Algorithm for optimizing a PID controller model based on a digital filter using a genetic algorithm

Abstract

The widespread use of digital signal processing distinguishes the current stage of development of science and technology. However, there are many developments for continuous signal processing. Such developments include methods for tuning the PID controller, so improving the digital PID controller model remains relevant. The problem of constructing a model of a digital PID controller, which can be used in robotic systems based on microcontrollers and programmable logic integrated circuits, is considered. It is proposed to use digital filtering methods as the basis for the regulator. The digital filter coeficients are calculated using a genetic algorithm. This approach makes it possible to improve the accuracy of the model, to ensure the calculation of the PID controller coeficients using classical methods for an analog PID controller. The software has been developed in the Python programming language that implements the proposed method. The modeling demonstrated the efectiveness of the developed model.

Keywords

• digital filter
• PID controller
• genetic algorithm
• model optimization algorithm

1. Introduction

The problem of efective control of technological processes, robotic systems, aircraft and other technical means remains relevant for many industries. For this purpose, regulators are used in many areas of science and technology. The most popular is the PID controller [1].

In recent years, the role and importance of computer technology in the life of modern society has increased dramatically and continues to grow, therefore, modern technical means are mostly implemented on the basis of microprocessors and microcontrollers, and many problem solutions are adapted to work in digital devices [2]. The PID controller did not escape its fate either [3].

Controller tuning can be done in several ways, including obtaining controller parameters in analytical form [1, 4, 5]. However, most of these methods are designed for analog PID control and are not suitable for digital because its model does not exactly match the PID controller, and, accordingly, the optimization of the digital PID controller model is relevant.

2. Theoretical background

In general, the control system for any object has the form shown in figure 1. where , , – proportional factor, constant of integration and constant of derivation of the controller, respectively.

In some cases, the following expression is used ( 2 ): () = () + ∫︁ 0

() () + , where , , – PID controller coeficients.

Thus, the PID controller includes three components: proportional, integral and diferential. The proportional component generates a control signal counteracting the deviation (mismatch) of the output signal from the set value. The greater the mismatch, the greater the impact on the control object. If the output signal is equal to the set value, then the error signal is zero, and therefore the control action of the proportional component is zero. The integrator is used to eliminate the static error. The diferentiating component takes into account the rate of change of the output signal, which allows you to get better control of the object by predicting the output value of the signal [1, 3].

There are several groups for assessing the quality indicators of object management: direct, root, frequency, integral. In practice, direct quality indicators have found the greatest application. This is due to the fact that direct indicators of the quality of object management are determined directly by the transient characteristic [1]. The following quality indicators can be distinguished: ( 1 ) ( 2 ) • steady-state output value; • static error; • regulation time; • overshoot; • attenuation rate; • etc.

The choice of control quality indicators depends on the task in which the PID controller is used.

To ensure the required performance of regulation, it is necessary to calculate the coeficients of the PID controller.

There are many methods for calculating quality indicators. One of the first methods for calculating the parameters of PID controllers was proposed by Ziegler and Nichols [6]. This technique does not give very good results, but it is very simple, therefore it is still often used in practice. After calculating the parameters of the regulator, manual adjustment is required to improve the quality of regulation.

In work of Sablina and Markova [4], other methods for calculating the parameters of PID controllers are also considered, namely: Chien-Hrones-Reswick, Kuhn. Relay methods are also widely used [5, 7].

If the methods considered were developed relatively long ago, then the methods below are quite recent.

In [8, 9], methods for optimizing the parameters of a PID controller using a genetic algorithm are considered. In these works, the choice is analyzed: fitness functions, the main operators of the genetic algorithm, quality indicators.

The possibility of using neural networks to optimize the PID controller coeficients was considered by Kadu and Patil [10]. The main focus of the article is on the analysis of the stability of such systems.

Many works are related to the determination of the optimal parameters of the PID controller for specific control objects [9, 10, 11, 12].

A large number of works are linked to the development of the digital PID [13, 14, 15, 16, 17, 18, 19]. However, in fact, all the articles cited can be divided into two groups.

The first group of works [15, 16, 17, 18] is based on expression ( 3 ) given in [13]: where – sampling period, = , = /.

Expression ( 3 ) is often written in a recurrent form to reduce computational costs ( 4 ): () = ( − 1) + ( () − ( − 1)) + () +

+ ( () − 2 ( − 1) + ( − 2)) .

