Design of PID Controller for Linear Dynamic Continuous Systems Using IGLSM and DE Algorithm

Third International Conference on Advances in Control and Optimization of Dynamical Systems March 13-15, 2014. Kanpur, India

Design of PID Controller for Linear Dynamic Continuous Systems Using IGLSM and DE Algorithm J.Nancy Namratha1*

Y.Hema Latha2

1. Assistant professor, Dept. of Electrical Engg., ANITS College of Engg., Visakhapatnam, A.P, India 2. M.E. Student, Dept. of Electrical Engineering, ANITS College of Engg., Visakhapatnam, A.P, India. * E-mail of the corresponding author: [email protected] one, approximate models are obtained by minimizing certain approximation error criteria, such as the 𝐻2 -norm [10]

ABSTRACT— This paper proposes a new computational simple scheme for Model Order Reduction to design a continuous PID controller for higher order linear time invariant continuous systems. The combined advantages of the improved generalized Least squares method (IGLSM) and error minimization by Differential Evolution technique (DE) is employed for both order reduction and controller design. First the denominator of the reduced order model is obtained by improved generalized least squares method and the numerator coefficients are determined by minimizing the integral square error between the transient responses of original and reduced order models using DE technique, pertaining to unit step input. Then a PID controller is designed for reduced order model, based on the minimization of integral square error between the desired response and actual response, pertaining to a unit step input using DE. Finally the designed PID controller is connected to the original higher order continuous system to get the desired specifications. The validity of the proposed method is illustrated through a numerical example. Keywords: Improved Generalized Least Squares Method, Integral square error, Order Reduction, Differential algorithm, PID controller.

Design of controller based on reduced order model is called process reduction technique. During the past decades, the process control techniques in the industry have made great advances. The process approach is computationally simpler as it deals with reduced order models. The computational and implementation difficulties involved in design of optimal and adaptive controller for higher order linear time invariant system can also be minimized with the help of reduced order models. Several methods have been developed for designing of PID controller [8]. Recently Evolutionary algorithms have been suggested to improve the PID tuning, such as those using Genetic Algorithm (GA) [11] and Particle Swarm Optimization algorithm (PSO) [12]. With the advance of computational methods in the recent times, optimization algorithms are proposed to tune the control parameters in order to find the optimal performance. In this paper, the design of a controller for a higher order continuous system is presented employing process reduction approach. The original higher order continuous system is reduced to a lower order model using both improved generalized Least squares method and the error minimization by DE. In this method the reduced denominator is obtained by the improved least squares method and the numerator of the reduced model is determined by minimizing the Integral Square Error between the transient responses of original and reduced model by DE pertaining to a unit step input. The Integral square errors between the original and reduced models are also determined and time responses for unit step input are plotted. The quality of a formulated reduced order model is evaluated by designing a PID controller.

I. INTRODUCTION The mathematical description of dynamic systems is carried out using theoretical considerations. In time domain representation, the modeling procedure leads to a high order state space model and a high order transfer function model in frequency. It is often desirable for analysis, synthesis and controller design for higher order systems and other purposes to represent such models by equivalent lower order state variable or transfer function models.

II DESCRIPTION OF PROBLEM

Numerous methods [1]-[4] have been proposed for solving the model approximation problem and they may be grouped into two major categories, the performance-oriented and the nonperformance-oriented approaches. In using a model approximation method that is not performance-oriented, the original system model is first transformed into a canonical form, e.g., a balanced state-space model or a certain continued-fraction representation. For a performance-oriented

