The paper discusses the problem of a popular industrial controller tuning in automatic control systems. This controller is named after their components: proportional, integral, and differential units. The problem with using such a controller is its configuration, which must consider the contradictory properties of the individual units' behavior. In practice, the approximate manual tuning technique proposed by Ziegler and Nichols is usually used. This technique is based on the system's response to a typical impact. The approximate tuning is not the best and can be improved. The proposed study proposes an improved methodology using transient endpoints such as overshooting, mean squared reference deviation error, and transient settling time. The proposed indicators create a criterion for the quality of tuning. Since analytical methods in this formulation of the problem are not suitable, it is therefore proposed to use a genetic algorithm that searches for the best parameters of the PID controller. A feature of the proposed algorithm is the introduction of restrictions on the desired parameters of the controller and the principle of elitism to eliminate losses of the best sets from previous search iterations. The proposed approach showed satisfactory results for stabilizing unstable control objects.
In automatic control systems, controllers of different types are used to increase productivity. These controllers directly influence the coordinates of the control object, generating a control signal that ensures the execution of the task with the required quality indicators. The best quality indicators are achieved in systems with feedback, where the controller is located in a forward circuit, and its input receives an error signal or deviation from the value on the system input generated by the feedback and input circuit signals. Since the input signal can change, it can be an additive mixture of desired and interfering signals, and the controller must ensure the stability of the control system, minor processing errors, and compensate for changes in the input signal. These properties have relatively simple PID controllers, which contain proportional and integral control elements that reduce errors in the static mode and differentiating units that affect the error in the dynamic mode.
Despite the existence of various types of control algorithms, the PID controller has found practical
application in industrial production. According to Astrom and Hagglund [
0000-0002-4334-4175 (D. Kucherov); 0000-0001-5456-0618 (O. Poshyvailo); 0000-0002-7810-2678 (I. Myroshnychenko) © 2023 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
The PID controller automatic tuning assumes that the mathematical model of the control object is
known, and the weak point of these methods is that it can correspond to the object only in a narrow
range of controller parameters [
Currently, a sufficient number of algorithms have appeared that use the properties of
“reasonableness” in the parameter selection, which are based on adaptation, fuzzy logic, and neural
networks. These approaches also include heuristic methods, which use the search for the best sets of
parameters when solving a multi-criteria problem. These methods are borrowed from observations
of natural selection, which occurs in nature when mechanisms of random selection, crossing, and
mutation are involved and are called genetic algorithms [
Tuning PID controllers based on genetic algorithm (GA) has been widely reflected in publications. However, the main problem of genetic algorithms is convergence, which can be both premature convergence, which leads to falling into a local minimum, and low convergence speed, which may not lead to obtaining the desired result within the specified time.
The genetic algorithm simulation for tuning a PID controller for an unstable object using a multicriteria objective function that ensures sufficient convergence in an acceptable time is paper objective.
Further, the paper is organized as follows: Section 2 presents a review of the well-known literature in the area of research; Section 3 formulates the research problem; a general approach to finding a solution using a genetic algorithm is given in Section 4; Section 5 is devoted to the simulation of the search for PID controller parameters for objects with zero and imaginary roots of the characteristic equation; and at the end of the paper, a discussion of the results obtained is given.
Whitley presents canonical and experimental forms of the genetic algorithm in the handbook [
Most publications devoted to tuning PID controllers use GA. Thus, A. Mirzal et al. [
A. A. Aly [
Recently, intellectual approaches to attunement have been developing. Katoch et al. present a
review of recent advances in GAs and a discussion of future research directions in the genetic
operators’ area, fitness functions, and hybrid algorithms [
Integrating a GA with an artificial immune system via a scheme incorporating diversity functions,
distributed computing, adaptation, and self-control for tuning a PID controller is presented in the
paper by Khoie et al. [
Zhou et al. [
An analysis of the presented literature showed that the problem of controlling unstable objects is not well studied, which is of interest for constructing a genetic algorithm for tuning the PID controller of these objects.
Some object with unstable dynamics is considered. Instability is characterized by the impossibility of establishing the output coordinates of an object to the required values in an acceptable time. The object is in the initial state. This state is known and characterized by these coordinates: state, speed, and acceleration of its changes.
To stabilize this object coordinates to a given position, a closed-loop controller with a proportional-integral-derivative (PID) controller in a feedback loop is used (shown in Figure 1).
