This paper proposes a new version of Water Swirl Algorithm namely Improved Water Swirl Algorithm for lower order model formulation of SISO continuous systems. Water Swirl Algorithm (WSA) is a swarm based optimization technique that mimics the way by which water finds a drain in a sink. It observes the flowing and searching behavior of water for drain and proposes a suitable strength update equations to locate the optimum solution iteratively from the initial randomly generated search space. The strength of a water particle is governed by three components namely, Inertia, Cognitive and Social. In the proposed Improved WSA, the cognitive component of water particle is spitted into good experience component and worst experience component. Due to the inclusion of worst experience component, the particle can bypass the previously visited worst position and try to occupy the best position. A weighted average method is proposed in this paper to reduce the higher order model formulation to lower order form. The result shows good performance of the Improved WSA in solving SISO continuous system problems as compared to other existing techniques. The proposed method is illustrated through numerical example from literature.
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Keywords: Water Swirl Algorithm, Weighted Average method, linear time invariant, SISO Continuous System, Swarm Optimization and Lower Order model formulation.
During the past few decades, there is massive development in the field of model order formulation. Model order formulation is the process of deriving the lower order model from the higher order model. Lower order models aid to carry out the process operations in real time by reducing the implementation and computational difficulties involved in the design of optimal and adaptive controller, compensators and observer design for higher order linear time invariant system. A lower order model is preferred because in the analysis and design, simulation of a compensator or controller for stabilization of the output response of the given system, higher order models are much more difficult to handle.
In general, a given Linear Time Invariant System (LTIS) can be represented in the form of open loop transfer function having numerator polynomial and denominator polynomial in s-variable. The degree of numerator polynomial should be less than or equal to the degree of denominator polynomial. It is important that lower order model maintains the characteristics of the higher order model. This is achieved by minimizing the error for each iteration. The main aim of formulation is to find the best possible approximation of the output of the original system.
In recent years, various techniques have been proposed for deriving the lower order model from higher order form. Among these, the first method is developed by Davison (1996) for lower order model reduction in which the dominant eigen values were retained and excited in the same proportion as in the original system. During seventies, numerous methods were suggested to solve the stability problem in the coefficient matching techniques. Shamash (1975) combined the features of both mode-retention and coefficient matching approaches. "Routh approximation method" was proposed by Maurice and Friedland (1975) to approximate the transfer function of a higher order linear system by a lower order model. Chen et al. (1980) developed the stability equation method to obtain a suitable lower order model. Bistritz (1980) presented a scheme for deriving lower order models by mixed pade approximation technique. Hwang (1983) formulated reduced order models, which matches some time moments and Markov parameters of the original model using continued fraction expansion.
Numerous other approaches have been proposed for model reduction in which information concerning the original model over the mid-frequency range is included in the lower order model (Xiheng, 1987; Stahl and Hippe, 1988). Pade simplification of squared magnitude functions was used by Aguirre (1994) for building reduced order models. Further, Aguirre (1994a) derived a scheme which exactly retains the poles and zeros of the original system, while computing the rest of the coefficients of the lower order model by least squares pade approximation. Jayanta Pal et al. (1994) developed a mixed method for model reduction using stability preserving methods and approximate generalized time-moments technique. Sastry and Chittamuri (1995) proposed a computer based approach for the stability analysis of higher order nonlinear systems using modeling and describing function approach.
Prasad et al. (2003) proposed a linear model reduction using the Mihailov criterion and factor division and have eliminated the calculations of initial time moments and Markov parameters. The above discussed model reduction methods pertain to linear time invariant continuous systems. Most of these techniques can be extended to linear time invariant discrete systems using suitable transformations.
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Khatwani and Bajwa (1978) extended the Routh approximation method proposed by Maurice and Friedland (2003) to discrete systems. Reduced order modeling of linear discrete systems based on a modified Routh stability criterion was proposed by Shamash and Feinmesser (1978), Lucas (1993)(1993a)(1996) used multipoint pade approximation to find optimal and sub-optimal lower order discrete model for the given higher order discrete systems. Prasad (1993) presented a new mixed method using stability equation approach and weighted time moments to reduce the order of linear time invariant high order discrete time scalar systems. A Routh approximation method for the order reduction of linear time invariant discrete systems using only one routhian array and avoiding the use of bilinear transformations was proposed by Sastry and Srinivasa Reddy (1995a); Mukherjee et al. (1997) developed a method of order reduction for linear discrete systems with real poles. Tewari and Bhagat (2004) presented a methodology for simplification of discrete time systems using improved Routh stability criterion. Each of these methods has its own advantages and disadvantages. In spite of these methods, there is no exact method which gives the best solution for all model formulation.
