Controllers of the form given by either Eqs. Considering the proportional, integral and derivative error, respectively in Eqs. In consequence, Eq. A compact form of Eq. Where C r s , is the set-point controller transfer function and C y s is the feedback controller transfer function. Where M yr s is the transfer function from the set point to the controlled output servo-control closed-loop , and M yd s is the transfer function from load-disturbance to the controller output regulatory control closed-loop.

The usual criterion for tuning a controller is directly related to the desired closed-loop system response. Integral performance indexes allow quantifying the closed-loop system performance due to a unit step load disturbance. Most common employed indexes are integral of absolute error IAE see Eq. For the PID case, e t and e u t , Eqs.

The load disturbance may enter at many different places, and extreme cases occur when it enters at the process input or output. However, when a feedback error appears, integral performance indexes evaluate the controller performance indistinctly, whereas load disturbance appears [ 14 ]. The abovementioned fact is the main motivation to adopt integral performance indexes.

IAE and IAU performance indexes are initially adopted in this work because of its interpretability from the PID controller parameters, but other criteria could also been used. The controller tuning process that minimizes an integral performance index can be seen as an optimization problem where the ultimate goal is to find a controller parameters combination such that the value of an Integral Performance Index is minimized.

Subject to a process model see Eqs. Similarly, for the 2DoF PID controller, the optimal problem consists of minimizing the objective function given by Eq. From the observation of living beings, we can see that these reproduce, adapt, and evolve in relation to the environment where they develop. Some of the characteristics acquired during life may be inheritable by the next generation.

The synthetic theory of evolution has been able to explain these processes and biological variations in detail [ 15 ].

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This theory bases on genes as units of inheritance transfer, that is, functional units of basic information for the development of an organism. The genetic material of an individual is in its genotype. The genotype consists of an organization of hierarchical structures of genes. The complex information contained in the genotype is expressed in the phenotype, that is, the visible characteristics and functionality of individuals.

In the evolutionary process, the occurrence of small variations in the phenotypes, apparently random and without a clear purpose, is recognized. Such variations which are usually called mutations prove their efficacy in the light of the environment and prevail through the selection of the individual, or otherwise they disappear. The natural needing to produce offspring motivates the selection of individuals. Because of a severe competition for reproduction, which only the fittest individuals achieve, it is assumed that the offspring overcome their parents by inheriting their mixed characteristics.

When resources in the environment become insufficient, only the fittest individuals will have a better chance of survival and reproduce. The selective pressure on individuals of a species makes them continually improve with respect to its environment. Evolutionary algorithms EA emulate the synthetic theory of evolution. As natural evolution, an EA begins with an initial set of potential solutions to a specific problem. This set can be composed at random in a delimited searching space or using information of the problem. EA include operators that select and create new individuals.

EA encompass different approaches that transfer the behavior of adaptation and evolution of species, giving rise to several methods. Among the most popular approaches are genetic algorithms, genetic programming, evolutionary strategies, and evolutionary programming. Nowadays, EA are not only based in the biological evolution, rather EA are identified as algorithms that search iteratively for a solution through a population in evolution.

Some of the main reasons for new optimization heuristics are the need to identify the interrelationships between the variables used to represent individuals according to the coding applied and the need to reduce the own parameters of the classical EA. New EA use operators different to the genetic ones. Some of those algorithms are differential evolution, estimation of distribution algorithms, and the multidynamics algorithm for global optimization.

Differential evolution at first glance is not based on any natural process. The proportional difference of two randomly chosen individuals from the population is added to a third individual, also randomly chosen. From this differential mutation, a fourth individual appears. This individual is compared against its parent, the third one. The best of them is selected to the next generation.

The process is repeated until a stop criterion [ 16 ].

## Digital Control Past, Present and Future of PID Control - 1st Edition

Estimation of distribution algorithms also bases their search on populations that evolve. The new population is recreated in each generation from the probability distribution obtained from the best individuals of the previous generation. The interrelationships between the variables are expressed explicitly through that distribution. There are no crossing or mutation operators.

The process is repeated until a stop criterion [ 17 ]. In MAGO, a differential crossover is applied between the target individual and its mutant coming from a numerical derivation. A tournament chooses the best of them. The interrelationships among the variables are explicit through a distribution of the population in each generation. The new population is sampled from a set composed of the best individuals until now, the historical trend and other individuals completely new.

Next, this algorithm is explained in detail.