The PID controller generates a control signal from the error signal using proportional, integral, and differentiating units in the form: ( ) = ( ) + ( ) + ( )
.
In (1) kp, ki, and kd are the transmission coefficients of the proportional, integral, and differential units, respectively; e(t) is the error signal,
( ) = ( ) − ( ), (2) where x(t) is the input signal, y(t) is the output signal, t is a variable that has the sense of time, and T is the observation time of the process. (1) We assumed that the parameters of the PID controller are subject to restrictions in the form: ≤ ≤ , ≤ ≤ , ≤
≤ .
In (3), values , are indicated as the upper and lower values of the controller parameters.
A master action of a known type is applied to the input of the object, for example, a unit step function x(t) = 1(t), generating the output signal y(t) in the form of a transient process. Sensors record the following indicators of the control process: overshooting OS, settling time tst, and steady-state error emin. Noise-free sensors measure object parameters.
The task is posed to configure the parameters of the PID controller that satisfy constraints (3) and ensure system stability with minimal values of overshooting, establishment time, and steady-state error of the transient process. This problem we can represent by a complex criterion in the form:
Tp is the duration of the adjustment process, i.e. when the output value y reaches the set value yst. Other designations: ymax is the maximum value of the initial value; , , and are weighting coefficients that determine the contribution of each term to the overall productivity of the GA; where OS is calculated by the formula MSE (mean squared error) is the mean squared error, which is calculated as follows
, where Imin is the minimum value of criterion (4).
At the first step of applying the GA procedure, a random set of chromosomes is introduced, which are associated with the parameters of the PID controller, the effect of which on the system is further evaluated. After the simulation, the manipulated variable is immediately returned to the GA, and the selected error criterion is used to assess the performance and determine the fitness value of this chromosome for further search for PID controller parameters that can satisfy the system designer.
GA performance is evaluated by the objective function I (4).
Traditionally, GA includes the following steps: selection, crossing, mutation, and quality assessment of the resulting chromosome set. If the stopping criterion is not met, the GA procedure is repeated the given number of times.
In the developed algorithm, the maximum number of generations of the new generation P is set. A considerable number of iterations leads to an unjustified increase in the operating time of the algorithm when obtaining a repeated result, which can be interpreted as looping in a local minimum.
After the initialization of a chromosome set consisting of N randomly selected individuals, the selection procedure is. This procedure is necessary for sampling the most successful generation of individuals according to the objective function (2). Since the initial data are given by real numbers, those data that correspond to the minimum value of the objective function rounded to the nearest larger whole get into the new chromosome set. It allows you to preserve the best generation of parameters and guarantees the asymptotic convergence of the GA.
The crossing operation was carried out according to the principle of arithmetic crossing, which is the most successful for finding the optimum of the function of many real variables. According to this principle, two offspring h1s and h2s are created based on the values of the genes of the parent chromosomes h1 and h2. Values of genes of offspring h1si and h2si are calculated according to the formulas: ℎ = ℎ + (1 − )ℎ , (3) (4) (5) (6) (7) (8) where ∆ℎ( , ) = (1 − ( / ) ); r is a random number from the interval [0; 1]; p is the current iteration number; k is a non-uniformity parameter. All chromosomes that do not meet the specified level of objective function are subject to mutations.
At the last stage of the algorithm, a new generation is formed. To prevent the best chromosomes from being lost, they are guaranteed to be included in the new population.
The specified sequence of operations is repeated until the algorithm is considered converged. The algorithm terminates when the specified performance is satisfied or all assigned generations are completed. In the latter case, the chromosome set with the best performance is chosen.
In the interests of modeling, we use a mathematical model described by a differential equation of the second or third order, which will reproduce the dynamics of the controlled process with sufficient accuracy.
To study the dynamics of such a model, the space of states is usually linked to the phase
coordinates of a dynamic system [
( ) = ( ),
In (12), x(t), y(t), and u(t) are vectors of phase variables, output coordinates, and control, respectively, depending on time t. Vectors dimensionality are x Rn, y Rm, and u Rp. Matrices A, B, and C have these forms A Rnn, B Rnp, and C Rmn.
A system with feedback provides the given dynamics of an object. A PID controller in the feedforward circuit of the system is installed in series with the object.