Most problems in this area are heuristics in nature which can find good solution in a reasonable amount of time (ko et al. 2008). Heuristics are rules to search for optimal or near-optimal solutions. Some of the heuristic tools include Simulated Annealing (SA), Ant Colony Optimization, Evolutionary Computation methods like Genetic Algorithm, Fuzzy Logic, Neural Networks (Sivanandam et al. 2005, 2007) and Swarm Intelligence techniques like Particle swarm Optimization, Water Swirl Algorithm etc., (Guoliang et al. 2012). In this paper, novel attempt is made to simulate the motion of water in the sink is presented. This behavior of water is studied by Menser and Hereford (2006) as Water Swirl Algorithm based on the method of fluid flow around a drain. In the proposed Improved Water Swirl Algorithm, the search behavior of water for drains and proposes suitable update equation to locate the optimum solution iteratively from the initial randomly generated search space.
This paper is organized as follows: Section 2 presents the problem identification, Section 3 focuses on the overview of Water Swirl Algorithm, Section 4 gives the proposed Improved Water Swirl Algorithm, Section 5 focuses on lower order model formulation of linear time invariant systems, Section 6 explains the results and discussion and lastly Section 7 concludes the work.
DESCRIPTION OF THE PROBLEM
Consider an nth order linear time invariant continuous system represented in Krishnamurthy et. al. (1978),
where N(s) is the numerator polynomial and D(s) is the denominator polynomial.
For the given original higher order system G(s) represented in equation (1), the problem is to find an mth lower order model R(s), where m < n in the following form represented by equation (2), such that the reduced model retains the characteristics of the original system and approximate its response as closely as possible for the same type of inputs with minimum integral square error.
where Nm(s) and Dm(s) are the numerator polynomial and denominator polynomial of the lower order model respectively. Also, Bi and bi represent the constant coefficients of s-terms of numerator and denominator of R(s).
The main objective of lower order model formulation is to minimize the integral square error of the unit step time responses of the lower order system and the original higher order system within a specified time interval with the constraint: the transient and steady state gain ratios of the higher order system should be maintained in the designed lower order system.
The fitness function for the considered problem lower order formulation of SISO Continuous Systems is formulated as given below. Mathematically, the integral square error (ISE) Nagrath et al. (2003) can be expressed as,
Yt is the unit step time response of the given higher order system at the tth instant in the time interval 0< t<.
yt is the unit step time response of the lower order system at the tth time instant.
is the time period in seconds over which the integral square error is calculated and is to be chosen.
Minimization of integral square error will ensure a good lower order approximant. The problem defined for linear time invariant continuous system holds the same for linear time invariant discrete system also. The weighted average method is used for approximating the lower order model from the higher order model which is explained in Appendix.
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OVERVIEW OF WATER SWIRL ALGORITHM
Water Swirl Algorithm is a swarm based optimization technique that mimics the way by which water finds a drain in a sink. This algorithm considers a certain number of water particles that represents the number of possible solutions for a variable in the search space. The sink holding the water particle represents the boundary conditions limiting the search space.
Water is an inevitable substance in the nature for all the livings. Water has peculiar characteristics that when it is poured or contained in a sink, it tries to find the drain (hole) in the sink continuously to leave out. The answer for this excellent natural behavior of water can be found in the field of Fluid Dynamics (2006) that deals with the nature of fluid flow.
When the drain of the sink is opened, a swirling motion is started in the water mass near the drain leading to the release of water though the drain. The swirl motion of water leads to an important phenomenon called vortex formation as depicted in Fig.1. (Rani et. al., 2011).
Fig.1 Illustration of Vortex Ring
During swirling motion, the vortex is not stationary, but rather continually moves towards the highest value in the search space. Since the particles are drawn towards the vortex, the search is concentrated in areas that have previously yielded good results (Previous Best). The drain would signify the best among the previous best known as the Global Best. Water particles exist for a certain amount of time, or iterations, and update the previous and global best position until an overall best solution is found. In view of this, strength update equation (4) and position update equation (5) are modified as below:
Here xq,ref is randomly generated using the range given for the solution variable. is a random number generated between zero and one. and are the strength vectors of water particles at ith and (i+1)th iterations respectively. Similarly, and are the positions of water particles at ith and (i+1)th iterations. , and denotes the reference position, previous best position and global best position of the water particle respectively.