The second group of works [3, 14, 19] is based on the expression ( 5 ):

() = ( − 1) + 1 () + 2 ( − 1) + 3 ( − 2) , where 1 = + + , 2 = − − 2, 3 = .

The analysis showed that expressions ( 4 ) and ( 5 ) are practically identical (the control signal () depends on the last three readings of the error signal). The main diference between them is that expression ( 4 ) allows you to determine the coeficients of a digital PID controller based on an analog prototype. For expression ( 5 ), the coeficients 1, 2, 3 must be selected when manually adjusting the control system. It may seem that these coeficients depend on the ( 3 ) ( 4 ) ( 5 ) controller coeficients , , , so they can be calculated, but this is not the case. This is easy to see if you pay attention to the fact that these coeficients 1, 2, 3 do not take into account the sampling rate.

Taking into account the above, it follows that digital and analog PID controllers are not considered as diferent entities, therefore, methods for calculating the controller coeficients are considered regardless of whether it is digital or analog. However, as the analysis has shown, there are at least two implementations of a digital PID controller, so the calculation methods must take into account the structure of the controller. In addition, as will be shown later, the digital controller ( 4 ) is not a complete analogue of the classic analog PID controller. Thus, the problem of the algorithm for optimizing the digital PID controller is relevant.

3. Results and discussion

This section will discuss the implementation of a digital PID controller. The controller will be based on a digital filter [ 20]. The method for calculating the filter coeficients will be performed using a genetic algorithm [21, 22].

3.1. Digital filter

Digital signal processing is used wherever it is necessary to perform tasks such as filtering, compressing, recovering, controlling, measuring a signal: audio, video, or any signal coming from any source [23].

Filtering is the most common digital processing task, which is implemented using digital iflters: filters with a finite impulse response (FIR filters); filters with infinite impulse response (IIR filters). In general, a digital filter is understood as a hardware or software implementation of a mathematical algorithm, the input of which is a digital signal, and the output is another digital signal modified by the filter.

The main operations of information filtering include: noise suppression, smoothing, prediction, diferentiation, signal separation, etc.

The main advantages of digital filters over analog filters: • may have parameters that are impossible to implement in analog filters, for example, linear phase response; • do not require calibration, because their performance does not depend on the destabilizing factors of the external environment, for example, temperature; • input and output data can be saved for later processing; • accuracy of digital filters is limited by the capacity of the filter coeficients. • can be easily rearranged to filter a diferent frequency range, for example, by changing the data sampling rate.

In general, the digital filter is described by the following expression ( 6 ):

− 1 − 1 () = ∑︁ · ( − ) + ∑︁ · ( − ), =0 =0 where ℎ () = – impulse response of an FIR filter.

The amplitude-frequency response (AFR) of such a filter will have the following form ( 8 ): where , – filter coeficients; () , () – input and output signal; , – number of iflter coeficients , respectively.

Expression ( 6 ) is also called IIR filter. Such a filter is often used when you need to perform ifltering with a minimum number of arithmetic operations. If all the coeficients are equal to zero, then such a filter is called an FIR filter. In this case, the digital filter will be described by the following diference expression ( 7 ): where – circular frequency.

In many digital signal processing applications, the use of FIR filters is preferable because they have the following advantages: • filter group delay constant (linear phase FIR filters); • FIR filters are always stable.

For FIR filters to be linear phase, the impulse response must be symmetric or antisymmetric [20]. In this case, four types of FIR filters are possible (table 1).

Here (0) = ℎ ︀( 2− 1 )︀ , () = 2ℎ (︀ 2− 1 − ︀) , (0) = 0, () = 2ℎ (︀ 2− 1 − ︀) , = 1, 2, 3, . . . , 2− 1 , () = 2ℎ (︀ 2 − ︀) , = 1, 2, 3, . . . , 2 .

2 − ︀) , () = 2ℎ (︀

An example of an antisymmetric FIR filter with an even number of coeficients is a diferentiating filter, the AFR of which corresponds to figure 2.

3.2. Genetic algorithm

Genetic algorithm is a heuristic algorithm, which is a kind of evolutionary algorithms, with the help of which optimization problems are solved using methods of natural evolution, similar to natural selection [21, 22].

The range of tasks solved using the genetic algorithm is very wide: ( 7 ) ( 8 )

• numerical optimization problems; • traveling salesman tasks; • scheduling; • function approximation; • artificial neural network training; • etc.

