PID-Controller Tuning Optimization with Genetic Algorithms in Servo Systems

Abstract

Performance improvement is the main goal of the study of PID control and much research has been conducted for this purpose. The PID filter is implemented in almost all industrial processes because of its well-known beneficial features. In general, the whole system's performance strongly depends on the controller's efficiency and hence the tuning process plays a key role in the system's behaviour. In this work, the servo systems will be analysed, specifically the positioning control systems. Among the existent tuning methods, the Gain-Phase Margin method based on Frequency Response analysis is the most adequate for controller tuning in positioning control systems. Nevertheless, this method can be improved by integrating an optimization technique. The novelty of this work is the development of a new methodology for PID control tuning by coupling the Gain-Phase Margin method with the Genetic Algorithms in which the micro-population concept and adaptive mutation probability are applied. Simulations using a positioning system model in MATLAB and experimental tests in two CNC machines and an industrial robot are carried out in order to show the effectiveness of the proposal. The obtained results are compared with both the classical Gain-Phase Margin tuning and with a recent PID controller optimization using Genetic Algorithms based on real codification. The three methodologies are implemented using software.

Keywords

Frequency Response (FR)Gain-Phase Margin Genetic Algorithm (GA)Positioning System PID control

1. Introduction

Positioning control systems control systems with motion capabilities in a quick, smooth and precise way, playing a key role in the manufacturing industry [1]. These systems are wide-spread in automated industries and are used for quality improvement, the reduction of production time and profit increase, despite the initial investment [2]. The principal and most common industrial applications are in CNC machinery and industrial robotics. When designing control systems in the industry, approximately 95 percent of these systems are Proportional-Integral-Derivative (PID) controllers [3] and 85 percent of these are implemented on closed-loop systems with a single input and a single output configuration [4]. The reasons for the wide use of these controllers are the easiness of their implementation in digital systems, their robustness, low cost maintenance, and so on [5, 6]; their effectiveness has been demonstrated in positioning control applications as shown in [7, 8]. Now, the positioning systems are normally unstable when they are implemented in a closed-loop configuration, so when the controller is introduced into the closed-loop it needs to be effectively tuned. However, PID controller tuning for positioning control systems is a time-consuming task. Typically, the adjusting of the controller parameters is carried out using formulas to provide a performance, which, although not deficient, is far from optimal. Therefore, much effort has been given to analysing the complex plant models of these systems in order to obtain an efficient tuning.

The most important tasks in control design are the system model identification, the law or the control algorithm, the tuning and the control implementation. Asides from the proper operation of all the elements within a control-loop, the performance of the whole control system mainly depends on the controller parameters which are obtained by its tuning [9, 10]. For many years much research has been done with the aim of adjusting the controller parameters. The most popular methodologies used for PID controller tuning, in industrial applications, can be classified as i) empirical methods such as Ziegler-Nichols (ZN) and Cohen-Coon (CC) [11], and ii) analytical methods such as root locus (RL) and frequency response (FR) analysis [12, 13]. The empirical methods have the advantage of applying easy rules to simple mathematical models, but do not provide as good results as expected. The analytical methods can deal with some higher order models; they are more precise and have the advantage of improving some specific parameters of the system response such as stability, rising time, steady state error, overshoot, etc. Both empirical and analytical methods are time consuming; however, the Gain-Phase Margin (GPM) method based on FR analysis has been demonstrated to be more efficient than the other methods when it is applied to PID controller tuning for positioning control systems such as for CNC machines and industrial robots, as shown in [14, 15]. This method has the advantage over time domain analysis that it provides information about the relative and absolute system stability. Additionally, the generality and versatility of this method allows it to be used to automatically compute the parameters of a PID controller, as some available commercial controllers like Galil Motion Control do [16]. Nevertheless, this method has some problems with its design and optimization; sometimes the adjustment of the PID controller parameters does not go as expected. Recently, modern optimization techniques such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO) and Ant Colony Optimization (ACO) are applied in the parameter optimization of PID controllers; a comparative study is presented in [17]. Nonetheless, the actual tendency in parameter optimization is the application of the GA due to their powerful searching capabilities and heuristic characteristics, more specific details about the GA can be reviewed in [18]. The effectiveness of this technique has been demonstrated in positioning control systems for robotics path generation in [19 –21]. Therefore, GA has also been used in control tuning applications, being shown to obtain better results than classical techniques. Besides, as can be seen in [22, 23] the micro-GA, a relatively new concept, could be useful for reducing computational resources and for providing appropriate solutions in many optimization problems when they are carefully designed and genetic diversity is ensured.

