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A robust computational technique for model order reduction (MOR) of multi-time-scale discrete systems (single input single output (SISO) and multi-input multioutput (MIMO)) is presented in this paper. This work is motivated by the singular perturbation of multi-time-scale systems where some specific dynamics may not have significant influence on the overall system behavior. The new approach is proposed using genetic algorithms (GA) with the advantage of obtaining a reduced order model, maintaining the exact dominant dynamics in the reduced order, and minimizing the steady state error. The reduction process is performed by obtaining an upper triangular transformed matrix of the system state matrix defined in state space representation along with the elements of

Model order reduction (MOR) of multi-time-scale systems has been an important subject area in control engineering for many years [

To obtain a model of lower order, a significant number of methods have been proposed in recent and earlier years, some for continuous time systems [

A numerous number of MOR methods are available for continuous systems, but very few have been devoted to the discrete-time systems MOR. The discrete-time system MOR may be performed in two different ways. The first one is performed based on transforming a continuous time model into another form using different types of transformation as seen in [

GA-based MOR, on the other hand, has received some of the researchers’ attention as well. Recently, Ponda et al. [

The work in this paper is organized as follows: Section

In this paper, MOR is investigated for discrete LTI systems of both SISO and MIMO type models. For SISO systems, a transfer function model is used, while the state space representation is used for MIMO systems.

For the SISO systems, consider the discrete-time system described by

GAs are based on principles inspired from the genetic and evolution mechanisms observed in natural systems. Their basic principle is the maintenance of a population of solutions to the problem that evolves in time. They are based on the triangle of genetic reproduction, evaluation, and selection [

In this paper, using the computational intelligence of GA, we will obtain the reduced order model based on only the dominant dynamics of the system. For dynamic decoupling, the GA will set the reduced order model state matrix

The GA will determine the parameters of the

It is to be noted that all of the parameters that the GA will have to find are restricted to be real values. Now, based on the number of unknown parameters (

The genetic algorithm used in this work will operate as follows.

An initial population comprising

The GA starts by randomly initializing a binary matrix with

A set of individuals in a GA chromosome.

The initial population matrix consists of random numbers within the lower and the upper bounds of the parameters. The population will be split into rows (chromosomes), each constituting one solution set. These solution sets are each taken to have their fitness evaluated, as seen next.

The fitness, a nonnegative measure of quality, is used as a measure to reflect the degree of goodness of the individual and is calculated for each individual in the population. This measure of quality is calculated based on minimizing the following related cost function:

In the selection process, chromosomes are chosen from the current population to enter a mating pool devoted to the creation of new children (offspring) for the next generation. The chance of a given individual to be selected to mate is proportional to its relative fitness That is, the larger the fitness value of a chromosome, the higher the probability of the chromosome to contribute one or more children in the next generation. First, the population fitness and associated chromosomes are ranked from highest fitness to lowest fitness. Then, only the best are selected to continue, while the rest are left. The selection rate (

There are several methods for choosing the chromosome pairs to be mated from the mating pool. In the proposed algorithm, random pairing was chosen, which uses a uniform random number generator to select chromosomes that enter the mating pool from kept chromosomes. The algorithm randomly chooses chromosomes from the mating pool for mating, while making sure that all pairs are unique. After the chromosome pairs are chosen, the next operation will be crossover.

Crossover provides the means by which valuable information is shared among the population. In the crossover operation, a pair of chromosomes is mated to produce two offspring in the process that inherit genes from their parents. To perform crossover, the chromosomes need to be in their gene format (i.e., binary representation). The two parent chromosomes are crossed over at random points to segment each chromosome into a number of parts. These parts of each chromosome pair are swapped between each other to produce two new chromosomes, the offspring. This is shown in Figure

Chromosomes random crossover operation.

Crossover is not applied to all pairs of chromosomes selected for mating. For every parent chromosome pair, a crossover rate decides whether or not crossover occurs. If crossover is not applied, children are produced simply by duplicating the parents. This gives each chromosome a chance of passing on its genes without the disruption of crossover. In the proposed algorithm, the crossover rate is set to 0.80, meaning that 80% of parent pairs produce offspring. The number of crossover points for the parent chromosomes is chosen as a random integer for each generation. As a crossover result, a new children population will be performed with the same size as the parent population.

