This study examines the possibility of applying the A.A. Krasovsky method of complex coordinates and complex transfer functions for multidimensional element types modeling. As it is demonstrated, the method can be taken as the basis for multidimensional regulation system design along with the extended version of classification suggested by A.A. Krasovsky

Klimanova, E.V.

Maksimov, A.V.

Samara University

Секция: Математическое моделирование физико-технических процессов и систем VI Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-2020) Classification of multidimensional element types in automatic regulation systems E.V. Klimanova1, A.V. Maksimov1 1Bauman Moscow State Technical University, Bazhenova 2, Kaluga, Russia, 248000 Abstract. This study examines the possibility of applying the A.A. Krasovsky method of complex coordinates and complex transfer functions for multidimensional element types modeling. As it is demonstrated, the method can be taken as the basis for multidimensional regulation system design along with the extended version of classification suggested by A.A. Krasovsky. 1. Introduction Currently, developers of modern multidimensional multi-connected technical regulation systems have to solve the problem of physically real connections present between the elements of developed systems even when designing mathematical models. These connections for some systems significantly affect their dynamics.  Opto-electronic angle and range tracking system of moving objects is an example of above-mentioned complex system [1]. The most common methods for the mathematical description of multidimensional systems are matrix methods [2, 3]. Matrix methods are convenient for representation of high degree complexity in multi-connected technical regulation systems and perfectly suited for numerical solution by computer.  However, with these methods there is a problem to obtain qualitative estimation of matrix equations solution if applied to the system as a whole. Moreover, the greater the dimension of the system, the greater the number of connections between channels in the system, the more difficult it is to study such systems. When conducting research on two-dimensional systems, Academician A.A. Krasovsky first suggested the method of complex coordinates and complex transfer functions [4]. In these works, he was also the first one to suggest a classification of two-dimensional systems with cross-connections. The method of A.A. Krasovsky was successfully applied and further developed in [1, 6].  The purpose of this study is to explore the possibility of reducing the complexity of modeling and research of multi-connected four-dimensional automatic regulation systems by applying the A. A. Krasovsky method of complex transfer functions. 2. Method description In [7–9] the authors presented an obtainment of complex-valued and quaternion transfer functions for two-dimensional and four-dimensional systems respectively. Consider the scheme given in [6] of generalized matrix-complex-valued unit of the control system from the  A.A. Krasovsky classification point of view  (Figure 1). Here 𝑦1(𝑡), 𝑦2(𝑡) - are input signals; 𝑥1(𝑡), 𝑥2(𝑡) - are output signals; 𝑊1(𝑝), 𝑊2(𝑝) – are transfer functions.  Секция: Математическое моделирование физико-технических процессов и систем Classification of multidimensional element types in automatic regulation systems VI Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-2020)  395  1W p2 ( )y t 2 ( )x t 1W p1( )y t 1( )x t 2W p 2W p-+ Figure 1. Generalized marix-complex-valued element.  In terms of the A.A. Krasovsky classification it is a two-channel element of control system with identical channels and antisymmetric cross- connections. The matrix equation of the dynamics for this element will take form as shown below:                                                                      X(t)=W(p)Y(t)                                                           (1) Here X(t)= [x1(t)x2(t)] - is a column-vector of output signal; Y(t)= [y1(t)y2(t)] - is a column-vector of input signal; W(p)= [W1(p) W2(p)−W2(p) W1(p)] - is a transfer function matrix; W1(p),  W2(p) - are transfer functions of the 1-st and the 2-nd channels respectfully. The equation of given element in complex coordinates can be written:                  𝑥(t) = 𝑊(p)𝑦(𝑡) = (𝑥1(𝑡) + 𝑖𝑥2(𝑡)) = (𝑤1(𝑝) + 𝑖𝑤2(𝑝)(𝑦1(𝑡) + 𝑖𝑦2(𝑡))                (2) Here 𝑥(t), 𝑦(t), 𝑊(p) – are output, input signals functions and transfer function in complex coordinates  (this means that the  name “complex-valued” can be applied in this case). After calculations (2) can be represented as:  𝑥1(𝑡) + 𝑖𝑥2(𝑡) = (𝑤1(𝑝)𝑦1(𝑡) − 𝑤2(𝑝)𝑦2(𝑡)) + 𝑖(𝑤1(𝑝)𝑦2(𝑡) + 𝑤2(𝑝)𝑦1(𝑡)) or coordinate form: 𝑥1(𝑡) = 𝑤1(𝑝)𝑦1(𝑡) − 𝑤2(𝑝)𝑦2(𝑡), 𝑥2(𝑡) = 𝑤1(𝑝)𝑦2(𝑡) + 𝑤2(𝑝)𝑦1(𝑡)                                              (3)    It is easy to see that the coordinate forms of the matrix equation (1) and the equations in complex variables (2) are identical and make it possible to build an element based on them as in Figure 1. The complex-valued equation (2) can be taken as the basis for the development of both a separate two-dimensional element of the projected system of any complexity, and the entire system as a whole. It should be noted that a special case may occur when: 𝑤1(𝑡) = 𝑤2(𝑡) Then the system of coordinate equations (3) can be written in two forms: the first one is:                            𝑥1(𝑡) = 𝑤(𝑝)𝑦1(𝑡) − 𝑤(𝑝)𝑦2(𝑡), 𝑥2(𝑡) = 𝑤(𝑝)𝑦1(𝑡) + 𝑤(𝑝)𝑦2(𝑡)  (4-a)    the second one is: 𝑥1(𝑡) = 𝑤(𝑝)𝑦1(𝑡) − 𝑤(𝑝)𝑦2(𝑡) = 𝑤(𝑝)(𝑦1(𝑡) − 𝑦2(𝑡)), 𝑥2(𝑡) = 𝑤(𝑝)𝑦1(𝑡) + 𝑤(𝑝)𝑦2(𝑡) = 𝑤(𝑝)(𝑦1(𝑡) + 𝑦2(𝑡))  (4-b)    These two different equation forms have two different structural schemes for the same control system element as in Figure2 and in Figure 3. Given classification can be applied for the less known complex number system such as hyperbolic complex numbers [9].  