Due to the advantages and despite the disadvantages of the FR analysis, many works have used this technique for PID control tuning. For instance, simulation software developed using C++ for PID tuning was presented in [24]. To obtain the PID coefficients the open and closed loops models, including the PID structure in a Laplace form, were evaluated in the interest frequencies. The software was tested on simulated models up to the third order. Meanwhile, a controller design methodology was proposed in [25] based on process FR data, identified through relay feedback tests and using multiple points on the Nyquist curve. The controller model was obtained by using the least squares method and the proposal was validated with simulations over linear systems and compared with classical ZN and GPM methods. On the other hand, an approach for an adaptive control for PI controllers was presented in [26]. The proportional and integral gains were adjusted using a Nelder-Mead algorithm. Also, open-loop gain and phase margin constraints were included in it. Simulations and experimental tests were carried out on a simple pilot column distillation loop with a commercial controller but using large sample periods for the tuning process. Recently, the FR methodology was used in [27] to efficiently design a feed forward compensator for an industrial motion stage in a transmission electron microscope. The approach does not necessarily provide optimal tracking behaviour; additional control action may be required. Simulations using a commercial controller for data acquisition and MATLAB/SIMULINK validated the proposal. Also, the GPM method has been applied in tuning applications tackling a specific problem in positioning control systems, as seen in [7, 28]. However, this method was applied offline because the controller tuning took longer. In contrast, GA is a powerful searching algorithm that mimics natural behaviour based on natural genetics and natural selection [29, 30]. Many works have proposed the use of GA to optimize the PID gains in control systems, but the standard GA has the drawback of pre-maturity and stagnation, while it looks for a global optimal solution. Recently, in [31] parameter optimization was carried out for a small helicopter based on GA focusing on the stability and adaptability of the system. The performance of the system is validated for the model simulations. A methodology for a PID controller design was presented in [10], where the antibodies concept was used to improve a simple GA providing population diversity and searching speed. The validation of this approach was performed using simulations for an artificial leg. In [32] an adaptive niche GA was proposed to find the optimal PID parameters, an elitist strategy ensured the stable convergence, the niche ideology kept the population diversity, and the adaptive mutations and crossover probabilities enhance the local search ability. The methodology was tested on four proposed models of first and second order by simulations. The GA and chaos optimizing concept were integrated in [33] for PID parameter optimization. Here, the initial population was generated by a logic sequence and adaptive mutation probability was used to jump the local optima. Simulation tests were performed over a third order proposed plant model. A Real-coded Genetic Algorithm (RGA) approach using floating point variables representation presented in [34] was used to optimize the PID parameters of a temperature system model, obtained by software identification tools; the system response was simulated compared to ZN and CC methods. Recently in [35] the RGA technique was applied for system identification and control tuning for a model reference adaptive control for a hybrid tank system. The model is obtained using recollected data through an acquisition card and the performance of the proposal was simulated using MATLAB. Moreover, an approach was developed in [36] of a self-tuning PID controller based on GA. A dominant selection and cyclic mutation operators were used to improve the fitness of the GA population. The method implementation was carried out using software and was validated by simulations. A tuning approach for PID controllers based on GA using adaptive mutation and crossover probabilities to avoid premature convergence was developed in [37]. Simulations of this approach were conducted in MATLAB for two simple third order models. Also, in [30] an adaptive GA with variable mutation probability was applied to optimize the PID controller gains for an ideal second order model representing a coal transportation system. Simulations of the proposal were carried out in SIMULINK. Meanwhile, in [38] a fuzzy PID controller was used for high order plants with time delays. The controller parameters were tuned by using the GA and two variants of PSO techniques offline. The proposal validation was executed by simulations in MATLAB. Finally, a multi-objective non-dominated sorting GA was developed in [39] for the tuning of a PID controller applied to a robotic manipulator. The results of simulations demonstrated the performance of the approach. From the analysis of these works, several aspects about control tuning methods can be concluded. For instance, the FR analysis has notorious advantages over time domain analysis and it can be implemented as an automatic tuner for positioning control systems. The GA is a powerful optimization technique when it is combined with other methodologies to avoid premature local convergence on tuning applications. PID control tuning has been used for simulated models in the majority of the works presented up to now. Yet, an integration of GA, using the micro-GA concept, and FR analysis have not been carried out for the optimal adjustment of the PID control parameters; besides, it would be desirable to implement this approach in real positioning control systems, which is the proposal of this work, beyond just simulations.

Given the necessity of increasing the performance in positioning control systems, the contribution of this paper is a new methodology for the optimization of PID controller parameters. This methodology couples the GPM method which is based on the FR analysis and the Adaptive Micro Genetic Algorithm (AMGA). One interesting characteristic of the proposal is the adaptive mutation probability which is implemented in order to increase the searching speed and avoid losing genetic information. On the one side, the GPM method is used to calculate the controller gains for any order model plant of real processes. The input parameters of the GPM method are the response time and the phase margin desired of the positioning control system. On the other hand, the AMGA technique ensures the GPM method will have optimized input parameters; the approach makes use of adaptive mutation probability to avoid local stagnation and uses micro-population concept to reduce computing resources and increase searching speed. The gains obtained with this approach are implemented in a proprietary controller board for three servo systems: a CNC lathe with two axes, a CNC milling machine with three axes and a six DOF industrial robot PUMA (Programmable Universal Manipulation Arm). Experimental results are compared with the classic GPM tuning method applied manually for the adjusting of the PID controller gains and with PID parameters optimization using a RGA method as presented in [35]. A simulation using a theoretical model of a positioning system is also performed to compare the methodologies implemented.

2. Background

2.1 PID Control scheme

The PID controller has been widely used since its invention in 1910 [40]. There are many PID control configurations, but the most common implementation of this controller, as shown in Figure 1, is the feedback-loop with a single input and a single output, as explained in [41].

Figure 1.