=

𝑛−1 𝑖 𝑖=0 𝑎𝑖 𝑠 𝑛 𝑗 𝑗 =0 𝑏𝑗 𝑠

978-3-902823-60-1 © 2014 IFAC

A. Model order reduction: Consider an 𝑛𝑡ℎ order linear time invariant continuous system represented by: 𝐺𝑛 𝑠 =

𝑁 𝑠 𝑎0 + 𝑎1 𝑠 + …………. +𝑎𝑛−1 𝑠 𝑛−1 = 𝐷 𝑠 𝑏0 + 𝑏1 𝑠 + ………..+𝑏𝑛−1 𝑠 𝑛−1 + 𝑏𝑛 𝑠 𝑛

… (1) 676

10.3182/20140313-3-IN-3024.00201

2014 ACODS March 13-15, 2014. Kanpur, India

The objective is to find an 𝑟 𝑡ℎ order model that has a transfer function 𝑟 < 𝑛 : 𝑅 𝑠 =

Where 𝑘𝑝 =Proportional gain, a tuning parameter, 𝑘𝑖 =Integral gain, a tuning parameter, 𝑘𝑑 =Derivative gain, a tuning parameter, 𝑒 = error between actual output and reference input, 𝑡 = Time or instantaneous time, 𝜏 =Variable of integration; takes on values from time 0 to the present 𝑡.

𝑁𝑟 𝑠 𝑐0 + 𝑐1 𝑠 + ………+𝑐𝑟−1 𝑠 𝑟−1 = 𝐷𝑟 𝑠 𝑑0 + 𝑑1 𝑠 + ………+𝑑𝑟−1 𝑠 𝑟−1 + 𝑠 𝑟 =

𝑟−1 𝑖 𝑖=0 𝑐𝑖 𝑠 𝑟 𝑗 𝑗 =0 𝑑𝑗 𝑠

… 2

Where 𝑎𝑖 0 ≤ 𝑖 ≤ 𝑛 − 1 ; 𝑏𝑗 0 ≤ 𝑗 ≤ 𝑛 & 𝑐𝑖 0 ≤ 𝑖 ≤ 𝑟 − 1 ; 𝑑𝑗 0 ≤ 𝑗 ≤ 𝑟 are scalar constants. The derivation of successful reduced order model coefficients for the original higher order model is done by minimizing the error index „E‟, known as ISE, which is given by [9]:

The transfer function of a PID controller is found by taking the Laplace transform of above eq. 𝐶 𝑠 =

∞

[𝐺𝑛 𝑡 − 𝑅 𝑡 ]2 𝑑𝑡

𝐸=

𝑘𝑑 𝑠 2 + 𝑘𝑝 𝑠 + 𝑘𝑖 𝑈(𝑠) 𝑘𝑖 = 𝑘𝑝 + + 𝑘𝑑 𝑠 = 𝐸(𝑆) 𝑠 𝑠

… (3) IV. DIFFERENTIAL EVOLUTION

0

Where 𝐺𝑛 (𝑡) and 𝑅(𝑡) are the unit step response of original and reduced order systems.

A. Overview Let 𝑓(𝑥) be a function of 𝑛-parameter vector

B. Compensator Design: The proposed method for design of a controller by process reduction technique involves the following steps: Step-1: Determine the lower order model to a given original higher order discrete system by minimizing the integral square error (E).

𝑥= [𝑥 1, 𝑥 2… 𝑥 n]T

Determine the values of the parameter vector within the intervals

𝑥 k ∈ 𝑥𝑘− , 𝑥𝑘+ , 𝑘=1,2,…,𝑛

Differential evolution (DE) is a simple evolutionary algorithm for parameter optimization [6]. The most distinct feature of DE is that it mutates vectors by adding weighted, random vector differentials to them. Usually, the performance of a DE algorithm depends on three variables: the population size𝑁𝑝 , the mutation scaling factor 𝐹 s, and the crossover rate𝐶𝑟 . In applying a DE algorithm to solve the above parameter optimization problem, it starts by generating a population of 𝑁𝑝 real valued 𝑛-dimensional vectors whose initial parameter values being chosen at random from within bounds set by the user. This population undergoes evolution in a form of natural selection. In every generation, each vector in the population becomes a target vector .Each target vector crossovers with a donor vector, which is generated by mutating a randomlyselected population vector with the difference between two randomly-selected population vectors, in order to produce a trial vector. If the cost of the trial vector is less than that of the target, the target is replaced by the trial vector in the next generation

