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ABSTRACT:-A New Model Order Reduction Technique For The Reduction Of The Linear Time Invariant System Has Been Proposed In This Paper. An Improved Intelligence Search Evolutionary Algorithm (ISEA) Is Employed In The Proposed Method To Tuning The Reduced Order Model. In the proposed algorithm, a two stage initialization process is used and No need of mutation operation, faster convergence rate. Also, it gives optimal solution with less number of generations due to reduce the computation time. The feasibility of the proposed algorithm is demonstrated for a Himmelblau function and model order reduction. The obtained results using proposed method are compared with the existing other model order reduction techniques. The Improved ISEA Algorithm Guaranteed The Stability In The Reduced Order Model And Also Preserves The Characteristics Of The Original System In The Approximated One.
.Introduction
The system is large in scale when its dimensions are so large that conventional techniques of modeling, analysis, control, design and computations fail to give reasonable solutions with reasonable computational efforts.The analysis of all physical systems starts by building up of a mathematical model.
In most practical situations a comprehensive description of a physical system is expressed in terms of high order differential equation which leads to a high order transfer function model in frequency domain or a large dimensional state space model in time domain. The analysis of all physical systems starts by building up of a mathematical model. In most practical situations a comprehensive description of a physical system is expressed in terms of high order differential equation which leads to a high order transfer function model in frequency domain or a large dimensional state space model in time domain. The analysis and design of such high order system is tedious and requires more hardware and computational effort. It is not cost effective and poses great challenge for on line implementation. Therefore it is desirable that a high order system be replaced by low order one that approximates the high order system in some sense and reflects the key qualitative dominant properties of the system under consideration and helps in understanding the behaviour of the system at reduced cost. Need of model reduction is To Have A Better Understanding Of The System, Reduce Computational Complexity ,Reduce Hardware Complexity To Improve The Methodology Of Computer Aided Control System Design. Most Of The Model Reduction Methods Available In The Literature Start Either With A Transfer Function Description And Give The Reduced Order Model In The Transfer Function Form, Or They Start With A State Space Model And Give The Reduced Order Models In The Same Form.
Some Of The Model Order Reduction Techniques are Modal analysis approach, Subspace projection methods, Optimal order reduction Balanced Realization Method, Aggregation Method, Henkel Matrix Approach ,Henkel-Norm model reduction, Continued Fraction Expansion and Truncation, Pade Approximation Techniques, Moment
Matching Method, Matching Frequency Response, Routh Stability Criterion Method, Stability Equation Method, Mihailov Stability Criterion Method, Error Minimization Method, Dominant Pole Retention, Polynomial Differentiation Method, Polynomial Truncation Method, and Factor Division Method. But those methods exhibits some demerits
Recently many evolutionary algorithms such as Genetic algorithm(GA), chaotic ant swarm(cas) ,artificial be colony(abc),chaotic optimization algorithm, and partical swarm optimization , differential evolution(de)algorithm[7] are used to tuning. In proposed method intelligence search evolutionary algorithm (ISEA) is used to tune the reduced order model.
In this paper after the introduction of model order reduction, review of problem statement is in section II. Intelligence search evolution algorithm is in section III. Example of model order reduction in section IV. Example himmelblau function with ISEA is in section V. overall Conclusion in section VI.
Proposed method [6]
In this proposed method, intelligent search evolution algorithm explained below for our model order reduction.
The intelligent search evolution algorithm calculation is a population based evolution technique for global enhancement. It has the capacity to take care of a wide class of issues. The proposed technique tries to approach the objective in an ideal way to find the ideal answer for any mathematical optimization issue. The key elements of the proposed strategy when contrasted with other evolutionary strategies are:
Two stage initialization process, No need of mutation operation, faster convergence rate. Reduction in computation timeless memory for population requires less parameter tuning. The starting population is randomly created with the control parameter limits in two stages. At that point, the evolutionary operators like crossover or recombination and choice are performed to all people until a stopping criterion is come to. The major phases of the proposed strategy are quickly depicted as takes after.
Step1. Single stage installation
The population is generated by using the following equation shown below.