Block diagram of the simplest closed-loop system.

2.1.1 Typical structure

The typical implementation of the PID control is shown in Figure 2. The signal error, e(t), enters the PID control block and the resulting excitation signal is the sum of the error signal affected by the proportional, integral and derivative actions.

Figure 2.

Block diagram of typical PID control structure.

The output excitation signal of the PID controller, where e(t) = r(t) - y(t), is:

u (t) = K_{p} e (t) + K_{i} \int e (t) d t + K_{d} \frac{d}{d t} e (t)

(1)

Also, the equation (1) can be rewritten in Laplace form as:

u (s) = e (s) (K_{p} + \frac{K_{i}}{s} + K_{d} s)

(2)

Finally, under the above scheme the transfer function of the PID controller or the control law is specified by

C (s) = \frac{u (s)}{e (s)} = K_{p} + \frac{K_{i}}{s} + K_{d} s

(3)

The equation (3) can be implemented on a digital system as an equation in differences for a second order infinite impulse response (IIR) filter specified by

\begin{array}{l} y (k) = a_{0} u (k) + a_{1} u (k - 1) + a_{2} u (k - 2) \dots \\ - b_{1} y (k - 1) - b_{2} y (k - 2) \end{array}

(4)

The corresponding discrete transfer function of (3) considering (4) is:

C (z) = \frac{a_{0} + a_{1} z^{- 1} + a_{2} z^{- 2}}{1 - b_{1} z^{- 1} - b_{2} z^{- 2}}

(5)

Therefore, the relationship between the PID gains and its implementation in a digital system is shown in Table 1.

Table 1.

Equivalence of parameters for PID controllers.

C(z)	a ₀	a ₁	a ₂	b ₁	b ₂
C(s)	K_p + K_i + K_d	–(K_p + 2K_d)	K _d	1	0

2.2 System frequency response analysis

According to Tal in [42], system analysis is a method for predicting the system response. With regards to frequency, this analysis considers two main aspects:

The system stability

The system response speed

The nature of the response is indicated by the system stability. The stability grade is usually expressed as the phase margin, θ_m. On the other hand, the response speed measures the time in which the system performs a small correction without saturation. Usually the crossover frequency parameter, ω_c, is directly related to the system response time, τ, which is approximately

τ = \frac{1}{ω_{c}}

(6)

The first thing to perform in the FR analysis is to obtain the open-loop transfer function of the system, L(s). Then, the open-loop frequency response, L(jω), is obtained by substituting “s” with “jω”. Here, the main parameters to be considered for the system design are θ_m and ω_c. Finally, the system stability is guaranteed by satisfying the following restrictions:

| L (j ω) | = 1 a t ω = ω_{c}

(7)

θ_{m} = 180 + \arg [L (j ω)]

(8)

In order to ensure the system stability for almost all the positioning control systems and to satisfy (7) and (8), there is a useful rule in the FR analysis. This rule establishes that phase margin values in the range of 30° − 45° provides a well-damped system response, higher values indicate larger overshoot and lower values indicate an under-damped system response.

2.2.1 Gain-phase margin tuning method

Consider the closed-loop system shown in Figure 3.

Figure 3.

Block diagram of the elements for GPM tuning process.

Where:

C(s), is the control law as stated in (3).

$D (s) = e^{- \frac{T s}{2}}$ , is the time delay associated with the sampling process of the digital controller.

K, is the actuator constant.

G(s), is the plant model.

L(s) = C(s)D(s)G(s)K, is the open-loop transfer function.

Then, the tuning process consists of solving equations (7) and (8) for the system frequency response. Therefore, the general expressions for the GPM tuning method are:

| L (j ω_{c}) | = | K_{p} + \frac{K_{i}}{j ω_{c}} + K_{d} j ω_{c} | \cdot 1 \cdot K \cdot A

(9)

θ_{m} = 180 ° + Arg [K_{P} + \frac{K_{i}}{j ω_{c}} + K_{d} j ω_{c}] - \frac{180 ω_{c} T}{2 π} + α

(10)

From equation (9), A = | G(jω_c)|, and the delay function has a magnitude of unity and hence has no effect in the equation. On the other hand, in equation (10), the phase shift associated with the delay is expressed as the term (180ω_cT) / (2π)deg, and α=Arg[G(jω_c)]. Now, it must be emphasized that the integrator effects mostly occur in the low frequency range, so it can be assumed that Ki has no effect on the magnitude. However, although the integrator makes the system response more precise it slightly degrades the system stability, so according to Tal in [42] the effect of Ki in the phase can be limited to a shift of −5°. Then, equation (9) and (10) can be rewritten as:

| K_{p} + K_{d} j ω_{c} | \cdot A \cdot K = 1

(11)

θ_{m} = 180 ° + \tan^{- 1} (\frac{ω_{c} K_{d}}{K_{p}}) - \frac{180 ω_{c} T}{2 π} - 5 ° + α

(12)

Equation (11) is the relationship between the magnitudes of the PID controller and the plant model according to restriction (7), so a more useful expression can be defined by

| K_{p} + K_{d} j ω_{c} | = \frac{1}{A \cdot K}

(13)

Equation (12) is the sum of all the contributed phases in the open-loop frequency response according to restriction (8). Now, β is defined as the contributed phase of the proportional and derivatives actions and is specified by

\tan^{- 1} (\frac{ω_{c} K_{d}}{K_{p}}) = θ_{m} - 175 ° + \frac{180 ω_{c} T}{2 π} - α = β

(14)

Therefore, the relationship between the PID controller gains and the plan model frequency response in the complex plane is shown in Figure 4.