Mutation is often introduced to guard against premature convergence. Generally, over a period of several generations, the gene pool tends to become more and more homogeneous. Therefore, further mutation is introduced to the offspring to guarantee that the offspring will have new qualities while retaining a similar overall structure. Mutation is applied to each child individually after crossover, where it randomly flips any bit (gene) with a small mutation probability (between 0.1 and 0.001). Mutation provides a small amount of random search to guard against premature convergence. In the proposed methodology, the mutation rate is set to 0.0125.

Replacement operation takes place once the crossover of all parent chromosome pairs is performed. The parent population is totally or partially replaced by the children population depending on the replacement scheme used. This completes the life cycle of the population. At this stage, the population is ready to enter the next generation and undergo a new round genetic operation. The decision in which offspring replaces which population individual is made based on the fitness evaluation. Fitness is evaluated for all the resulting offspring as well as for all of the original population individuals and a replacement factor decides how many of the offspring with the highest fitness values are to replace the main population individuals. In the proposed approach, a replacement factor of 0.70 is used. This process continues until reaching the end of the population size or best desired fitness.

The GA is terminated when some convergence criterion is met. The termination condition could be considered as specified fitness value, reaching maximum number of generations, or a set progress limit. As one generation has gone by, depending on the genetic algorithm’s termination condition, the algorithm could stop any time and identifies the chromosome with the highest fitness value as the optimal solution set. On the other hand, it may repeat the entire process from the selection procedure to continue another generation of the genetic algorithm. The termination condition for the proposed algorithm is reaching a fitness value of 99.999% and an average fitness value of at least 99.9% in the entire population set. When the algorithm terminates, the highest fitness chromosome is distributed across the

In this section, we consider a few discrete system examples, which have been investigated by different researchers in recent years. The examples are given as SISO and MIMO linear time invariant systems. Results of investigations are compared with the existing ones to show the potential of the new approach.

As a first example, we consider the following 7th order SISO system investigated by Telescu et al. [

Impulse responses for the full and reduced order models.

Our fourth advantage is clearly seen when simulating the full and reduced models to a step input. As seen in Figure

Step responses for the full and reduced order models.

In this example, we consider the following 8th order SISO system investigated by Yadav et al. [

Model order reduction method comparison.

System | Dominant dynamics |
---|---|

Original 8th order | 0.8797 ± 0.2442 |

0.0773 ± 0.1078 | |

Proposed 2nd order | 0.8797 ± 0.2442 |

Artificial bee colony [ |
0.8778 ± 0.2432 |

Differential evolution [ |
0.8768 ± 0.2484 |

Genetic algorithm [ |
0.8854 ± 0.2398 |

As a second comparison, the full and reduced order models were simulated for a step input with results as seen in Figure

Step responses for the 8th and reduced 2nd order models.

Error comparison of the reduced order models with the full order.

In this example, we consider the 5th order MIMO discrete system investigated by Li [

Full and reduced order model’s responses for mixed input signals.

Error of the reduced order models for the system first output response.

Error of the reduced order models for the system second output response.

In Figures

As seen in the three previous examples, the advantages of the new approach are present in all three of them. That is, the reduction process provides new reduced order models, maintains the exact dominant dynamics of the original model in the reduced order model, and provides reduced order models with the least response errors as compared with other methods. In addition to that, an advantage that the new approach is applicable to SISO and MIMO discrete type systems.

In this paper, a robust computational intelligence approach is proposed using GA for MOR of discrete MIMO type systems with the advantage of dominant dynamic preservation. The reduction process is performed based on transforming the system state matrix (in a state space model) while decoupling the multi-time-scale dynamics. The dominant dynamic preservation is performed by retaining the dominant poles of the original system as a subset in the reduced order model utilizing the transformed system state matrix. Once the reduced order model state matrix is designed, the GA MOR new technique will search for the

The authors declare that there is no conflict of interests regarding the publication of this paper.