=Paper=
{{Paper
|id=Vol-2922/paper013
|storemode=property
|title=Algorithms for switching technical systems modeling
|pdfUrl=https://ceur-ws.org/Vol-2922/paper013.pdf
|volume=Vol-2922
|authors=Olga N. Masina,Alexey A. Petrov,Olga V. Druzhinina,Sergey V. Chernomordov,Daniil Y. Openkin
}}
==Algorithms for switching technical systems modeling==
https://ceur-ws.org/Vol-2922/paper013.pdf
Algorithms for switching technical systems modeling*
Olga N. Masina1[0000-0002-0934-7217], Alexey A. Petrov1[0000-0002-3011-9278],
Olga V. Druzhinina2[0000-0002-9242-9730], Sergey V. Chernomordov1[0000-0001-6773-8737] and
Daniil Y. Openkin1[0000-0003-1519-1615]
1
Bunin Yelets State University, 28, Kommunarov st., Yelets, 399770, Russian Federation,
2
Federal Research Center “Computer Science and Control” of Russian Academy of Science,
44, building 2, Vavilov st., Moscow, 119333, Russian Federation,
olga121@inbox.ru
Abstract. The paper is devoted to the design of algorithms for switching tech-
nical systems modeling. These algorithms are based on the application of nu-
merical methods for solving ordinary differential equations, methods of control
theory, and methods of global parametric optimization. An algorithm for
switching using a PID controller according to a mismatch signal is developed.
An algorithm for global parametric optimization based on artificial neural net-
works with training is proposed. Model examples are considered. For the devel-
opment of the corresponding software the Python language with a set of
mathematical libraries is used. The obtained results can be used in solving vari-
ous applied problems, in particular, problems of transport dynamics, problems
of aircraft control, and problems of energy systems control.
Keywords: neural network modeling, switching technical systems, PID con-
troller, machine learning, numerical optimization methods, Python.
1 Introduction
The switching system can be represented as a multi-mode dynamic system with a
mode switching law that determines the activity intervals of each mode [1]. Such
systems find numerous applications in applied problems of mechanical systems con-
trol, processes in switching power converters [2-3], in controlling the movement of
unmanned aerial vehicles, automated transport systems, robotic systems, and in other
fields [4-5]. The actual problems in this direction are the analysis of these technical
systems models.
Within the framework of the direction related to the study of switching systems,
the problems of constructing models and synthesizing controls are of significant inter-
est [6]. Subsequent modeling is associated with the development of appropriate algo-
rithms and with software implementation in high-level languages.
*
Copyright c 2020 for this paper by its authors. Use permitted under Creative Commons License Attribu-
tion 4.0 International (CC BY 4.0).
In [7-8], a number of problems of analysis and synthesis of two-dimensional and
three-dimensional models of controlled switching systems are considered. These
problems are solved on the basis of numerical optimization various methods using
neural network modeling. The most important problems of neural network modeling
are the problems of developing the effective methods of machine learning of neural
networks [9-10]. The algorithms of global parametric optimization used in the prob-
lems of constructing optimal trajectories and in machine learning problems are con-
sidered in [7-8; 11–13] and in other papers.
The purpose of this paper is to develop algorithmic software for controlled switch-
ing systems modeling based on the use of intelligent technologies and numerical op-
timization methods. Mathematical models of technical systems with switching are
considered. The models are described by means of ordinary differential equations
considering the given initial conditions and multivalued right-hand sides. A switching
algorithm based on the PID controller is proposed, as well as a neural network opti-
mization algorithm. The analysis of model examples based on the results of computa-
tional experiments is carried out.
2 Models and methods
The mathematical model of the technical system under study is given by an ordinary
differential equation of the form
mx QT mG D ( x, x ,
x, t ). (1)
In (1) the following notations are accepted: x R n is the phase vector, t is the
time, D ( x , x ,
x, t ) is the perturbation in the system, G is the vector-column of the
potential gravitational field, m is the mass. Through T (t 0 , t1 ,...t n 1 )* we denote the
column vector corresponding to the time. The matrix Q in (1) is a matrix of coeffi-
cients the search for which is carried out by means of solving the initial boundary
value problem for the following equation:
mx QT mG. (2)
We note that model (2) is a special case of the model (1) and is characterized by
the absence of a perturbation vector. The switches in the model (1) are determined by
the program control algorithm. We develop program control algorithms considering
the need to reduce the number of switches.