Figure 4.

Frequency response of the plant and controller gains relationship.

Finally, the PID controller gains are obtained from the following expressions:

K_{p} = \frac{\cos (β)}{A \cdot K}

(15)

K_{d} = \frac{s i n (β)}{A \cdot K \cdot ω_{c}}

(16)

K_{i} = \tan (5 °) \cdot K_{p} \cdot ω_{c}

(17)

2.3 Genetic algorithms scheme

The Genetic Algorithm, GA, according to Rao in [29] is a powerful optimization searching technique based on the principles of natural genetics and natural selection. A flow chart of the general scheme of the implementation of the GA is shown in Figure 5.

Figure 5.

General flowchart of the Genetic Algorithm.

In the GA, normally the design variables, corresponding to the genomes in the natural genetic, are represented as binary strings and they are concatenated to form an individual, corresponding in turn to a chromosome in natural genetics. Other representations can be used. However, a binary representation is more adequate if an implementation in a digital system has to be carried out. The main elements of natural genetics, according to Rener in [18], used for the searching procedure are:

Reproduction

Crossover

Mutation

The reproduction operation consists of selecting individuals from the present population without changes to form part of the new population, in order to provide the possibility of survival for already developed fit solutions. Meanwhile, the crossover operation consists of creating new individuals (offspring) from the present individuals (parents), according to a crossover probability, Cp, by selecting one or more crossover points within the chromosome of each parent at the same place. Then, the parts delimited by these points are interchanged between the parents. On the other hand, the mutation operation makes modifications to a selected individual according to a mutation probability, Mp, by modifying one or more values in the binary representation. It is worth noting that if Mp is too large the GA is a purely stochastic search, but if Mp is too small it will be difficult to create population diversity in the GA.

In order to obtain the optimal solutions, the population of the GA must be evaluated. This evaluation consists of calculating the fitness value of each individual by minimizing or maximizing an objective function, J. The objective function is defined based on the optimization problem; the fitness function, f, is then specified by

f = 1 / J

(18)

The objective of the GA is to search for the global optimal solutions in the optimization problems. The fitness value is an indicator of the best and the worst individuals in the population. Therefore, the optimal solutions are obtained by selecting the best individuals in each generation of the optimization process; a generation is a complete cycle in which the population is generated, the genetic operators are applied and finally the population members are evaluated. The iterative cycles finish when the termination criterion is finally satisfied. There are many methods for selecting the best individuals according to their fitness value. For instance, the most popular methods applied in the GA are the proportional selection, the roulette wheel selection, the ranked selection and the tournament selection.

2.3.1 The micro genetic algorithm

The Micro-Genetic Algorithm (MGA) is a relatively new concept applied to improve the performance of Standard Genetic Algorithm (SGA). According to Coello in [23], the MGA refers to a GA with a very small population. This population is randomly generated and then the genetic operators are applied until nominal convergence is reached; just at the moment when individuals have similar or identical characteristics. The best individual selected from the evaluated present population is transferred to the new population and the rest of the new population is randomly generated in order to inject population diversity. The MGA is applied to avoid the premature convergence and the stagnation problem of the SGA, as seen in [22]. Also, this concept can reduce the use of computational resources and increase the searching speed in software implementations. However, in some cases this technique does not guarantee a global optimal solution, but for applications such as that in the present work, this technique can provide appropriate local optimal solutions for the optimization problems which other deterministic methods find difficult in accordance with Hultmann in [39].

3. PID control frequency response tuning optimization based on Adaptive Micro-GA

In this section the methodology for tuning the optimization of PID controllers is described. As mentioned in the introduction section, the optimization approach couples the GPM method based on FR analysis and the AMGA technique.

3.1 Methodology description

Positioning control systems generally consist of three mains blocks: the PC-software, the controller board and the positioning system. In Figure 6 the general scheme of the proposed methodology is shown in the PC-software block. In the following paragraphs the complete description of the proposed methodology is presented.

The PC-software block contains the system model identification, which is not the scope of this paper, but it is worth mentioning that the Least Squares Method (LSM) as presented in [3] is used to carry out the identification task. In order to obtain the plan model for each servo system mentioned in the introduction section, little random references are followed by the positioning system and the recollected data is employed in the LSM. Then, the plant model is provided in the discrete form as:

G (z) = \frac{n c_{n} \cdot z^{n} + n c_{n - 1} \cdot z^{n - 1} + \dots + n c_{0}}{d c_{n} \cdot z^{n} + d c_{n - 1} \cdot z^{n - 1} + \dots + d c_{0}}

(19)

The AMGA technique is implemented to optimize the input parameters of the GPM tuning method. Finally, when the optimization process ends, the output parameters of the GPM tuning are the optimized PID controller gains. The optimization process is implemented offline. The obtained parameters are sent to the controller board of the positioning system.

Figure 6.