Step-3: Test the designed PID controller for the reduced order model for which the PID controller has been designed. Step-4: Test the designed PID controller for the original higher order model. III. Proportional Integral Derivative (PID) Controller A proportional-integral-derivative controller (PID controller) is a generic control loop feedback mechanism (controller) widely used in industrial control systems. A PID controller calculates an "error" value as the difference between a measured process variable and a desired set point. The controller attempts to minimize the error by adjusting the process control inputs. The PID controller algorithm involves three separate constant parameters: the proportional, the integral and derivative values. „P’ depends on the present error, „I’ on the accumulation of past errors, and „D’ is a prediction of future errors, based on current rate of change. The proportional, integral, and derivative terms are summed to calculate the output of the PID controller. Defining 𝑢(𝑡) as the controller output, the final form of the PID algorithm is: 𝑢 𝑡 = 𝑘𝑝 𝑒 𝑡 + 𝑘𝑖

𝑒 𝜏 𝑑𝜏 + 𝑘𝑑 0

(5)

which minimize the cost function 𝑓(𝑥).

Step-2: Design a PID controller for the reduced order model. The parameters of the PID controller are optimized using the same error minimization technique between the desired and actual response pertaining to a unit step input, Employing DE & IGLSM Algorithm.

𝑡

(4)

B. Population Initialization The DE algorithm is started with generating a population of 𝑁p real-valued 𝑛-dimensional vectors 𝑥𝑗 =[𝑥𝑗 ,1 , 𝑥𝑗 ,2 … 𝑥𝑗 ,𝑛 ]𝑇 , j=1,2,… 𝑁𝑝 (6) whose initial parameter values are chosen at random from within user-defined bounds 𝑥 𝑗 ,𝑘 ∈ [𝑥𝑘 , 𝑥𝑘 ],𝑘=1,2,…,𝑛 (7) It

𝑑 𝑒(𝑡) 𝑑𝑡 677

is

noted

that

the

search

region

or

interval

2014 ACODS March 13-15, 2014. Kanpur, India

[𝑥𝑘 , 𝑥𝑘 ] set by a user for the parameter 𝑥𝑘 may be smaller than, but within, the allowable interval 𝑥𝑘− , 𝑥𝑘+ in the case 𝑥 k-=-∞ of and/or 𝑥 k+=∞. Once the initial population is

allowable lower bound 𝑥𝑘 <0 and positive allowable upper bound 𝑥 k>0 is described as follows. At the end of every 𝑁c generations of population evolution, we examine each * parameter in the vector 𝑥 that has the lowest cost. If the current upper bound 𝑥 k of the 𝑘-th parameter 𝑥𝑘∗ is positive and 𝑥 k≥ 𝑥𝑘∗ >𝑥 k/2 , then the upper bound 𝑥 k get doubled. In the case where is still greater than the doubled upper bound, the upper bound 𝑥 k is set to 𝛾𝑥𝑘∗ , where 𝛾 > 1 is a user-set expansion factor. The selection of the search-space expansion factor 𝛾 depends on the size of the initial search space. Usually, it is set to 2 for a small initial search space and 1.2 for a large initial search space.

generated, the cost of each population vector is evaluated and stored for future reference. C. Mutation

Mutation is an operation that adds a vector differential to a population vector. In the DE algorithm, a population vector 𝑥∝ is mutated into 𝑠 = [𝑠1 , 𝑠2 , … 𝑠𝑛 ] 𝑇 by adding to 𝑥∝ the weighted difference of two randomly selected but different population vectors 𝑥𝛽 and 𝑥𝛾 , i.e., (8)

𝑠=𝑥∝ +𝐹𝑠 *(𝑥𝛽 + 𝑥𝛾 )