As an unstable control object, a double integrator is chosen, which has a double zero root and is described by a system of differential equations in the first-order form:
where x1(t) and x2(t) are the phase variables of the integrator containing its output and its derivative, u(t) is the control signal, and k is the amplification factor of the control system. The variable t is a time. At the control start (t=0), the initial state of integrators is unexcited, i.e. x1(0) = x2(0) = 0.
The object can be stabilized by applying a control device. Such a device is a PID controller, which at the space of phase variables, can be represented by a system of equations
( ) = ( ) + ( ) + ( ), ( ) = ( ), ( ) = ∫ ( ) , ( ) = ( ).
In (10), y1(t), y2(t), and y3(t) are the outputs of the proportional, integral, and differential units, respectively. Values kp, ki, and kd are the parameters of the corresponding units of the PID regulator, and variable e(t) is the error signal at the input of the PID regulator. Restrictions (3) are applied to the parameters of the PID controller. system.
A standardized single signal in the form of a step, i.e. r(t)=1(t), is applied to the input signal of the In the interests of modeling and to prevent complex integration algorithms, we present the considered system in the extended phase space Xext R5, which takes into account the phase coordinates of the considered objects, which allows us to present the system in the form of a matrix differential homogeneous equation of the first order of the form: 5.1. Double integrator 5.1.1. Math model (13) (14) (15) (16) with the initial vector Xext(0) = (0, 0, 0, 0, 1)T. The system of equations in (14) has the form: ̇ ( ) = ( ), ̇ ( ) = − + 1 ( ) − ( ) + ( ) +
+ 1 ( ), ̇ ( ) = ( ) −
( ) + ( ), ̇ ( ) = − ( ) + ( ),
̇ ( ) = 0
In (16), x1 and x2 are the phase coordinates that coincide with (12), x3 and x4 are the outputs of the differential and integral links of the PID controller, respectively, and x5 is the input signal.
Limit parameters (3) of the PID controller are given by inequalities: 10 10 10 ≤ ≤ ≤ ≤ 1, ≤ 1,
GA parameters are given in Table 1. Coefficients of the objective function , , The number of generations of the new generation
Initial chromosomal set Arithmetic crossover coefficient Coefficient r mutation
The PID controller's initial and final parameters are given in Table 2. The PID controller parameters, at the beginning of the study, were selected according to the Ziegler-Nichols method. Median of the generated random series
Transient processes in the system according to Table 2 are presented in Figures 2 and 3.
Analysis of Figures 2 and 3 shows a general decrease in oscillations and transient transients. System performance increased by approximately 6%.
An oscillating object is chosen as an unstable control object, the characteristic equation of which
has conjugate imaginary roots j and is described by a system of first-order differential equations
of the form [
̇ ( ) = ( ), (17) ̇ ( ) = −ω ( ) + ω ( ), where x1(t) and x2(t) are the phase variables of the object, which have the content of its output and its derivative, u(t) is the control signal, k is the parameter of the control object, and is the oscillation frequency. The initial state of the control object (t=0) is not excited, as in the case of the previous object x1(0) = x2(0) = 0.
The system response to a standardized unit signal is in the step form, i.e. r(t)=1(t). The initial values of the coefficients of the PID regulator are set according to the Ziegler-Nichols algorithm and have the values kp = 0.31, kd = 1.023 s, and ki = 0.021 s-1, performance I = 4.027. The graphs of the functions x(t), r(t), and e(t) before and after setting the PID controller by GA have the form shown in Figures 4 and 5. After the GA action, the following values of the coefficients of the PID regulator were obtained: kp = 0.0548, kd = 0.6446 s, ki = 0.0222 s-1, and I = 3.9434. A productivity increase of 2.1% was obtained.
The paper presents the GA implementation technology for setting the PID controller parameters for an unstable object with multiple zero roots of the characteristic equation. A multi-criteria function was used as the objective function, which includes the overshooting, the root mean square error, and the duration of the transition process. The total increase in adjustment compared to the ZieglerNichols algorithm is 6% for the double integrator and 2.1% for the oscillating object, which allows us to conclude the feasibility of using GA to improve the adjustment of the PID controller.
The work has been carried out on an initiative basis. The authors thank both the authorities of the National Aviation University and the especially leadership of the Faculty of Computer Science and Technologies for their support during the preparation of this paper.