General diagram of proposed methodology; coupling GPM method and AMGA technique to optimize the controller gains.

The AMGA technique works as follows. First, the initial parameters are specified. In order to avoid infinitely cycled loops, the iterative optimization process has a termination criterion defined by a maximum number of generations, G. As the micro-population concept is applied for the GA, a small number of individuals have to be defined, PSize; in this work four individuals are generated. Besides, due to the micro-population concept, the crossover operation is applied to all the individuals by choosing two crossover points randomly. An adaptive mutation probability is used, so the initial mutation probability is also specified, Mp. The string length of each variable must be specified, Svl. Then, the string length of the chromosome, Scl, is the concatenation of each string variable. The searching space range must be defined for each variable, which are the minimum, v_min, and maximum, v_max, values that the corresponding design variable can assume. Finally, the weighted values, ω_i, of the objective function, J, are established.

The next step is to generate the initial population. Normally, when a large population size is defined, the initial population is generated randomly and the searching space distribution of the individuals is generally accepted. However, for this work a good distribution of the individuals must be ensured because a small population is used. Therefore, the individuals initially generated were distributed according to each one's variable range in a fixed way. Notice that the optimization variables (genomes) of the AMGA technique are τ and θ_m, so their defined ranges, shown in Table 2, are used to distribute the initial individuals. The next step in the optimization approach is the population evaluation. To calculate the fitness value of each individual, system response simulations are carried out. For this, the variables of the four individuals are decoded into their corresponding numerical values, taking into account each variable range. The general expression for binary string decoding to its real value is:

Table 2.

Defined parameters for the AMGA technique.

AMGA parameters		Specified value
Maximum generations number, G		30
Variables number, Vn		2
Population size, PSize		4
Variable string length, Svl		16
Variable range [v_min v_max]	Response time, τ	[0.01, 1]
Variable range [v_min v_max]	Phase margin, θ_m	[30, 70]
Weighted value for ISE, w₁		0.3
Weighted value for OS, w₂		0.3
Weighted value for MSE, w₃		0.4

V_{d e c} = v_{\min} + \frac{v_{\max} - v_{\min}}{2^{S v l} - 1} \sum_{i = 0}^{S v l} 2^{i} V_{b i n} i

(20)

In which, V_dec is the real value of the optimization variable and V_bin is the corresponding bit value in the binary string representation. Then, the decoded values are used to simulate the system response in MATLAB as follows. The plant model and the PID control model are used to evaluate the closed-loop system response. However, the control model requires the PID gains in order to perform the simulation. For this reason, the “GPM Method Tuner” function is used. This function, according to equations (11) and (12), requires the magnitude, A, and the contributed phase of the plant model, α. In order to calculate these values for any order model plant, the following procedure is used. The complex form of z is specified by

z = A e^{- j ω} = A \cos (ω) + j A \sin (ω)

(21)

Therefore, the real and the imaginary components of the discrete transfer function specified by (19) can be obtained by applying:

r e = \sum_{i = 1}^{n + 1} c_{n - i + 1} \cdot \cos [(n - i + 1) \cdot ω]

(22)

i m = \sum_{i = 1}^{n + 1} c_{n - i + 1} \cdot \sin [(n - i + 1) \cdot ω]

(23)

Where n is the order of the numerator or denominator in the discrete transfer function, the coefficients C_n–i+1 are specified by nc for the numerator and by dc for the denominator in expression (19). Then, the magnitude and the contributed phase of the plant can be obtained by calculating the modulus and the inverse tangent using the real and the imaginary components, respectively. Finally, the PID controller gains are obtained by solving equations (15) to (17). When the model plant and the model of the PID controller are known, the step response of the closed-loop, as shown in Figure 1, is simulated. Furthermore, the system response performance criteria defined in this work are the Integral of the Squared Error (ISE), the Mean Squared Error (MSE) and maximum overshoot value (OS). The ISE and MSE are defined by

I S E = \int_{0}^{\infty} e^{2} (t) d t

(24)

M S E = \frac{σ^{2}}{n_{e}}

(25)

From equation (25) σ is the standard deviation of the error and n_e is the length of the error vector. The objective function, J, must be minimized in order to avoid larger overshoot values and to obtain satisfactory dynamics in the transient response. This function is then specified by

J = w_{1} \cdot I S E + w_{2} \cdot O S + w_{3} \cdot M S E

(26)

From equation (26), w₁, w₂ and w₃ are the weighted values used to enhance the corresponding performance criteria, ISE, MSE or OS. Then, the fitness function is specified by equation (18). The selection of the best individuals is the next step. The “tournament selection” is applied in this work. As the number of individuals in the population is small, the tournament selection is applied for all the individuals. The individual with the highest fitness value is selected as the best individual and the individual with the worst fitness value disappears.

After the selection of the best individuals, the termination criterion of the optimization process is evaluated. Therefore, if the maximum number of generations is reached, the AMGA function designates the optimized GPM parameters. Meanwhile, if the maximum number of generations is not reached yet, a new population is generated. The criterion for generating the new population is as follows. The worst individual will not appear in the new population. The best individual is selected and directly introduced into the new population. The rest of the offspring is generated by applying the crossover operation between the best individual and the two remaining individuals of the present population, and between the two remaining individuals only. The mutation operation is applied to the new population, except for the best individual, according to the adaptive mutation probability, Mp, in order to infuse material diversity. Three mutation points for each individual are selected for this process. When the AMGA starts, the Mp value is large in order to increase the searching speed, but when the generations elapse, this value is decreased to avoid losing genetic information. The adaptive mutation probability is implemented according to the next expression.