TABLE 1: VALUES OF THE DE ALGORITHM VARIABLES USED IN EXAMPLES 1 AND 2

where 𝐹𝑠 is a scaling factor in the interval (0,2). In the event that mutation causes a parameter 𝑠𝑘 to shift outside the allowable interval 𝑥𝑘− , 𝑥𝑘+ then 𝑠𝑘 is set to 𝑥𝑘− if 𝑠𝑘 < 𝑥𝑘− , and 𝑥𝑘+ if 𝑠𝑘 > 𝑥𝑘+ . The mutated vector 𝑧 will be used as a donor vector for producing a trial vector. It is noted the mutation scheme (8) makes the DE a local optimizer because the vector differentials generated by a converging population eventually become infinitesimal. D. Crossover Crossover is used to generate a trial vector by replacing certain parameters of the target vector by the corresponding parameters of a randomly generated donor vector. The crossover rate 𝐶𝑟 determines when a parameter should be replaced. The process of producing a trial vector 𝑥𝑡 =

Parameters

Value

Population Size (𝑁𝑝 )

100

Maximum no. of generations (𝑁𝑖 )

200

Search-Space checking period, (𝑁𝑐 )

10

Search Space expansion factor, γ

1.2

Crossover constant, (𝐶𝑟 )

0.6

Mutation Scaling factor, (𝐹𝑠 )

[0-2]

𝑇

𝑥𝑡,1 , … 𝑥𝑡,𝑛 from the target vector 𝑥 = 𝑥1 , … 𝑥𝑛 𝑇 and the donor vector 𝑧 is begin with generating a set of 𝑛 random numbers 𝑟1 , 𝑟2 , … 𝑟𝑛 which are distributed uniformly in the interval (0,1). Next, a set of non uniform binary sequence 𝑏1 , 𝑏2 , … 𝑏𝑛 is generated by letting

𝑏𝑖 = 1 𝑖𝑓 𝑟𝑖 ≤ 𝐶𝑟

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

𝑖 =1,2,…𝑛

In the event that any parameter search interval has been expanded, 𝑁𝑝 population vectors are generated from the expanded portion of search space and their costs are evaluated. A new initial population of 𝑁𝑝 population vectors is then formed by picking up the better population vectors among the 𝑁𝑝 evolved population vectors and the 𝑁𝑝 newly generated population vectors. Then this new initial population evolves with the expanded search space.

(9)

Then each element 𝑥 t,k of the trial vector 𝑥 t is taken as 𝑥𝑡,𝑘 =

𝑥𝑘 𝑓𝑜𝑟 𝑏𝑘 = 1 𝑠𝑘 𝑓𝑜𝑟 𝑏𝑘 = 0

(10)

F. Selection

Once the trial vector has been determined, its cost is evaluated and compared with that of the corresponding target vector. The target vector will be replaced by it in the next generation if its cost is larger than that of the trial vector.

The cost of each trial vector 𝑥𝑡 is compared with that of its parent target vector 𝑥𝑖 . If the cost of the target vector 𝑥𝑖 is lower than that of the trial vector, the target is allowed to advance to the next generation. Otherwise, the target vector is replaced by the trial vector in the next generation.

E. Search-Space Expansion Scheme In the case that the initial search region is set excessively large, the convergence of DE search may become very slow and is prone to get trapped in a local optimum. On the other hand, if the initial search region is set too small, it may fail to locate the true optimum, though the mutation formula with a proper scaling factor 𝐹𝑠 allows the DE algorithm to locate the minimum that lies outside the initial search region. Hence, in the absence of the exact knowledge about the proper and finite interval within which the true optimum parameter locates, it is desired to allow the DE algorithm to expand the search space dynamically. Toward this end, we incorporate in the DE algorithm a search space expansion scheme. The implementation of this scheme for the cases of negative