EA emulates the mechanisms of natural selection and genetic inheritance inspiring from the Neo-Darwinian theory of biological evolution. EA have evolved themselves to treat with artificial evolution processes. MAGO is a good example of this evolution.

MAGO does not work with genetic operators [ 18 ]. MAGO starts with a random initial population on a search space bounded by the problem. To guarantee diversity and increase the exploitation of the search space, MAGO creates new individuals by means of three different subgroups of the population simultaneously. Each group has its own dynamics: a normal distribution over the searching space, a conservation mechanism of the best individual, and a strategy for maintaining diversity [ 19 ].

Because introducing statistics operators, MAGO provides a strong way to demonstrate the evolution. The mutation based on numerical derivation, generalizes the searching space where MAGO can acts. MAGO takes advantage of the concept of control limits [ 20 ] to produce individuals on each generation simultaneously from the three different subgroups. The size of the population is fixed, but the cardinality of each subgroup changes in each generation according to the first, second and third deviation of the actual population, respectively.

The exploration is performed creating new individuals from these three subpopulations, individuals that are governed by any of their dynamics, the exploration is performed. For the exploitation, MAGO, looking for the goal, uses a greedy criterion in the first subset. MAGO is evolutionary in the sense that works with a population of possible solutions randomly distributed throughout the searching space approaching iteratively to the final solution. MAGO is autonomous in the sense that it regulates its own behavior and does not need human intervention.

In each generation, MAGO divides the population into three subgroups. To know how many individuals will belong to each subgroup, the actual entire population is observed as having a normal distribution. The average location, the first, second and third dispersion of the whole population are calculated to form the three groups.

To each subgroup is assigned many individuals as the cardinality of each different level of standard deviation. Each group has its own evolution. The cardinality of these subgroups changes autonomously in each generation. The subgroup named Emerging Dynamics G 1 creates a subpopulation of individuals around the individual with better characteristics; this group is the evolutionary elite of each generation, that is, the fittest individuals contributing with their genes to the next generation. The Crowd Dynamics G 2 creates a group of individuals but around the current population mean, configuring the historical trend.

This dynamic is applied to the largest portion of the population, and it is always close to the emerging dynamics, but never close enough to be confounded. These two dynamics could be merged within a same territory only until there are sufficient and necessary conditions to ensure a full exploration of the searching space, usually at the end of the evolutionary process.

The Accidental Dynamics G 3 is a small group created by quantum speciation. It is established in isolation from individuals of the other two dynamics generation after generation. This portion of the population is always formed spontaneously and contains entirely new individuals. MAGO uses the covariance matrix of the population of each generation to establish a distribution of exploration.

With the Accidental Dynamics, the main diagonal of the covariance matrix is different from zero, ensuring numerical stability of the evolutionary process. Because in each generation, the population is treated for its division by a normal distribution for its division according to the first, second, and third deviation, the subgroups G 1 , G 2 , and G 3 not interbreed.

Emerging Dynamics: This subset is created with the N1 fittest individuals in each generation. The N1 fittest individuals within the first standard deviation of the average location of the current population of individuals move in a line toward the best one of the entire population, in a kind of mutation that incorporate information from the best of all. The mutation and selection of individuals who have obtained the best values in their objective function is based on the simplex search method of numerical derivation [ 21 ].

MAGO uses only two individuals for this mutation, the best one and the trial one. If this movement generates a better individual, this one passes to the next generation; otherwise, its predecessor passes on with no changes. This method does not require gradient information for the derivation. The fittest individuals are ordered from the best one. Test individuals are created bringing them closer to the best one, following the rule in Eq. F j is a matrix that includes information about the covariance of the problem variables, Eq. The covariance matrix of the current population considers the effect of the evolution, and Eq.

Crowd Dynamics : The number of individuals of this subgroup corresponds to the cardinality of the second deviation of the normal distribution of the actual population. This subgroup has the role of exploring the searching space in a neighborhood close to the population mean. If the population mean and dispersion matrix for generation j are x M j and S j , then the Crowd Dynamics individuals are created from a uniform distribution on the hyper-rectangle L B j U B j , see Eqs.

The diagonal of the population dispersion matrix of the generation j , described by Eq. Initially, the neighborhood around the mean may be large, but as evolution proceeds, this neighborhood is reduced, and the population mean is getting closer to the optimal but following on another path. Accidental Dynamics : This group is a smaller one in relation to its impacts on the population. N3 new individuals are created from a uniform distribution over the whole search space, as in the initial population. The two dynamics mentioned above concentrate the population around their local optima.