M p = 0.4 - [j \cdot (a / G)]

(27)

Where 0 < j ≤ G, is the current generation cycle. From several tests it was found that the value of a when proposed as 0.006 provides good results. When the optimization process ends, the optimal GPM parameters are designed. Then, the “GPM Method Tuner” function is called upon one last time to obtain the optimized PID control gains. The parameters defined for the AMGA technique are summarized in Table 2. Also, the parameters of the RGA optimization technique for the purposes of comparison are summarized in Table 3.

Table 3.

Defined parameters for the RGA technique.

RGA parameters		Specified value
Maximum generations number, G		100
Variables number, Vn		3
Population size, PSize		20
Variable range [v_min v_max]	Proportional gain, K_p	[0.01, 100]
Variable range [v_min v_max]	Integral gain, K_i	[0.01, 50]
Derivative gain, K_d		[0.01, 300]
Weighted value for ISE, w₁		0.3
Weighted value for OS, w₂		0.3
Weighted value for MSE, w₃		0.4

3.2 Simulation test

In order to validate the effectiveness of the optimization approach, a simulation test is performed in MATLAB. The results are compared with the classical tuning using the GPM method and with a PID optimization based on RGA as presented in [35]. The theoretical model used is that simplified by Tal in [42] for positioning systems:

G (s) = \frac{K_{a} K_{t} K_{f}}{J s^{2}}

(28)

Where, K_a is the amplifier gain with a specified value of 1 [A/A], K_t is the torque constant with a specified value of 0.2 [Nm/A], K_f is the position sensor gain with a specified value of 636 [counts/rad], and J_m is the inertia moment with a specified value of 1.1⁻⁴ [kg.m²]. The results are shown in Figure 7.

Figure 7.

Simulation of the step response; comparing the optimization techniques with classical GPM tuning.

The features of the Laptop used for the implementation of the three methodologies are a 64-bits operative system, 8 GB of RAM, an Intel Core i7-6312QM CPU operating at 2.1 GHz. From Figure 7 it can be seen that the classical GPM tuning provides a good response in general for positioning systems, but the improvement in the system response is notoriously bad when using the optimization techniques. The GPM-AMGA optimization and the RGA optimization present a significant reduction in the overshoot when compared to the classical GPM tuning. However, the GPM-AMGA optimization reduces the overshoot even more when compared with the RGA optimization. The response time is also a significant improvement for both optimization techniques when compared with the classical GPM tuning. In this case the response times for the GPM-AMGA optimization and for the RGA optimization are very similar. The computing time required for the RGA optimization is 21.4786 seconds while the respective computing time required for the GPM-AMGA optimization is 1.3783 seconds.

4. Experimentation and results

The efficiency of the optimization approach is validated by experimental tests performed over three servo systems.

4.1 Experimental setup

The optimization approach was implemented using PC-software offline using MATLAB in function blocks. Three servo systems, shown in Figure 8, are used for the experimental tests with the following characteristics. For the retrofitted CNC lathe the y-axis was selected to perform the tests. In this axis a servomotor 705-0-101 from EZG with a 4000 counts/rev incremental encoder is used. The servo motor is driven by a servoamplifier 413 from Copley Control Corp. In the CNC milling machine the z-axis was selected to perform the tests. The axis machine uses a brushless servo motor 3485-ME8137 from MCG with a 4000 counts/rev incremental encoder. The driver of the motor is a brushless PWM servo amplifier from Advanced Motion Controls. Finally, for the 6-DOF industrial PUMA robot the first robot link is selected to perform the tests. A standard servo motor with a 4000 counts/rev is used. The driver is a servoamplifier 403 from Copley Control Corp. The controller used in all the servo systems is a proprietary controller board based on a high-performance FPGA technology. This controller includes the PID control law and the necessary complementary modules; for specific details about the controller board see [43]. The plant models are obtained by using recollected data from the encoders of each servo system using the LSM approach, and for each real plant model the GPM-AMGA optimization, the RGA optimization and the classical GPM tuning method for the PID controller are applied. For the three tuning methods, ten motion tests are conducted. Each motion test consists of six references sent to the controller as step functions in order to determine the system dynamics. The motion commands for the CNC machines are 1000, 4000, 10000, 4000, 1000 and 0 counts for a total tracking of 20000 counts. In the case of the industrial robot the motion references are 100, 400, 1000, 400 100 and 0 counts for a total tracking of 2000 counts. The following subsection presents the results considering the high repeatability obtained.

Figure 8.

Servo systems used, a) CNC Lathe, b) CNC milling machine, and c) Industrial PUMA robot.

4.2 Study cases

The first case corresponds to the CNC lathe, for this system the identified plan model is specified by equation (29). The second case corresponds to the CNC milling machine and its corresponding plant model is specified by equation (30). Finally, the third case corresponds to the industrial PUMA robot; the plant model identified for this system is specified by equation (31).