For example, consider obtained in [8] generalized matrix-complex-valued control system for the hyperbolic numbers. Секция: Математическое моделирование физико-технических процессов и систем Classification of multidimensional element types in automatic regulation systems VI Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-2020)  396 W(p)W(p)y1(t)y2(t) x2(t)x1(t)+− Figure 2. Two-dimensional element with output cross-connections. x2(t)x1(t)+−W(p)W(p)y2(t)y1(t) Figure 3. Two-dimensional element with input cross-connections.           Figure 4. Generalized matrix-complex-valued control system for the hyperbolic numbers.  The result of complex coordinates method applied to given matrix-complex-valued control system is shown below: W(p)W(p)y1(t)y2(t) x2(t)x1(t)++ Figure 5. Two-dimensional element with output cross-connections.  Figure 5 demonstrates two-dimensional element with output cross-connections and equal in signs transfer functions. This is an expansion of the A.A.Krassovky classification, because A.A.Krassovky considered only antisymmetric cross-connections. 3. Conclusion Complex coordinates method has been applied to the generalized matrix-complex-valued unit of the control system and the result of such approach is reducing complexity of multidimensional control systems modeling. Given method allows using generalized structural schemes for the first stage of modeling, without examining the features of typical element. Nevertheless, if the developer chooses to apply, for example an oscillatory element as a typical element, then accordingly to the classification of A.A. ++y1(t) x1(t)y2(t) x2(t)W1(p)W1(p)W2(p)W2(p)Секция: Математическое моделирование физико-технических процессов и систем Classification of multidimensional element types in automatic regulation systems VI Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-2020)  397 Krasovsky in Figure 2 will be a structural scheme of two-dimensional oscillatory element with cross-connections at the output, and in Figure 3 - a two-dimensional oscillatory element with cross-connections at the input and so on. Same development approach can be applied for other element types such as integrating and differentiating. In addition, there are quaternion elements were presented in [7-9], but they require further study. 4. References  [1]  Barsky, A.G. Two-dimentional and three-dimentional automatic control system theory / A.G. Barsky –  Мoscow: Logos, 2015. – 192 p.  [2]  Pupkov, K.A. Matrix Methods of Calculation and Design of Complex Automatic Control Systems for Engineers / K.A. Pupkov, N.D. Egupov – Moscow: Bauman Moscow State Technical University Press, 2007. – 664 p. [3] Pupkov, K.A. Methods of Classic and Modern Control Theory / K.A. Pupkov, N.D. Egupov – Moscow: Bauman Moscow State Technical University Press, 2012. – 272 p. [4] Krasovky, A.A. Of two-channels automatic control systems with antisymmetric connections // Autom. And Telemech. – 1957. – Vol. 2. – P. 126-136.  [5] Kazamarov, A.A. Two-dimentional automatic control systems dynamics / A.A. Kazamarov, A.M. Palatnik, L.O. Rodnyansky – Moscow: Nauka, 1967. – 308 p.  [6] Klimanova, E.V. Complex-valued and hypercomplex models of typical control system elements. Part 1. Mathematics for two-dimentional control systems / E.V. Klimanova, A.V. Maksimov, E.V. Maksimova // J. Elect. W. andElect. Sys. – 2016. – Vol. 21(9). – P. 66-72.  [7] Klimanova, E.V. Complex-valued and hypercomplex models of typical control system elements. Part 2. Mathematics for four-dimentional control systems / E.V. Klimanova, A.V. Maksimov, E.V. Maksimova, S.V. Stepanov // J. Elect. W. and Elect. Sys. – 2017. – Vol. 22(3). – P. 22-29.  [8] Klimanova, E.V. 4-d models of typical control system units / E.V. Klimanova, A.V. Maksimov // V International Conference on Information Technology and Nanotechnology (ITNT) – Samara: Novaya Technica. – 2019. – Vol. 3. – P. 418-422.  [9] Klimanova, E.V. Four-dimensional models for control system typical units / E.V. Klimanova, A.V. Maksimov // Journal of Physics: Conference Series. – 2019. – Vol. 1368. – P. 042033.  

Classification of multidimensional element types in automatic regulation systems

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Classification of multidimensional element types in automatic regulation systems

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