Differential Evolution Algorithm 1) Choose population size 𝑁𝑝 , mutation scaling factor 𝐹𝑠 , search-space checking period𝑁𝑐 , space expansion factor 𝛾, and the maximum number of iterations 𝑁𝑖 . 2) Specify the initial search intervals [𝑥𝑘 , 𝑥𝑘 ], 𝑘=1,2,…𝑛 . 3) Generate randomly𝑁𝑝 , vectors 𝑥 j, from the specified search space 𝑋 = 𝑥: 𝑥𝑘 ∈ [𝑥𝑘 , 𝑥𝑘 ] , 𝑘=1,2,…𝑛 and evaluate their costs 𝐶𝑗 , 𝑗 =1,2… 𝑁𝑝 . 678

2014 ACODS March 13-15, 2014. Kanpur, India

4) Initialize the iteration index 𝐼=1. 5) Set the initial target index 𝑗=1. 6) Choose at random three different integers𝛼,𝛽 and 𝛾 from the set { 𝑗=1,2…𝑁𝑝 }

where, the 𝑐𝑗 and 𝑚𝑗 are the time moment proportionals and Markov parameters of the system, respectively. Elimination of the 𝑑𝑗 ( 𝑗 = 0,1, ...,r −1) in (14) by substituting into (13) gives the reduced denominator coefficients as the solution of

7) Mutate the vector 𝑥∝ to 𝑠=𝑥∝ + 𝐹𝑠 *(𝑥𝛽 − 𝑥𝛾) . 8)

Generate

a

non

uniform

binary

𝑐𝑡−1 𝑐𝑡−2 … … … … 𝑐𝑡−𝑟 e0 𝑐𝑡−2 𝑐𝑡−3 … … … 𝑐𝑡−𝑟−1 e1 ⋮ ⋮ ………⋮ ⋮ ⋮ 𝑐𝑟−1 𝑐𝑟−2 … … … 𝑐1 𝑐0 x ⋮ 𝑐𝑟−2 𝑐𝑟−3 … … … 𝑐0 -M1 ⋮ ⋮ ⋮ ……⋮ ⋮ ⋮ ⋮ −𝑀𝑚 −𝑟 −𝑀𝑚 −𝑟−1 –Mm-1 er-1

n-sequence

𝑏={𝑏1 , 𝑏2 , … 𝑏𝑛 }. 9) Obtain the trial vector 𝑥𝑡 from the mutated vector S and the target vector 𝑥𝑗 using crossover scheme (10). 10) Evaluate the cost of the trial vector 𝑥𝑡 and denote it by𝐶𝑡 . If 𝐶𝑡 > 𝐶𝑗 then 𝑥𝑗𝑛𝑒𝑤 = 𝑥𝑗 and 𝑐𝑗𝑛𝑒𝑤 = 𝑐𝑗 ,otherwise 𝑥𝑗𝑛𝑒𝑤 = 𝑥𝑡 and 𝑐𝑗𝑛𝑒𝑤 = 𝑐𝑡

⋯ (15)

11) Increment the target index 𝑗 by 1. If 𝑗 < 𝑁𝑝 then go to Step 6. 12) Do the replacement 𝑥𝑗 = 𝑥𝑗𝑛𝑒𝑤 and 𝑐𝑗 = 𝑐𝑗𝑛𝑒𝑤

Or , 𝐻𝑒 = 𝑚 in matrix vector form.

for

If the denominator given by 𝑒 in (15) is unstable, or has a singularity, then the next Markov parameter 𝑚𝑟−𝑡+1 can be assumed to be matched by extending (14) with the equation:

𝑗=1,2…𝑁𝑝 . 13) If 𝑖 is a multiple of 𝑁𝑐 , then execute the search-space expansion scheme.

𝑑𝑡−1 = 𝑚1 𝑒𝑡+2 + 𝑚2 𝑒𝑡+1 + ⋯ + 𝑚𝑟−𝑡+1

14) Increment the iteration index 𝐼 by 1. If 𝐼 <𝑁𝑖 , then go to Step 5. 15) Stop.

