# Numerical Optimization of Regulators for Automatic Control System

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Новосибирский государственный технический университет

Год издания: 2019

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ВО - Бакалавриат

ISBN: 978-5-7782-3802-2

Артикул: 777808.01.99

The book is intended for students and PhD students in the field of automatics, mechatronics, robotics, digital or analog feedback control. Particullary, 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.

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- 621: Общее машиностроение. Ядерная техника. Электротехника. Технология машиностроения в целом
- 681: Точная механика. Автоматика. Приборостроение

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- 27.03.04: Управление в технических системах

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V.A. ZHMUD, L.V. DIMITROV, J. NOSEK NUMERICAL OPTIMIZATION OF REGULATORS FOR AUTOMATIC CONTROL SYSTEM NOVOSIBIRSK 2019

DC 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

ontent 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 3

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 4

. 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 5

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 8

opular 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 9

xchange 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 10

nputs as usual peripheral device, such as a stepper motor. The tasks of this book include control of objects, which operate in real environment and gets the impact of this environment. Along with the fact that the control signal can change the output value (it can also be called output signal), this value is also influenced by some external factors (disturbances). Therefore, the control task is to provide complete dependence of the output values on the prescribed value and the complete absence of its dependence on external factors (disturbances). If the dependence of the output value from the input signal is linear, at least approximately, it is possible to describe it with the transfer function, which describes the ratio of the output value to this of the input signal. When describing this ratio, we use Laplace mathematical apparatus in order to take into account the dynamic properties of the object and other elements of the system. If the object model is non-linear, the transfer function is not suitable. In this simulation, this feature does not play a significant role, since software tools allow simulating both linear and non-linear elements. Let the controlled object is linear and it can be described by a transfer function W2(s), which represents the ratio of the output signal y(t) to the input signal u(t) in Laplace transformation, i. e. W2(s) = Y(s)/U(s). It is necessary to provide that the output value become equal to the prescribed value. Let the prescribed value is given by the signal v(t) and called “prescription”. It is required to convert the prescribed signal into control signal in such way that the output signal would become equal to the prescribed signal, at least approximately. If the mathematical relationship between the output signal of the object and its input is known, then it is not difficult to calculate the desired signal, which would lead output value to the desired value. Since signals are best considered in the Laplace transformation where the output of a linear link is described as the product of the transfer function of the link to the transform of the input signal, it can be assumed that the necessary input signal in Laplace transform is the result of the dividing of the necessary output signal transform to the transfer function of the object. If the object would not be affected by any other influence except the control signal, it would be fair. However, in reality, the output signal of an object is equal to the sum of the signal calculated in this way and the results of the action of all uncontrolled influences on it, which together can be described as some additive disturbance at the output of the object. In Fig. 1.1 this fact is displayed by the structure of the object model, namely: the action of the control signal is described by the transfer function of the object W2(s), the result of the action 11

f the control signal is indicated by the virtual signal X(s) corresponding to the signal in the form of the time function X(s). In addition immeasurable disturbance h(t) effect on the object. This signal is added to the virtual signal x(t), so that at the output of the object we do not have the signal x(t), therefore the outputs signal of the object is not signal X(s), but the sum of this signal and the disturbance. Since the disturbance is unknown, the output value of the object is also unknown. Example 3. We would like to control the heating of an aquarium, and we know that if we apply for a heater voltage of 10 V, then the temperature will increase by 1 °C. The transfer function of the object (excluding its dynamic properties) can be treated as gain of KO = 0,1°C/V. If it is necessary that the aquarium temperature would be higher than the temperature in the room by five degrees, the prescription should be equal to v(t) = +5° C. The regulator must convert this value into the appropriate voltage and the required conversion factor is the inversion of the according transfer function of the object: KR = 1/KO = 10 V/°C. In this case, the control signal is supplied to the heating element and will be equal to the product of the prescription to this coefficient, namely: u(t) = KRv(t) = 50 V. That is, the regulator should apply to the heater the voltage of 50 V. At the same time, we do not take into account which temperature is in the room. In addition, other external or internal disturbances may act and that is increasing or lowering the temperature in the aquarium. It may also be that the mentioned ratio is not constant or dependent on the temperature or other conditions. Therefore it is impossible to know accurately even the temperature increment. Therefore, such system does not control the temperature, but only adds some effect with uncertain result. Example 4. If in Example 3 we can measure the temperature in the aquarium and on the basis of this measurement change the voltage applied to the heater to provide the desired temperature at the outlet, then even ignorance about the transfer function of the heater would not prevent to solve the problem accurately. For example, we could heat the water up until it reaches the desired temperature, after which we would switch off the heater. Once the water temperature drops, we would again switch on, and so on. This operation can be made of automatically, not using a person by combining the temperature sensor directly with the switch. This principle is in the base of the action of the automatic electric heaters with the simplest elements of the automatic adjustment. Thus, even if heating is not as efficient as expected, the sensor will still not disconnect the heating power up until the temperature reaches the desired value. 12