The generalized model (1) allows us to form trajectories that can satisfy a set of
phase constraints, considering a given quality criterion. For a model of an aircraft, we
can set a quality criterion at which the time or energy costs will be minimal. In this
case, the phase restrictions depend on the need to be in the specified air corridor and
change the destination during the motion. In particular, one of the model examples is
a model of the aircraft dynamics with the condition of minimum fuel consumption.
Further we consider the optimal control problem for the model (1). A control qual-
ity criterion has the following form:
t1
U (t ) dt min .
0
(3)
In (3) the following notations are used: U is the control vector, t1 is the specified
time to reach the phase subspace.
We use methods of numerical optimization [14], methods of control theory (the
method of constructing PID controllers [15] and the method of sliding modes [16]). In
addition, global parametric optimization methods (including methods that simulate
biological processes: neural network methods, bee swarm methods, ant colonies, etc.)
can be used for searching of optimal trajectories of the model (1) [17].
In this paper we solve the control problem by synthesizing regulators based on arti-
ficial neural networks (ANNs). The synthesis of regulators with the use of ANNs
makes it possible to form models of weakly formalized systems [9]. It should be noted
the effectiveness of ANNs in the problems of finding optimal trajectories for dynamic
models of switching systems. In particular, it is possible to change the parameters for
the nonlinear approximation of multidimensional functions using the ANNs.
3 Results
We consider the issues of constructing a feedback control algorithm using a PID con-
troller. The use of PID controllers reduces the time for debugging the system and
allows to set the necessary parameters. Under the condition that there are no large
noises in the control loop with feedback, the design of the PID controller becomes a
simple, but quite effective way of control, and for the implementation of such control,
a simplified algorithm for tuning the coefficients is sufficient [18].
The scheme of the generalized switching algorithm using the PID controller is
shown in Figure 1. At the stage of this algorithm activation it is necessary to set up
such a trigger threshold that leads to switching. Namely, instead of a continuous sig-
nal we fixate a specific value (0 or 1) which is used to control the switching of the
system.
Fig. 1. Scheme of the generalized switching algorithm using the PID controller.
We configure the PID controller using a neural network optimization method. As
neural networks for configuring the PID controller, multi-layer networks of direct
signal propagation with one hidden layer and with a linear activation function are
most suitable. Note that when tuning the coefficients of the PID controller, there is an
analogy with the selection of the weight coefficients of a neural network.
To solve the problem of constructing a switching algorithm, a method is proposed
that is based on switching the matrices of the control parameters Q taking into ac-
count the feedback [7].
Further we present the algorithm developed for the model (1) “Configuring the PID
controller” (Algorithm 1), which consists of the following steps.
─ Step 1. To initialize the PID controller with random coefficients.
─ Step 2. To perform the switching algorithm and to find the trajectory.
─ Step 3. To calculate the value of the quality criterion.
─ Step 4. If the stop condition is carried out, the algorithm is finished.
─ Step 5. To optimize the coefficients P, I, D and the threshold value λ.
─ Step 6. To proceed to step 2.
The configured PID controller is then used to find the optimal switching moments
in the controlled system (2). In this algorithm, we use optimization methods to obtain
the PID controller coefficients. In this case, the principle of applying optimization
methods is that the coefficients will be selected taking into account the minimum
number of switches.
The flow chart of Algorithm 1 is shown in Figure 2.
Fig. 2. Flow chart of the algorithm “Configuring the PID controller”.
Algorithm 2 includes Algorithm 1. We present Algorithm 2 “Switching generation
based on the PID controller” for the two-dimensional case of the model (1), which
consists of the following steps.
─ Step 1. To configure the PID controller (Algorithm 1).
─ Step 2. To perform one iteration of the numerical integration algorithm.
─ Step 3. To check the stop condition. If the stop condition is carried out, the algo-
rithm is finished.
─ Step 4. To calculate the switching generator outputs. If the activation function is on
the output is x>Ω, then the switch is performed, otherwise go to step 2.
─ Step 5. To calculate new values of matrix Q.
─ Step 6. To proceed to step 2.
The flow chart of the algorithm “Switching generation based on the PID control-
ler” is shown in Figure 3.
Fig. 3. Flow chart of the algorithm “Generation of switches based on the PID controller”.
As a software tool for prototyping software we use the high-level language Python
with the libraries NumPy, SciPy, SymPy, and Matplotlib for scientific computing,
data processing, and visualization [19].
A prototype of software for interpreting the output value of the PID controller has
been developed. The corresponding implementation code is shown in Figure 4.
Fig. 4. A program code fragment for the realization of PID controller.