𝑐𝑡−1 𝑐𝑡−2 … 𝑐0 − 𝑚1 − 𝑚2 … − 𝑚𝑟−𝑡 and [ 𝑚𝑟−𝑡+1 ] respectively. Calculation of 𝑒 from this non square system of equations can only be done in the least- squares sense, i.e.: 𝑒 = 𝐻𝑇 𝐻

−1

𝐻𝑇 𝑚

(17)

If the denominator polynomial is still not adequate, then the H matrix and the 𝑚 vector may again be extended by assuming a matching of the next Markov parameter in the sequence and (17) is solved for the new estimate of 𝑒.

For a reduced 𝑟th order model of 𝐺𝑛 𝑠 in (11), given by: 𝑑0 + 𝑑1 𝑠 + ⋯ + 𝑑𝑟−1 𝑠 𝑟−1 𝐺𝑟 𝑠 = 12 𝑒0 + 𝑒1 𝑠 + ⋯ + 𝑒𝑟−1 𝑠 𝑟−1 + 𝑠 𝑟 Which retains (𝑟 + 𝑡) time moments and (𝑟 − 𝑡) Markov parameters (0 ≤ 𝑡 ≤ 𝑟) the coefficients 𝑒𝑘 , 𝑑𝑘 in (12) are derived from following set of equations

VI. ILLUSTRATIVE EXAMPLES Example 1: Consider an fifth order system transfer function 𝐺(𝑠) = 3.75𝑠 5 +5.6625𝑠 4 +2.953131 𝑠 3 +0.668732 𝑠 2 +0.06617 𝑠+0.002262 𝑠 6 +1.75𝑠 5 +1.1125𝑠 4 +0.323126 𝑠 3 +0.045216 𝑠 2 +0.002896 𝑠+0.000066

Denominator by improved generalized least square method: (13)

No. of Time moments = 4 t[0]=34.272728, t[1]=-501.269989, t[2]=8647.485352, t[3]=159075.3125 No. of Markov parameters = 0 The second order reduced denominator using improved generalized least square method in S-domain is

𝑑𝑟−1 = 𝑀1 𝑑𝑟−2 = 𝑀1 𝑒𝑟−1 + 𝑀2 ⋮ ⋮ ⋮ 𝑑0 = 𝑀1 𝑒1 + 𝑀2 𝑒2 + ⋯ + 𝑀𝑟 𝑀1 𝑒0 + 𝑀2 𝑒1 + ⋯ + 𝑀𝑟 𝑒𝑟−1 = −𝑀𝑟+1 ⋮ ⋮ ⋮ 𝑀𝑚 −𝑟 𝑒0 + 𝑀𝑚 −𝑟−1 𝑒1 + ⋯ + 𝑀𝑚 −1 𝑒𝑟−1 = −𝑀𝑚

(16)

This in effect adds another row to the H matrix and the m vector in (15), given by:

V. Order Reduction by Improved Generalized LeastSquares Method: Here, the model reduction by generalized least-squares method suggested in [5] is discussed in brief Consider the 𝑛th order system transfer function, given by: 𝑏0 + 𝑏1 𝑠 + ⋯ + 𝑏𝑛−1 𝑠 𝑛−1 𝐺𝑛 𝑠 = (11) 𝑎0 + 𝑎1 𝑠 + ⋯ + 𝑎𝑛 −1 𝑠 𝑛 −1 + 𝑎𝑛 𝑠 𝑛

𝑑 0= 𝑒0 𝑐0 𝑑1 = 𝑒1 𝑐0 + 𝑒0 𝑐1 ⋮ ⋮ ⋮ 𝑑𝑟−1 = 𝑒𝑟−1 𝑐0 + ⋯ ⋯ + 𝑒0 𝑐𝑟−1 −𝑐0 = 𝑒𝑟−1 𝑐1 + ⋯ ⋯ + 𝑒1 𝑐𝑟−1 + 𝑒0 𝑐𝑟 −𝑐1 = 𝑒𝑟−1 𝑐2 + ⋯ ⋯ + 𝑒1 𝑐𝑟 + 𝑒0 𝑐𝑟+1 ⋮ ⋮ ⋮ −𝑐𝑡−𝑟−1 = 𝑒𝑟−1 𝑐𝑡−𝑟 + ⋯ ⋯ + 𝑒1 𝑐𝑡−2 + 𝑒0 𝑐𝑡−1 and

