, business, and science. Wherever there is mechanical control of a physical value of an object, there are such systems.
    The main goal of this book is to develop methods of numerical optimization of ACS. Main tools are connected with software VisSim [19-20].
    The software complex Simulink + MATLAB (along with the set of packages for their expansion Toolbox and Blockset), as V. Dyakonov rightly notes [49], proved too cumbersome for such relatively simple applications as modeling and optimization of regulators. This package takes up a noticeably large amount of memory, an excessively large library of blocks, most of which are unnecessarily specialized and are not required in most of the tasks to be solved; the created files also occupy a fairly large amount of memory, in total this amounts to several gigabytes. For this reason, there has recently been a sharp increase in interest in a small-scale but sufficient universal system of block imitation visual-oriented mathematical modeling, such as VisSim. This software has been created by the corporation Visual Solution Inc. (USA). The main developer of the software and the Head of the corporation is Peter Darnell. Along with the system itself, a number of packages for its expansion have been released, significantly increasing the already tangible capabilities of this system.
    This “pearl” in the world of mathematical modeling programs has long attracted the interest of specialists in the field of mathematical modeling. For example, the corporation MathSoft, the creator of the famous and most

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popular computer mathematics Mathcad, not only ensured the docking of this system with the VisSim, but also began to supply VisSim with some versions of the Mathcad package [49]. The VisSim version can also be integrated with the Simulink + MATLAB system [49].
    A major contribution to the distribution of the VisSim was made by the site www.vissim.nm.ru, created under the direction of N. Klimachev from the South Ural University, some versions of this program, for example, Vis-Sim 3.0, are distributed free of charge, later versions can be used within 60 days for free, these versions can be obtained from the website www.vissim.com [69].
    The objectives of the work are description and discussion of the developed modeling techniques, the example of research and optimization of such systems, development of recommendations for their further modifications and their successful application [21-42].
    Modern production processes are inconceivable without ACS, providing stabilization of a number of important characteristics of these processes and their control by the prescribed rules of technology.
    These systems are dynamic because in the calculation of such systems one must take into account the dynamic properties of all their elements and relationships. The action of these systems evolves over time and essentially depends on the dynamic characteristics of the system as a whole, which complexly depends on the dynamic characteristics of its constituent elements. When taking into account the feedback action it is inadmissible to neglect the dynamic errors in the signal path delays, dynamic dependencies, because such neglect will give an erroneous result.
    We can illustrate this with the following two examples.
    Example 1. Let us assume that a car is moving along the road and the driver follows the route with the help of the global satellite navigation system. The system indicates the correct direction of travel. The driver adjusts the car's route, so that the car is moving closer to the goal. A new possible route requires a new direction of movement. The entire system, consisting of a vehicle (object), the driver (regulator) and the navigation system (sensor of the output value) is closed. However, it is not necessary to consider this system as dynamic system, although its action takes place in time. The time it takes to update the position of the object and to adjust the direction of travel is negligible compared with the time that is required to achieve such a new situation in which the previous instructions in the direction of movement are no longer relevant. Delay in the adjusting of the course is so insignificant in comparison with the time of movement on the

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exchange rate that the system can become unstable as the result of this delay.
    In addition, this system is not automatic, because a human is presented in the loop of its control. Systems with a human in the loop are automated but not automatic. This book does not consider such systems.
    Example 2. Let us assume that a two-wheeled motor axis is an autobalancing device. Such system is called balancing robot or Segway and includes position sensors devices, drives to the motor, the motor itself and the power supply of the entire system. The center of mass of this system is higher than the wheel axis of rotation. Therefore, in the absence of a control signal to the motor system capsize so that the center of mass located below the axis of rotation. If you want the center of mass to remain above the axis constantly, it is required to form the impact on the axis to this balancing robot to maintain a balance by moving forward or backward. If in such system the developer wrongly considers the time of receipt of the signal from the sensors, it will not be able to work steadily. The speed of all its components must be strictly coordinated with each other and match the pace of change random noise acting on the system. This system is dynamic. Such systems are the subject of this book.



            1.2.     Feature of ACS: negative feedback


    Let us consider management of a structure without feedback, as shown in Fig. 1.1 and explain why this structure is not applicable.


Fig. 1.1. Control without feedback

    Any device or physical object can be treated as control object if it has output value and input for changing this value. The output value Y(t) depends on the input controlling value U(t). If there is not such a relationship, then the object cannot be controlled. If the relationship is strictly deterministic and the output depends only on the input signal, the control of such object is not among the tasks we have to solve, as it is fully described by its

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