The solution search equation (4) for updating particle strength has well balanced exploration and exploitation abilities. Even though it resembles the velocity updating equation of PSO, the use of reference position xq,ref in second term performs very well in exploration. The generation of strength vector 'Î±' within the actual range of the variable avoids the problem of velocity clamping. There is no weight 'w' and parameters 'c1' and 'c2' in this search equation and it is completely parameter free.
While updating strength of a particle using (4), different variables have different value that provides larger search space and makes WSA always perform better due to independent updating of each variable. The position update equation given in (5) performs a local selection task by using new strength vector and reference position xq,ref for determining the next adjacent position around the vortex ring.
AN IMPROVED WATER SWIRL ALGORITHM
In the new proposed Improved water swirl algorithm, the strength update equation is splitted into three components namely inertia, cognitive and social component. The cognitive component is splitted into two components. The first component is good experience component which refers to the previously visited good position of the water particle and second component is worst experience component which refers to the previously visited worst position of the water particle. Due to the inclusion of worst experience component in the strength update equation, the particle can bypass the previously visited worst position and try to occupy the best position.
The modified strength update equation (6) is given by
Here xq,ref is randomly generated using the range given for the solution variable. is a random number generated between zero and one. and are the strength vectors of water particles at ith and (i+1)th iterations respectively. Similarly, and are the positions of water particles at ith and (i+1)th iterations. ,, denotes the reference position, previous best position, global best position of the water particle and previous worst position respectively.
The algorithmic steps for the proposed Improved WSA are as follows:
Step1: Select the number of water particles, range of water particle and maximum iterations
Step2: Initialize the particle position and strength.
Step3: For each water particle (xp), evaluate fitness function using equation (3)
Step4: Select the particle global best value and the particles
individual worst value.
Step 5: Compare the fitness value of (xp) with fitness value of (xprevBest). If it is greater value, then set xprevBest as xp. Otherwise, go to Step 3 for evaluating fitness value.
Step6: Update the particle individual best (XprevBest), global best (XgBest) and particle worst (XworstBest) in the strength update equation (6) and obtain the position of the particle.
Step 7: The optimal solution is obtained when the integral
square error is minimum.
Step 8: Stop
A detailed flow chart illustrating the various steps of the proposed Improved WSA algorithm is shown in Fig 2. The proposed Improved WSA starts with the initialization of control parameters like Number of Water Particles ('N'), Boundary or range of each water particle ('B') and the maximum iterations ('I'). Then the initial position, reference position and the strength of the water particle are randomly generated within the actual range of each particle. Fitness of each water particle is evaluated.
Calculate Transient gain and Steady state gain for the given problem
Read the coefficients of the numerator and denominator from
Until all the water particle exhaust, update the previous best (prevBest), previous worst (worstBest) and set the best of prevBest as global best (gBest). Then the strength and position of the water particle are updated using equation (6) and (5) respectively until the maximum iteration is reached. At last, the final gBest value is returned as an optimal solution.
Fig. 2 (Continued)
Apply proposed weighted average method to G(s) to get the lower order model of the problem R(s), scale it and tune it to maintain the transient and steady state gain of G(s).
Loop until maximum iteration
Initialize the number of water particles, maximum iterations and boundary of each particle. Randomly generate strength, reference position and particle's position
For each water particle(xp) Evaluate fitness
If fitness(xp) >fitness (xprevBest)
Set best of xprevBest as xgBest
Invoke Improved WSA for lower order model
Update particle's strength and particle's position in (6) and (5)
Return xgBest as optimal solution
Fig. 2 Flowchart for lower order model formulation of
single input single output linear time invariant continuous systems
LOWER ORDER MODEL FORMULATION OF SISO LTICS
The proposed methodology for lower order model formulation of SISO LTICS is as follows:
Step1: Consider an n-th order linear time invariant continuous system represented by the transfer function (TF) G(s) in general form as given in equation (1).
Step2: Calculate the transient gain (TG) and steady state gain (SSG) for the given higher order system in equation (6.1) as follows:
Step3: Applying proposed weighted average method as specified in appendix to obtain an initial lower order model of order 2. Thus, the transfer function of basic second order model obtained using weighted average approach is as given in equation (9) i.e.,
Step4: Scaling equation (9), R(s) becomes,
Step5: Equation (10) is tuned to maintain the transient and steady state gain obtained in equations (7) and (8), resulting initial second order model R(s) is,
Comparing equations (9) and (11), it can be noted that,
Step6: The coefficients of the initial second order model R(s) in equation (12) are fed as input to Improved WSA process. The main aim of Improved WSA is to minimize the objective function integral square error E given in equation (3). The Improved WSA algorithm in section IV is invoked to search for the better values of and in equation (11), so that the characteristics of the formulated second order model matches the given higher order system. The improved WSA is carried out within the constraint of maintaining the transient and steady state gain of the second order model in accordance with that of the given higher order system calculated in equations (7) and (8).