−𝑐𝑡−𝑟−1 −𝑐𝑡−𝑟−2 ⋮ = 𝑀1 ⋮ ⋮ 𝑀𝑚

(14)

𝐷(𝑠) = 𝑠 2 + 0.225215𝑠 + 0.009092

679

2014 ACODS March 13-15, 2014. Kanpur, India

Example 2: Consider an eighth order system transfer function

Numerator by DE technique: No. of iterations 𝑁𝑖 = 120

1.682𝑠 7 +12.89𝑠 6 +41.808002 𝑠 5 +74.711998𝑠 4 + 79.182007 𝑠 3 +49.064003 𝑠 2 +16.000002 𝑠+1.988 8𝑠 8 +58.953999𝑠 7 +185.330002 𝑠 6 +322.575989𝑠 5 + 335.864014 𝑠 4 +210.454025 𝑠 3 +76.807999𝑠 2 +16𝑠+2

𝐺(𝑠)=

Swarm size 𝑁𝑝 = 100

No. of Time moments = 4

𝑥𝑘 = 0.009092, 𝑥𝑘 = 3.5

t[0]=0.994, t[1]=0.048001, t[2]=-14.025581, t[3]=45.356579

The second order reduced numerator using differential evolution technique in s-domain is

No. of Markov parameters = 0 The second order reduced denominator using improved generalized least square method in s-domain is 𝐷(𝑠) = 𝑠 2 + 0.23521𝑠 + 0.0716

𝑁(𝑠) = 3.017862𝑠 + 0.306552 The proposed second order reduced model obtained is 3.017862 𝑠+0.306552

𝑅(𝑠) =

𝑠 2 +0.225215 𝑠+0.009092

Numerator by DE technique:

(ISE=0.608587)

No. of iterations 𝑁𝑖 = 100

Fig.1(a) Comparison of step responses of 𝐺(𝑠) and 𝑅 𝑠

Swarm size 𝑁𝑝 = 50 Step Response 35

𝑥𝑘 = 0.071675, 𝑥𝑘 = 0.4

30

The second order reduced numerator using differential evolution technique in s-domain is

Amplitude

25

20

𝑁(𝑠) = 0.244245𝑠 + 0.072315

15

The proposed second order reduced model obtained is

original reduced

10

0.244245 𝑠+0.072315 𝑠 2 +0.23521 𝑠+0.071675

𝑅(𝑠) = 5

(ISE=0.000814)

Fig. 2(a) Comparison of step responses of 𝐺(𝑠) and 𝑅 𝑠 0

0

20

40

60

80

100

120

140

Time (seconds)

Step Response

Step Response

1.4

35 1.2

original reduced

30

pade ram

Amplitude

25

Amplitude

1

tew ari

20

0.8

0.6 original

15

0.4

10

0.2

5

0

reduced

0

10

20

30 Time (seconds)

0

0

2

4

6

8 (sec)10 Time

12

14

16

18

Fig. 1(b) Comparison of step responses of 𝐺(𝑠) and 𝑅 𝑠 with other reduction techniques

680

40

50

60

2014 ACODS March 13-15, 2014. Kanpur, India

The authors proposed a mixed algorithm for reducing the order of linear dynamic SISO systems. In this algorithm, the concept of order reduction by generalized least squares method has been improved and employed to determine the coefficients of reduced denominator while the coefficients of reduced numerator are obtained by minimizing the integral square error between the transient response of original and reduced models using DE technique pertaining to unit step input. The algorithm is implemented in C-language. The matching of the unit step response is assured reasonably well in the algorithm. The quality of a formulated reduced order model is judged by designing the discrete PID controller. PID controller of the formulated reduced order system efficiently controls the original higher order system. This approach minimizes the complexity involved in direct design of PID controller. The algorithm is simple and computer oriented. A comparison of step responses for the proposed reduction algorithm and the other well known existing order reduction technique is shown from which it is clear that the proposed algorithm compares well with the other techniques of model order reduction.