The key concept of a genetic algorithm is an individual that encodes a possible solution to a problem. An individual is characterized by a chromosome or a set of chromosomes. The atomic unit of a chromosome is a gene (most often encoded by one bit). When solving the problem, a population of individuals is created. Each individual is assessed by the degree of fitness, which is determined in the task by the fitness function. Thus, individuals are determined that are better adapted to the "environment" (have the best solution).

The genetic algorithm is iterative, therefore, at each iteration, a new population of individuals is generated, which has better fitness than the previous one. This process continues until the desired results are achieved, or the number of iterations exceeds the threshold.

The peculiarity of the genetic algorithm is that the set of solutions is immediately improved, unlike many other optimization algorithms.

To create a new population, genetic operators are applied to current individuals: crossing, mutation, selection.

Crossover is an operator that applies to two parents. Most often, each of them is divided into two parts at the same random gene position. Formed individuals are a combination of the first and second parts of chromosomes from diferent parents (figure 3). The considered option is called the one-point crossing method. There are other crossover methods: multipoint, uniform, etc.

Mutation is an operator that makes a change in a gene at a random position on the parent chromosome. The mutation is designed to reduce the likelihood of optimization at the local maximum. There are the following mutation methods: bit inversion, exchange, permutation, etc.

Selection is an operator aimed at selecting individuals in accordance with a certain criterion. There are various selection methods: roulette method, tournament selection, ranking method, etc.

The following sequence describes how the genetic algorithm works: 1. Generating an initial population; 2. Calculation of the fitness of chromosomes; 3. Selection of initial chromosomes (solutions) with the best fitness values for creating a new population; 4. Performing the crossing operation; 5. Performing a mutation operation; 6. Calculation of the fitness of chromosomes; 7. If the stop condition is met, return the chromosome with the best fitness value, otherwise go to step 3 to process the new population.

As mentioned above, the genetic algorithm refers to heuristic search algorithms, so it is necessary to adjust the hyperparameters. To make sure that the hyperparameters used made it possible to obtain a solution close to optimal, it is essential to control changes in chromosome iftness from generation to generation.

3.3. Development of a PID controller model

Let’s define the transfer function of the analog PID controller. For this, it is necessary to perform the Laplace transform of formula ( 2 ) with zero initial conditions (0) = 0. As a result, we get the following expression: where – Laplace operator.

If in expression ( 9 ) we substitute = , then we obtain an expression for the frequency response of the analog PID controller ( 10 ): () = + 1

+ , () = − + .

( 9 ) ( 10 ) Let’s look at the frequency response of the regulator at = 10, = 1, = 1 (figure 4).

The AFR of the analog PID controller shows that the integrating component has an efect in the low-frequency range, and the diferentiating component in the high-frequency range.

Now let’s compare the AFR of the diferentiating components of the analog and digital PID controllers.

First, let’s write the frequency response of the derivative component of the analog PID controller. It can be seen from formula ( 10 ) that its frequency response is determined by the expression (11): () = . (11) (12)

Now let’s determine the AFR of the diferentiating component of the digital PID controller. From expression ( 3 ) it can be seen that in the time domain the diferentiating component has the following form (12):

() = ( () − ( − 1)) .

In the brackets of expression (12) there is a digital filter of the first order, therefore the frequency response can be determined from formula ( 8 ). As a result, we will have the following expression (13):

From table 1 it follows that there are 4 types of such filters. Filters of types III and IV have an imaginary part of the AFR. However, the type III filter cannot always be used as such. The reason is that the value of the transmission coeficient at the maximum frequency will be equal to zero () = 0 regardless of the filter coeficients (table 1). Such a filter can be used as a diferentiating filter only in the initial section. We need to use the entire range, so for our task the best solution would be to use a type IV filter.

From expression ( 7 ) and table 1, it follows that the diferentiating component for a digital PID controller can be represented in the form of expression (14): () = ∑︁ ℎ() · ( ( − ) − ( + − 2 + 1)), (14) where – number of independent coeficients. controller will be described by the expression (15):

In this case, taking into account expression ( 3 ), the algorithm for implementing a digital PID ∑︁ () + =0 − 1 =0 () = () + ∑︁ ℎ() · ( ( − ) − ( + − 2 + 1)).

(15)

For simplicity, we will call it PPID. By analogy, we can write an expression similar to the recurrence formula ( 4 ) or recurrently recorded only an integrating part (this will reduce the likelihood of overflow and reduce the number of arithmetic operations) in the following way (16): () = () + () + ∑︁ ℎ() · ( ( − ) − ( + − 2 + 1)), (16) where () = ( − 1) + () – integrating component.