Step7: The second order model corresponding to the minimum integral square error is declared as the best second order approximant for G(s) in equation (1) and is given in equation (2).
Step8: To ensure the effectiveness of the proposed procedure, the unit step response of the given higher order system in equation (1) and the response of the proposed second order model in equation (2) is compared. Also, the integral square error is computed and compared with that calculated for the second order models obtained by other methods. This formulated second order model R(s) in equation (2) is found to maintain the original characteristics of the given higher order model.
Step1: Consider a 7th order Interval system transfer function where the higher order system considered for lower order modeling is given by
Step2: The Transient Gain (TG) and Steady State Gain (SSG) for the given system G(s) for the above equation is calculated as below:
Step3: A weighted average method discussed in Appendix is used to obtain the lower order model R(s) from the given G(s), whose coefficients are used as initial seed valued for training in Improved WSA. On applying the weighted average method to G(s) in equation (13), the basic lower order model R(s) is given by,
Step4: The equation (15) should be scaled and tuned to maintain the transient gain and steady state gain.
Step5: Equation (16) is tuned to maintain the transient and steady state gain obtained in equations (14), resulting initial second order model R(s) is,
Step6: To maintain the characteristics of higher order system, the improved WSA algorithm is now invoked to search the value of 's' term (2.126) and the constant term (3.333 ) in the denominator of R(S) of equation (17).
Step7: The transfer function of the reduced second order model obtained using Improved WSA method is given below:
The results of applying Improved WSA algorithm to the lower order system modeling problem are tabulated in Table 1.
Table 1 Lower order system modelling results using Improved WSA approach
Reduced Order Model
Chen et al method
Krishnamurthy andSeshadri method
Gutman et al method
Jayanta Pal et al method
The salient points noted in the illustrations are brought out in this section. Illustration deals seventh order linear time invariant continuous system with transient gain and steady state gain of 35 and 20.26 respectively. Initially, the approximate lower order model was obtained using the proposed weighted average method. Improved WSA was invoked to minimize the integral square error for the approximate lower order model with the constraint of maintaining the transient gain and steady state gain of the given higher order system. Improved WSA algorithm was simulated and it searched for the best second order model with a minimal integral square error of 1.7226. Table 1 reveals that the proposed scheme yields better value for integral square error in comparison with other techniques. From Figure 3, it is observed that the unit step time response of the proposed second order model maintains the original characteristics of the given seventh order system compared to the second order models obtained using other known methods. The Improved WSA algorithm was coded in Intel Pentium processor 4.0, 2.8 GHz, 256 MB RAM.
The characteristics of the lower order systems obtained by the proposed Improved WSA optimization method are comparable and closely agree with that of the corresponding higher order systems. This is observed graphically with the help of unit step time response curves. Also, the integral square error measured between specified time intervals is minimum in proposed methodology when compared with the other model reduction methods. The above mentioned observations are substantiated with the given illustrative examples. The proposed approach can also be used for multi input and multi output systems. This can also be extended for further design of controllers and compensators as well as state variable controllers and observers for stabilization process.
Fig.3 Comparison of unit step response for illustration
Consider the transfer function of nth order linear time invariant continuous higher order system as given in equation (1). The weighted average method is proposed to approximate lower order model from the given higher order system is as follows:
Consider the numerator of equation (1), which is derived using weighted average method and is as tabulated in Table 2.
Table 2: Weighted average method for numerator polynomial
A B C
From Table 2. It can be noted that the given numerator polynomial is approximately reduced to a lower order numerator polynomial employing weighted average method.
Consider the denominator polynomial of equation (1), which is derived using weighted average method and is as tabulated in Table 3.
Table 3: Weighted average for denominator polynomial
P Q R
Generally, the denominator polynomial of transfer function decides the order of the system. Hence, the least lower order model deduced using the proposed weighted average method is of order i.e., a second order system.
From Table 2 and Table 3, the approximate lower second order model derived for the given higher order system is
In Table 3, the first row is of nth order, the second row corresponds to (n-1)th order, the third row is of (n-2)th order and so on, where the last row corresponds to second order.
Thus, using the proposed weighted average method and approximate lower order model as in equation (19) is computed.