G (z) = \frac{2.091e-007 z^{2} + 3.293e-007 z + 4.06e-005}{z^{2} - 1.78 z + 0.78}

(29)

G (z) = \frac{2.98e-005 z^{2} + 0.000306 z - 0.000233}{z^{2} - 2 z + 0.999}

(30)

G (z) = \frac{5.95e-005 z^{2} - 0.000104 z + 6.98e-005}{z^{2} - 1.96 z + 0.956}

(31)

In Table 4, only the optimized parameters obtained from the GPM-AMGA optimization and the proposed parameters for the classical GPM tuning are presented. It must be clarified that the RGA optimization directly modifies the PID controller gains and for this reason there are no RGA parameter in Table 4. Meanwhile, the optimized parameters finally obtained by the two optimizations techniques and the parameters obtained by the classical tuning are presented in Table 5.

Table 4.

AMGA optimized parameters and user proposed parameters for each servo system.

	GPM-AMGA Optimized parameters	GPM Proposed parameters
Study case: CNC Lathe
τ	0.4011	0.2
θ_m	40.0073	45
Study case: CNC Milling Machine
τ	0.2310	0.2
θ_m	46.3750	45
Study case: Industrial PUMA Robot
τ	0.3538	0.25
θ_m	32.5024	45

The performances obtained from the experimental tests for the three servo systems are shown in Table 6. The performance criteria are the OS, the ISE and the MSE. In order to demonstrate the reduction in the computing time (increasing the searching speed), the time elapsed for the PID parameter optimization process of the RGA and the GPM-AMGA optimization techniques is presented in Table 7. The system response using the two optimization techniques and the classical GPM tuning is shown in Figure 9, for the CNC lathe, the CNC milling machine and the industrial PUMA robot, respectively. Finally, in Figures 10, 11 and 12 the respective input, the error signal in counts, the output and the voltage command signal, at the PID controller are shown.

4.3 Results and discussion

The results obtained from the experimental tests show the GPM-AMGA optimization technique proposed provides excellent results when compared with the RGA optimization technique and with the classical GPM method for tuning applications. The computational time and resources for the proposed optimization technique are significantly reduced compared with the RGA optimization. This affirmation is based on the characteristics from each optimization technique shown in Tables 2 and 3. For the RGA optimization technique the minimum value for the population size is 20 and the minimum amount of generations for the optimization process is 100 in order to reach nominal convergence. Even when the RGA algorithm is based on a floating-point representation of the individuals and using a crossover and mutation operations with floating-point values, such characteristics being appropriate for software applications, the computing time and the resources are much higher than those used in the AMGA algorithm. On the other hand, the AMGA optimization algorithm requires a population size of four individuals, which means that only four evaluations are performed in order to obtain the fitness value of the population as opposed to the RGA where 20 evaluations are performed for the same goal. Besides, the minimum amount of generations needed in order to reach nominal convergence for the optimization process of the AMGA algorithm is 30. Therefore, the computational effort required for the AMGA algorithm is less in comparison with the RGA algorithm and the results shown in Table 7 confirm this affirmation. From Table 7, it can be observed that the time required for the AMGA algorithm is almost an order of magnitude less than the time required for a RGA algorithm in the PID parameters optimization. Meanwhile, in the classical GPM method the controller tuning task can take from several minutes to hours to find favourable controller parameters. For instance, the GPM parameters for the classical tuning were adjusted by hand several times in order to present a just comparison against the two optimization techniques. The GPM method was only implemented on software requiring only the proposed parameters as input. The optimized parameters of the AMGA algorithm and the proposed parameters for the classical GPM are shown in Table 4. In this table there is no RGA parameter because the RGA optimization algorithm modifies the PID controller gains directly. Based on the parameters of each tuning technique for the PID controller, shown in Table 5, the graphics of the system response of the CNC lathe, the CNC milling machine and the industrial PUMA robot are shown in Figures 9a, b, and c, respectively. In Figures 9d, e and f the zooms are shown for the larger motion reference of each servo system. The RGA algorithm does not consider the response time and the phase margin desired and the effects of this are observed in Figures 9d, e and f. Although the system response for the RGA optimization is improved, in general, large overshoots are seen. The GPM-AMGA optimization is based on the frequency response analysis considering as primordial parameters for the system design the response speed and system stability. For this reason, the effects in the system response are observed as minimal overshoot values and quick response at the same time. It must be remarked that although many tuning methods claim to produce overshoot values within a specified range, in practical cases this is not totally true, such as the case of the ZN method. On the other hand, in Figures 10, 11 and 12 the input and output signals at the PID are observed controller only for the GPM-AMGA and RGA optimization techniques for the purposes of comparison. As can be observed the signal error for all the cases is very similar, a deep analysis shows that the error signal is lower for the optimization technique proposed. However, the output signal at the PID controller in the proposed methodology has lower oscillations than the RGA method, requiring less effort for the control action. Finally, in Table 6, the performance of the optimization techniques and the classical tuning method are summarized. From this table it can be observed that the GPM-AMGA optimization proposal produces the minimal overshoot value for all the case studies. Also, for the CNC lathe and the industrial robot the proposed optimization approach presents better results in MSE and ISE indexes than the RGA optimization. Only in the case of the CNC milling machine are the results obtained from the GPM-AMGA optimization very similar to the RGA optimization results. However, considering that the requirements of the positioning control systems in the manufacturing industry are the quality increase of the products, which means overshoot reduction and quick response, the GPM-AMGA technique can provide the expected solution.