The graphical interpretation for the example implementation of Algorithm 1 is
shown in Figure 5, where (t ) is the PID controller output, (t ) is the mismatch
signal.
Fig. 5. Graphical interpretation of the PID controller output value.
Before implementing the PID controller the trajectories of the dynamic system of a
search are being performed. As an example, we give the result of the trajectory search
in the two-dimensional case when moving under the influence of gravity without
taking into account the action of other forces. Figure 6 shows a program code frag-
ment for searching of trajectories before further optimization.
Fig. 6. A fragment of the program code for finding a trajectory when moving under the
influence of gravity.
The program code shown in Figure 6 can be used for further expansion and imple-
mentation of algorithm 1.
An important problem is to develop aggregated algorithms that combine switching
algorithms and algorithms for finding optimal trajectories. Such an aggregated algo-
rithm is a new neural network optimization algorithm, the essence of which is as fol-
lows. In the space of the optimized function, k neural network automata with a given
topology are randomly initialized. A neural network automata is an artificial neural
network with certain state parameters (energy and position). The purpose of function-
ing of a neural network automaton is to find the function optimum. Neural network
automata form a closed system, the evolution of which proceeds according to a com-
bined heuristic algorithm [20].
For the two-dimensional case of the model (1), a global parametric optimization
algorithm is developed based on a combination of Hooke–Jeeves methods [21], artifi-
cial neural networks, and the swarm optimization method.
Further for the two-dimensional case of the model (1), we present the developed
Algorithm 3 “Search for the function optimum by the aid of neural network automata
swarm”.
─ Step 1. To initialize a k neural network automata ( k 2 ) in n-dimensional Euclid-
ean space.
─ Step 2. To calculate the outputs of neural network automata.
─ Step 3. To change the position of neural network automata in accordance with the
output vectors.
─ Step 4. To check the stop condition. If the stop condition is carried out, the algo-
rithm terminates. Otherwise, go to step 2.
We note that the proposed Algorithm 3 belongs to algorithms inspired by nature,
and is a zero-order search algorithm. Further we present the developed Algorithm 4
“Training of neural network automata”.
─ Step 1. To initialize a network of k neural network automata ( k 2 ) in n-
dimensional Euclidean space.
─ Step 2. To perform steps 2-4 of Algorithm 3.
─ Step 3. To find the value of the loss function.
─ Step 4. To optimize the weight coefficients of neural network automata.
The flow chart of Algorithm 4 is shown in Figure 7.
Fig. 7. Flow chart of the algorithm «Training of neural network automata».
The software based on the developed algorithms can be used to expand the optimiza-
tion module of the SSMC software package [22].
The algorithm using neural network automata is a universal optimization algorithm
and can be used in a number of applied problems of parametric optimization.
4 Discussion
For the developed flow chart (Figure 2), an example of the implementation of
Algorithm 1 is given. Figure 3 shows the flow chart of Algorithm 2, which includes
Algorithm 1. The program code shown in Figure 6 is a prototype of software for
modeling switchable systems based on a PID controller. The considered examples
demonstrate the effectiveness of the developed algorithms (see Figures 5). The
program code shown in Figure 6 requires further extensions and modifications for
various cases, in particular, for the case of increasing the system dimension and for
the case of the influence of indeterminate perturbations with the corresponding
complication of the model.
The implementation of Algorithm 3 and the performing of a series of compu-
tational experiments to find the optimal trajectories of switching systems are the
prospects for the development of this paper. The importance of the results obtained in
the development of optimization algorithms (see Algorithm 3) is that these results can
be applied in a number of different subject areas related to the mathematical modeling
of controlled systems.
To implement the flowchart of Algorithm 4 (Figure 7), it is possible to use
reinforcement learning based on evolutionary algorithms to find the function
optimum.
When studying and verifying switching high-dimensional models, along with algo-
rithms 1-4 and algorithms for the numerical solution of multidimensional differential
equations, it is advisable to use Data Mining methods [23] for the purpose of prelimi-
nary preparation of model parameters arrays.
5 Conclusion
In this paper we study an approach to computer research and development of algo-
rithms and software for technical switching systems.
A switching algorithm based on a PID controller on the basis of a mismatch signal
is proposed. The algorithm of global parametric optimization on the basis of artificial
neural networks with training is developed. These algorithms allow to analyze the
influence of various parameters on the qualitative characteristics of the switching
technical system functioning process.
The presented results can be used in solving various optimization problems, in
solving problems of mathematical control theory, as well as in designing and
improving switching systems taking into account uncertainties. In addition, the results
obtained can be used in machine learning problems and transport route optimization
problems.
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