Introduction

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 [1], more than 95% of process controllers are PIDs, and their popularity is growing. Results from testing hundreds of plants [2] showed that about 30% of regulators used manual tuning; 65% of regulators with automatic tuning did not provide a minimum variance error compared to manual mode. These results highlight the need to find effective tuning methods.

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 [3]. The actual control object is nonlinear; its parameters can change depending on time and operating conditions; there are time delays in the control circuits, which complicate the adjustment of the controller parameters. As a result, manual methods for tuning PID controller parameters are superior to complex calculations based on a given mathematical model.

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 [3].

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.

Paper review

Whitley presents canonical and experimental forms of the genetic algorithm in the handbook [4]. Here, we can be acquainted with the theoretical foundations of GA, including the design theorem and some precise models of the canonical GA.

Most publications devoted to tuning PID controllers use GA. Thus, A. Mirzal et al. [5] implemented a genetic algorithm (GA) in the canonical form to determine the parameters of a PID controller to compensate for the time delay in a first-order system. However, the work notes that the genetic algorithm does not always give the desired result. The studies performed in the Matlab environment showed negative results that exceeded 4% of all completed studies. A. A. Aly [6] presented setting up the PID controller GA in the SIMULINK environment for an electrohydraulic servo drive described by a system of four first-order differential equations and two control inputs. A feature of the controller tuning is the fitness function in the form of a mix of the integral of time multiplied by the absolute value of the error, overshoot, and steady-state error. Suyanto et al. [7] studied GA for tuning the PID controller in a water level control system in a miniplant with a three-phase oil separator. Lambora et al. [8] present the GA basic processes discussion, their features, and applications. T. A. F. K. Yusoff et al. [9], Meena, and A. Devanshu compared the performance of Ziegler-Nichols and GA tuning methods in [10]. GA for tuning fractional order PID controllers in an automatic voltage regulation (AVR) system, we use these controllers proposed by Sun et al. [11].

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 [12]. An adaptive GA for adjusting the parameters of the PID controller, which is based on the operations of crossing and mutation according to adaptive probability algorithms, was proposed by Zhao J. and Xi M [13]. A machine-learning algorithm for a PID controller for optimal tuning is presented in [14].

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. [15]. Yunan and Qu presented a combination of GA and iterative learning algorithms in [16]. Kawecki and T. Niewierowicz [17] proposed a hybrid algorithm based on genetic and classical algorithms for synthesizing time-optimal speed control of asynchronous motors. Flores-Morán et al. [18] present a combination of genetic and Fuzzy algorithms for tuning PID controllers in a DC motor control problem. Anh [19] proposes the Fuzzy Nonlinear ARX model for modeling, identification, and adaptive tracking of the angular position of the joint of a nonlinear robot manipulator in the article. The combination of the ant colony algorithm and GA for optimizing FID parameters is proposed in [20]. Bajarangbali et al. [21] show the integration of particle swarm optimization methods with GA for second-order objects.

Zhou et al. [22] proposed a scenario approach based on a genetic algorithm for investigating the events of entry and car tracking. Meng and B. Song [23] presented the improvement of genetic algorithms regarding population, selection, crossover, and mutation compared to simple genetic algorithms. A comparison of the structure and performance of a nonlinear variable PID controller (NL-PID) and a genetic algorithm (GA) PID controller is performed by Korkmaz et al. in [24]. Zhang et al. proposed the self-organization of genetic algorithms, in which new operators of dominant selection and cyclic mutation are introduced to optimize the PID controller parameters [25].

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.

Problem formulation

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:

𝑢(𝑡) = 𝑘 𝑒(𝑡) + 𝑘 𝑒(𝑡) + 𝑘 𝑑𝑒(𝑡) 𝑑𝑡 .(1)

In (1) kp, ki, and kd are the transmission coefficients of the proportional, integral, and differential units, respectively; e(t) is the error signal, 𝑒(𝑡) = 𝑥(𝑡) − 𝑦(𝑡), (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.