Conclusion 1. In systems with feedback, the control may be sufficiently accurate even if the object model is known accurately enough, and even in case when this model changes over time due to some unknown factors. Thus, only the negative feedback loop provides sufficient accuracy of control of the output values of the object. Fig. 1.2 shows this control loop in general. Automatic control theory is developed enough to control the linear stationary objects, but with the number of variables of the object and (or) with the growth of the order of the equations describing the relationship between the input and output signals, the analytical solution of this problem becomes extremely difficult. These problems arise when it is necessary to take into account nonlinear or delay element. In this case, it is much more successful to apply numerical methods based on simulation of such systems [20-42]. Simulation allows the selection, implementation and simulation test of the regulator, but without theoretically based and practically proven methods of these possibilities a developer cannot implement them effectively. Thus, it is relevant the design, development and justification of methods of numerical calculation of regulators for objects that contain high-order elements, nonlinear and transcendental elements. It is also necessary to test these methods to solve the regulator design problems for continuous technological processes with the objects characterized by the mentioned features. V(s) U(s) > Object Y(s) Feedback and Regulator Fig. 1.2. Control with feedback loop Design of the feedback system demands the development of a regulator and total structure of feedback. In addition, coefficients in these structures should be calculated or found with any other method. The selection of the structure is often extremely simple. The feedback is used with a unite coefficient. It is put through a subtractor, and device, which usually has three separate control channel, is used as a regulator. These three channels include channel with proportional converting of the signal (proportional channel), channel with an integrating converting (integrating channel) and channel with derivative converting (derivative channel). Hence, the name of the regulator, containing all three channels is PID 13

egulator, of PID-controller, or PID, according the first letters of the three channels. If the regulator does not contain any of these channels, the corresponding letter falls from its name, for example, remains the PI or PD. If some of the channels is presented twice, the letter can be doubled or given with an exponent, for example, PIDD or PID². If there is only proportional path, can be used historically established name "K-regulator", where "K" stands for easy gain, which, as a rule, should be large enough for the success of the tasks of the object control. As a rule, control objects in industrial process lines are characterized by a significant delay, which cannot be neglected even in comparison with the phase delay of the minimum-phase part of the transfer function. In addition, non-linearity of the object produces strong influence to the object actions. Along with the presence of cross-linking, it requires consideration of objects like multiply and non-linear at the same time. Papers of many scientists are devoted to research of the methods of control of such objects [21-42]. In particular, control techniques are developed for multichannel objects using the method of the highest derivative of the state vector, the method of analysis and design of adaptive systems based on the principle of localization have been successfully used, methods of separation movements and some other methods and techniques are also developed and used [20]. Cases of linear objects are considered and studied deeply enough. As a rule, the decision of the task of the regulator design in this case is solved analytically, mostly in consideration of the space object states. For objects that contain delay and (or) non-linearity, it is not always possible to use analytical methods due to significant increase in the complexity of the problem. Problems of control of higher order multivariable objects whose matrix transfer function has a greater dimension and thus contains elements of higher order links and objects that contain nonlinear transcendental or links, are extremely complex. Most often, they cannot be solved analytically or their resolving with the use of the known methods is extremely difficult, and it requires enormous computing resources or a lot of time, which makes these tasks insoluble by known techniques. The use of different simplifications of the problem leads to the fact that the resulting solution is far from the desired, or the difference of practical implementations from the theoretical results is unacceptably high. Numerical simulation and numerical optimization are increasingly used due to the development of mathematical methods and special programs, as well as due to increased computing performance. These methods allow find 14