(5)
Where i 1, 2... ps ; j 1, 2,.., ncv.
ps = population size.
ncv = number of control variables.
& are the lower and upper bounds of jth control variable.
rand ( 0,1 ) is a uniformly distributed random number Between 0 and 1.
Initialize the maximum number of iteration.
Step2. Evaluating fitness function
Take the performance measure integral square error (ISE) and in reduced order model transfer function we take the condition as the ratio of numerator coefficient and denominator coefficient of original plant is equal to the ratio of numerator coefficient and denominator coefficient of reduced order model
i.e.
Where =coefficient of numerator of original plant
=coefficient of denominator of original plant
= coefficient of numerator of reduced order plant
=coefficient of denominator of reduced order plant
Step 3. Two stage initialization
The population of a specified size is created for every control variable by utilizing the Eq.11. The two stage introduction process gives better probability of recognizing an ideal answer for the given target function. In the first stage, a population vector of size (spv ncv) is shaped. All the control variables in the population vector must fulfil the equity and imbalance requirements. Assess the estimation of cost capacity for every string in the population vector. Select the best string from the population vector comparing to least cost. Repeat the method for number of population vectors (n). In the second stage, consolidate all the best strings from every population vector to frame multi-dimensional vector [X] of size (n ncv) and this new population is utilized for developmental operations. For clear reference, the two stage introduction procedure is appeared in Fig. 3. The superscript in Fig. 3 speaks to population vector number.
Step4.Recombination [8]
The recombination or crossover operator is a critical operator which is in charge of the structure recombination. There are numerous hybrid plans in the writing, for example, single point, and multipoint and crossover operator. In this study, an effective recombination operator has been utilized so that inquiry along variables is additionally conceivable. In the event that x_i^((j) ) and x_i^((k) )are the estimations of variables x_(i )in two strings j and k. The crossover between these two qualities might create the following new values.
(6)
Here, the parameter λ is a random number between 0 and 1.
Step5.Selection
For the present work, sorting and ranking choice technique is utilized. In this strategy, at every generation, the cost and related strings are ranked from the least cost to the most elevated expense. At that point, the least cost and comparing string is considered as the best cost and best string for the next generation until the stopping criteria is come.
Step6. Stopping criteria
The accompanying are the conditions normally used to stop iterative process:
1. The quantity of generations (cycles) equivalent to the predetermined greatest number of generations.
2. The tolerance acquired during iterative procedure is not exactly determined tolerance.
In the present work, the principal condition i.e., the quantity of generation’s equivalent to the predefined most extreme number of generations is utilized as stopping criteria.
IV. Model order reduction
Consider an example eighth order system in Shamash (1975) [9] with a transfer function of
(7)
By using proposed method the reduced model is obtained as
(8)
The step responses of original model and reduced model transfer function are shown in figure. Table shows the proposed method compare with other existing methods. And also take the condition the ratio of numerator coefficient and denominator coefficient of original plant is equal to the ratio of numerator coefficient and denominator coefficient of reduced order model
i.e., (9)
V. Example Himmelblau Function with ISEA
Consider the Himmelblau function [10] is given by
(10)
The objective is to minimize the function in the interval (0≤x_1,x_2≤6)The true solution to this problem is (3, 2) having function value equal to zero.
Step 1: single stage Initialization
(11)
Where i=1, 2, ----,ps; j=1,2,---,ncv
Ps=10, sqv=2 & ncv=2
3.017 4.257 205.4
2.573 1.828 7.683
1.138 1.161 93.4
4.093 1.817 57.49
PV1= 3.25 0.905 8.807
4.187 2.27 83
5.16 5.122 1026
3.561 2.979 51.31
5.399 4.93 1048
3.87 4.908 518
Step2. Two stage initialization
2.573 1.828 7.683
3.25 0.905 8.807
PV 2= 3.561 2.979 51.31
4.093 1.817 57.49
3.87 4.908 518
Step 3: Recombination
If xi (j) and xi (k) are the values of variables xi in two stage j and k. The crossover between these two values may produce the new value