Table 5.
PID controller parameters obtained.

GPM-AMGA optimization RGA optimization Classical GPM tuning

Study case: CNC Lathe

Kp 29.6787 97.0836 40.8838

Ki 40.6772 466.4310 112.3706

Kd 0.9632 4.2785 0.5439

Study case: CNC Milling Machine

Kp 14.2863 49.5412 19.0010

Ki 33.9929 19.2312 58.0277

Kd 0.5365 0.9600 0.5052

Study case: Industrial PUMA Robot

Kp 152.7212 99.5051 148.4444

Ki 237.2675 315.9438 326.4040

Kd 4.6012 4.9393 4.5883

	GPM-AMGA optimization	RGA optimization	Classical GPM tuning
Study case: CNC Lathe
Kp	29.6787	97.0836	40.8838
Ki	40.6772	466.4310	112.3706
Kd	0.9632	4.2785	0.5439
Study case: CNC Milling Machine
Kp	14.2863	49.5412	19.0010
Ki	33.9929	19.2312	58.0277
Kd	0.5365	0.9600	0.5052
Study case: Industrial PUMA Robot
Kp	152.7212	99.5051	148.4444
Ki	237.2675	315.9438	326.4040
Kd	4.6012	4.9393	4.5883

Table 6.

Percentage average of the ten motion tests for each servo system of the OS, the ISE and the MSE.

Tuning method	OS (%)	ISE Counts	MSE (%)
CNC Lathe
GPM-AMGA	6.9	3.7470 ×10⁹	0.6660
RGA	19.28	4.1936 ×10⁹	0.7452
GPM	27.3	4.9102×10⁹	0.8729
CNC Milling Machine
GPM-AMGA	5.65	3.7436×10⁹	0.6649
RGA	29.26	3.6888 ×10⁹	0.6553
GPM	13.0	4.3736e ×10⁹	0.769
Industrial PUMA Robot
GPM-AMGA	17	4.5668×10⁷	0.0081
RGA	18.16	4.7432×10⁷	0.0084
GPM	37.16	5.0649×10⁷	0.0090

Table 7.

Computing time required for the PID parameters optimization.

Optimization technique	Computing time (seconds)
Study case: CNC Lathe
GPM-AMGA	3.138513
RGA	20.693711
Study case: CNC Milling Machine
GPM-AMGA	2.972523
RGA	20.259184
Study case: Industrial PUMA Robot
GPM-AMGA	1.722917
RGA	21.467128

Figure 9.

System response obtained, a) CNC lathe, b) CNC milling machine, c) industrial PUMA robot, d) zoom of the CNC lathe response, e) zoom on the CNC milling machine response, and f) zoom on the industrial PUMA robot response.

Figure 10.

PID controller input and output for GPM-AMGA and RGA optimizations applied to the CNC Lathe.

Figure 11.

PID controller input and output for GPM-AMGA and RGA optimizations applied to the CNC Milling Machine.

Figure 12.

PID controller input and output for GPM-AMGA and RGA optimizations applied to the industrial PUMA robot.

5. Conclusions

A new methodology for PID control tuning is presented in this paper which combines the GPM tuning method, which is based on the FR analysis, and the AMGA optimization technique, applying the micro-population concept and the adaptive mutation probability. The use of the GPM method ensures some characteristics of the system response, such as speed response and system stability. The final outputs of the proposed methodology are the optimized PID controller parameters. These parameters are used in real positioning control systems, considerably reducing the overshoot value without compromising the speed response as opposed to the RGA optimization in which the desired overshoot is not considered, even when the error and response time is reduced. The GPM-AMGA optimization significantly reduces the computational resources and the computing effort when compared with the RGA, demonstrating the effectiveness of the proposed approach. From the experimental results it can be observed that even when the optimized parameters are not guaranteed to be the global optimal solutions, they are at least local optimal solutions presenting an excellent performance for the positioning systems studied, improving the system behaviour satisfactorily. It must be emphasized that the classical GPM method for control tuning makes it harder and a time consuming task, taking several minutes in the best cases and up to hours in the worst cases. However, the proposed optimization methodology significantly reduces the time required for the controller tuning. Besides, this methodology is flexible in the sense that it can be applied to different positioning control systems. An interesting point is what happens if the parameters of the real system change. If these parameters change drastically after the offline tuning optimization the system could behave unexpectedly. However, in real systems the commercial controllers normally include tuning software which is applied at the beginning of large operation periods and then if system efficiency decreases because the system parameters change a re-tuning is made. The same situation applies for the optimization approach. Under normal operation conditions the parameters of the real system would change for many reasons (equipment changes, wearing, aging, etc); when these changes are detected a new identification and a tuning optimization process must be performed.

Footnotes

6. Acknowledgments

This project was partially supported by CONACyT scholarship 331141, FOFIUAQ-2012, and SEP-PROMEP F-PROMEP-74/REV-04 projects.

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