We assumed that the parameters of the PID controller are subject to restrictions in the form:

𝑘 ≤ 𝑘 ≤ 𝑘 , 𝑘 ≤ 𝑘 ≤ 𝑘 , 𝑘 ≤ 𝑘 ≤ 𝑘 .

In ( 3), values 𝑘, 𝑘 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:

𝐼 = α • 𝑂𝑆 + β • 𝑀𝑆𝐸 + γ • 𝑇 ,(4)

where OS is calculated by the formula

𝑂𝑆 = 𝑦 − 𝑦 𝑦 • 100%,(5)

MSE (mean squared error) is the mean squared error, which is calculated as follows

𝑀𝑆𝐸 = 1 𝑁 (𝑦 − 𝑦 ) ,(6)

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; 𝐼 ≤ 𝐼 , (7) where Imin is the minimum value of criterion (4).

Solution

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 = 𝑘ℎ + (1 − 𝑘)ℎ ,(8)ℎ = 𝑘ℎ + (1 − 𝑘)ℎ ,(9)

where k ∈ [0; 1] is some real coefficient. The geometric interpretation of this process is shown in Figure 2. In Figure 2 wmin,i and wmax,i denote the minimum and maximum value of the ith parameter. Mutation of selected chromosomes consists in changing the values of genes by some impurities Δhij according to the rule ℎ′ = ℎ + ∆ℎ , (10) where hij is the gene before the mutation; h'ij is the gene after the mutation. The algorithm uses an uneven mutation for the ith gene. According to [8], the new gene values in case of uneven mutation are calculated according to the formula ℎ = ℎ + ∆ℎ 𝑡, 𝑤 , − ℎ , 𝑖𝑓 𝑟 ≤ 0.5, ℎ − ∆ℎ 𝑡, ℎ − 𝑤 , , 𝑖𝑓 𝑟 > 0.5, (11) where ∆ℎ(𝑡, 𝑦) = 𝑦(1 − 𝑟 ( / ) ); 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.

Simulation

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 [7]. At the same time, the differential equation of the object model is represented by a system of differential equations, which for a stationary object is convenient to present in matrix form:

𝑥(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡) 𝑦(𝑡) = 𝐶𝑥(𝑡),(12)

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  R n , y  R m , and u  R p . Matrices A, B, and C have these forms A  R nn , B  R np , and C  R mn .

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.

Double integrator

Math model

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:

𝑥̇ (𝑡) = 𝑥 (𝑡), 𝑥̇ (𝑡) = 𝑘𝑢(𝑡),(13)

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 𝑢(𝑡) = 𝑦 (𝑡) + 𝑦 (𝑡) + 𝑦 (𝑡), 𝑦 (𝑡) = 𝑘 𝑒(𝑡), 𝑦 (𝑡) = 𝑘 ∫ 𝑒(𝜏)𝑑𝜏 , 𝑦 (𝑡) = 𝑘 ( ) . ( 14)

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.

A standardized single signal in the form of a step, i.e. r(t)=1(t), is applied to the input signal of the system.

In the interests of modeling and to prevent complex integration algorithms, we present the considered system in the extended phase space Xext  R 5 , 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: 𝑋 ̇ (𝑡) = 𝐴 𝑋 (𝑡).

(

)15

with the initial vector Xext(0) = (0, 0, 0, 0, 1) T . The system of equations in (14)

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 ≤ 𝑘 ≤ 1, 10 ≤ 𝑘 ≤ 1, 10 ≤ 𝑘 ≤ 1.

GA parameters

GA parameters are given in Table 1. 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.

Simulation results

Transient processes in the system according to Table 2 are presented in Figures 2 and 3.

Oscillating object

An oscillating object is chosen as an unstable control object, the characteristic equation of has conjugate imaginary roots j and is described by a system of first-order differential equations of the form [26,27] 𝑥̇ (𝑡) = 𝑥 (𝑡), 𝑥̇ (𝑡) = −ω 𝑥 (𝑡) + ω 𝑘𝑢(𝑡),

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.

Discussion

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 adjustment compared to the Ziegler-Nichols 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.