Step Response 1.4

1.2

1

Amplitude

0.8

0.6

0.4

original reduced ram

0.2

0

0

10

20

30

40

50

60

Time (seconds)

Fig. 2(b) Comparison of step responses of 𝐺(𝑠) and 𝑅 𝑠 with other reduction techniques

REFERENCES

Applications of DE for PID controller design:

[1] R.Genesio and M. Milanese, “A note on the derivation and use of reduced order models”, IEEE Trans. Automat. Control, Vol. AC-21,No. 1, pp. 118-122, February 1996. [2] M. Jamshidi, “Large Scale Systems Modeling and Control Series”, New York, Amsterdam, Oxford, North Holland, Vol. 9, 1983. [3] G. Parmar, R. Prasad and S. Mukherjee, “Order reduction of linear dynamic systems using stability equation method and GA”, World Academy of science engg. and tech Vol-26, 2007. [4]C.S.Hsien,C.Hwang 1989,“Model order Reduction of Continous time Systems using a modified Routh Approximation Method” IEE Proceedings control theory appl.136:pp no:151-156 [5] T. N. Lucas and A. R. Munro, “Model reduction by generalised leastsquares method”,Electronics Letters, Vol. 27, No. 15, pp. 1383-1384, 1991. [6] H.A.Abbass, Rahul.Sarker,&Charles Newton,“PDE: A Pareto-frontier differential evolution approach for multi-objective optimization problems”, proceedings, 2001 [7] R.Storn,”Differential Evolution,a simple and efficient heuristic strategy for global optimization over continuous spaces”,journal of global optimization, vol.2,Dordrecht,pp no:341-359,1997 [8] L. A. Aguirre, “PID tuning based on model matching “, IEEE electronic letter, Vol 28, No.25,pp 2269-2271, 1992. [9] S. K Mittal, Dinesh Chandra, “Stable optimal model reduction of linear discrete time systems via integral squared error minimization: computer-AIDED approach “, AMO-Advanced modeling and optimization, vol 11, number 4, 2009.

In this study, the PID controller has been designed employing process reduction approach. The original higher order continuous system is reduced to lower order model employing DE technique. Then the PID controller is designed for lower order model. The parameters of PID controller are tuned by using same error minimization technique employing DE as explained in section IV the optimized PID controller parameters are: 𝑘𝑝 =0.1789, 𝑘𝑖 =7.24471, 𝑘𝑑 =0.94696; The transfer function of the designed PID controller is as follows: Gc(s)=

0.94696 𝑠 2 +0.1789𝑠+7.24471

𝑠

The unit step responses of the reduced system with and without PID controller are shown in fig 2(c)

Step Response 1.4 w /0 comp w ith comp

1.2

Amplitude

1

0.8

0.6

0.4

0.2

0

0

5

10

15

20

25

30

35

40

45

50

Time (sec)

Fig.2(c) step response of reduced with and without PID controller

CONCLUSION: 681

2014 ACODS March 13-15, 2014. Kanpur, India

[10] E. G. Collins, S. S. Ying, W. M. Haddad, and S.Richter, “An efficient, numerically robust homotopy algorithm for H2 model reduction using the optimal projection equations,” Math.Modeling Syst.: Meth.,Tools Applicant. Eng.Rel. Sci., vol. 2, no. 2, pp. 101–133, 1996. [11] A.Varsek, T. Urbacic and B. Filipic, “Genetic Algorithm in controller Design and tuning “, IEEE transaction on sys. Man and Cyber, Vol 23, No.5 pp 1330-1339, 1993. [12] Z. L. Gaing, “A particle Swarm Optimization approach for optimum design of PID controller in AVR system”, IEEE transaction on energy Conversion, vol.19, no.2, pp 384-391,2004

682

2014 ACODS March 13-15, 2014. Kanpur, India

683