To obtain the final model of the PPID controller, it is necessary to determine the coeficients ℎ(), where = 1, 2, 3, . . . , − 1.

To synthesize the filter (14), a fitness function is required. The synthesis of the filter with the best uniform approximation will be performed in the form of the problem of minimizing the weighted Chebyshev norm (17):

︁( = () ⃒ () − ̂︀ ()⃒

→ , ⃒ ⃒ )︁ (17) − 1 =0 ⃒ ⃒ weight function.

3.4. Experiments

where (), ̂︀ () – AFR of the approximated and approximating filters, respectively, () – To test the model of the digital PPID controller, we will carry out a number of experiments. Let us synthesize a digital FIR filter (14). There are many methods for their design in the scientific literature [ 20]. The most widely used are the classical methods for calculating FIR iflters: weighing method; frequency sampling method; least squares method; best uniform approximation method. The first two are not optimization methods, but are fairly easy to use. The third and fourth methods are referred to as optimization methods. The fourth method allows obtaining the best results, but, as a rule, it is impossible to determine analytically the function of the best uniform approximation. However, in this case, the synthesis of the FIR iflter will be carried out using the genetic algorithm [ 24], which will allow us to obtain some advantages, for example, when searching for the values of the filter coeficients, we will take into account the efect of quantization.

When solving a problem with a genetic algorithm, it is necessary to isolate the phenotype that determines the real object. In our case, the filter coeficients that will form an individual will act as a phenotype (figure 7).

Fitness function will be described by expression (17). The AFR of the diferentiating part of the analog PID filter (11) at = 1 (figure 5, graph 1) will act as the AFR of the approximated iflter. The AFR of the approximating filter will be determined by a type IV filter (table 1).

The simulation was carried out using the Python programming language. To implement the genetic algorithm, it is necessary to tune the hyperparameters. In our case, they will have the following form:

POPULATION = 100 # number of individuals in the population SURVIVOR = 0.2 # survival probability MUTATION = 0.1 # possibility of mutating of an individual GENERATIONS = 250 # maximum number of generations

Below is the fitness function code in the Python programming language, which corresponds to the expression (17), where prototype is an instance of the PrototypeFIR class of the filter being approximated (figure 2); fir – an instance of the Fir1T class approximating the filter ( 7 ). def fitness(individual): # fitness function fmax = prototype.getSamplingFrequency() / 2 fir = fir1t.Fir1T(fmax, individual) emax = 0 for fi in prototype.getReferencePoints(): e = abs(prototype.getGain(fi)

fir.getGain(fi))*prototype.getWeight(fi) if e > emax:

emax = e return emax, Figure 8 shows the synthesis of a diferentiating component PPID controller.

Table 2 shows the calculated coeficients of filters of diferent orders, and also indicates the approximation error of the diferentiating component of the PPID controller.

If we compare the proposed model of the PPID controller with the original digital PID controller ( 3 ) at = 1, we can see that the expressions difer only by the factor ℎ(0) (table 2). However, due to him, the error was reduced by 15%. Figure 9 shows the relative error of this PPID controller.

Figure 10 shows the tendency of decreasing the error with an increase in the number of coeficients ℎ(). 6 is suficient for solving most control problems in robotic

It can be seen that 4 ≤ ≤ systems.

4. Conclusion

The problem of constructing a model of a digital PID controller, which can be used in robotic systems based on microcontrollers and programmable logic integrated circuits, is considered.

The regulator is based on digital filtering methods. It is proposed to use an FIR filter with a linear phase of the IV type as a filtering device. This made it possible to fairly accurately approximate the diferentiating component. So, for a classic digital PID controller, the introduction of one coeficient has reduced the relative frequency response error by 15%. In addition, the PID controller model was developed with the ability to use ready-made methods for calculating the PID controller coeficients.

The digital filter coeficients are calculated using a genetic algorithm. The phenotype is the iflter coeficients. The Chebyshev norm was used as a fitness function.

The simulation results were carried out using the Python programming language. Data for all filters up to 21 orders (up to 11 independent coeficients) has been analyzed.

As shown in the work, for most control problems in robotic systems, it is suficient to use iflters with 4-6 independent coeficients.

Perspectives for further research consist in testing the proposed methods on a wider range of problems, studying the efects of finite bit depth, and analyzing the structure of the PID controller.

Acknowledgments

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