To maintain diversity, MAGO introduces new individuals in each generation with the accidental dynamic, sampling a uniform distribution throughout the search space. This dynamic also ensures the numerical stability of the covariance dispersion matrix. The accidental dynamics always guarantees the diversity and dispersion of the population, even if the other two groups already have converged.

Following, the pseudo code of MAGO is presented. For control tables, if the process is outside the control limits, then it is assumed that the process is out of order. Consider the population dispersion matrix of generation j , S j and its diagonal diag S j. If Po b j is the set of possible solutions in generation j , then three groups can be defined as in Eqs 30 , 31 , and This way of defining the elements of each group is dynamical in nature and autonomous in MAGO.

Cardinalities depend on the dispersion of the whole population in generation j. A feedback controller is a device that automatically manipulates a predetermined variable to ensure the balance of the system around an operating point. It compares the actual value of the controlled variable to its desired value feedback obtaining an error signal to calculate the control action so that it maintains or returns the system to the point of operation [ 9 ]. The output, u t , control action is a composite of three effects, K p , the proportional action, T i , the integral time and, T d , the derivative time, which are calculated based on the error.

MAGO is a real-valued evolutionary algorithm, very efficient and effective instrument to solve problems in continuous domain. It has been chosen as a tool for estimating the parameters of a controller that minimizes an integral performance index. The representation of the evolutionary individual is a vector containing the controller parameters, as positive values in a continuous domain. See Table 1. The error is calculated for each point of time, t k , throughout the measurement horizon as the difference between the system output and the reference signal.

Each tuning rule for PID controllers has restrictions on the behavior of the plant, expressed in the range of validity, and has only been applied to a certain group of processes. Most methods for optimal tuning of SOSPD systems require, from experiments carried out directly on the plant, additional critical system information. Readers are referred to [ 25 ] for a good compilation of the PID tuning rules and [ 26 ] for a complete analysis of the different tuning rules characteristics and features. It is not always possible to perform experiments such as reaction curves and closed-loop tests because the extreme stress and oscillations may create instability and damage to the system.

This scenario shows that a general rule for tuning PID controllers must be sought.

## The Open Automation and Control Systems Journal

A tuning method that best satisfies the operation requirements of each problem and ensures optimal values for the controller parameters according to the 2DoF PID chosen criterion. This situation could be reduced to an optimization problem consisting of minimizing an objective function. A suitable combination of the three parameters required by the PID controller will be the result of minimizing a performance criterion.

This is the approach taken in this chapter. MAGO has shown a great capacity to optimize nonlinear dynamical problems in the continuous domain. Next section is concerned to optimal 2DoF PID controllers satisfying an integral performance index and not requiring additional system information coming from experiments on the plant. In this chapter, it is not included a convergence analysis of the MAGO; however, its convergence has been previously demonstrated in [ 18 , 19 ].

A 2DoF PID controller attempts simultaneously achieve good closed-loop servo-regulatory dynamical responses. In [ 22 ], a set of system models as a benchmark suitable for testing PID controllers was proposed. This set of system models presents different challenges of control because PID controllers are not well suited for all of them. From the best of our knowledge, none of the PID controller tuning rules applies to obtain suitable values for its parameters for all the systems in that benchmark.

In this chapter, nine different system models are taken from the benchmark and 2DoF PID controllers are designed, one for each system model. The 2DoF PID controller parameters obtained here are compared with parameters reported in [ 22 ] for the same system models. Then, MAGO is employed to solve the optimization problem. Consider the system models given by Eqs. In [ 22 ], the control problem was solved as an optimization process where the objective function was a combination of the IAE and IAU performance indexes.

The main features of the procedure used in to solve the control problem were: 1 not only IAE but also IAU are included in the objective function establishing a kind of trade-off between the system performance and robustness due to the restriction imposed in the controller action effort through the IAU index.

Traditionally, the tuning process for a 2DoF PID controller is carried out in two stages as follows: firstly, values of K p , T i , and T d are found such that closed-loop system achieves some dynamical behavior. These two stages tuning process imply that the closed-loop system responses are not optimal for any of the system operation modes. Fminsearch function was used to find a set of initial conditions for the controller parameters. Next, fmincon function was used trying to find the overall optimal controller parameters.

The authors highlight that with this optimization strategy exists the possibility that the problem solution is not the global optimum of the objective function although the closed-loop systems performance was satisfactory in all cases. In this chapter, a similar procedure for solving the optimization is adopted to facilitate the comparison of results. The objective function to be optimized is given by Eq. Access Restriction Subscribed.

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