Design of Sliding Mode Controller via Novel Model Order Reduction Technique

Design of Sliding Mode Controller via Novel Model Order Reduction Technique

M. Prajeeth M. Sudheer Kumar,M.E

M. E Student Anits Engg.College,

Sangivalasa, Bheemunipatnam Visakhapatnam

Asst. Professor Anits Engg.College,

Sangivalasa, Bheemunipatnam, Visakhapatnam

AbstractThis paper presents a mixed method for finding stable reduced order models using error minimization technique and modified pole clustering technique for single input-single output (SISO) large scale systems. The denominator polynomial of the reduced order model is determined by forming the clusters of the poles of original system and the cluster centers are identified by using inverse distance measure criterion and the numerator polynomial is obtained by the error minimization technique. For this reduced order model, sliding mode controller is designed. The performance of the system for both PID controller as well as SMC is compared. This reduced system with SMC is stable even when the original high order system is unstable.

KeywordsModified pole clustering, Order reduction, Error minimization technique, Stability, Transfer function, Sliding mode

control systems. Many reductiontechniques for linear continuous systems such as Padé approximations, Routh approximations and time moment matching based methods have been proposed [5-9].

In the proposed method, the denominator polynomial of the reduced order model is obtained by using modified pole clustering technique while the coefficients of the numerator are obtained by error minimization technique [10-12].

1. PROBLEM STATEMENT

   Let the transfer function of high order original system of the order n be

   controller.

   = = 0+1+2 2++ 1 1+

   1. I

      NTRODUCTION

      0 +1+2 2 ++ 1 1+

      Model order reduction (MOR) is a technique for reducing the computational complexity of mathematical models in numerical simulations. Many modern mathematical models of real-life processes pose challenges when used in numerical simulations, due to complexity and large size (dimension). Model Order Reduction aims to lower the

      Where ; 0 and ; 0 are known scalar constants.

      Let the transfer function of the reduced model of the order

      k be

      = =0 +1 +22…+ 1 1+

      computational complexity of such problems, for example, in simulations of large-scale control systems. By a reduction of the model's associated state space dimension or degrees of

      (2)

      0 +1 +2 2…+ 11+

      freedom, an approximation to the original model is computed. This reduced order model can then be evaluated with lower accuracy but in significantly less time.

      A model is often too complicated to be used in real life problems. It is an un-debated conclusion that, the development of mathematical model of physical system made it feasible to analyze and design. So the procedures based on the physical considerations or mathematical models are used to achieve simpler models than the original one. Whenever a physical system is represented by a mathematical model it may be a transfer function of very high order. Available methods for analysis and design may become cumbersome when applied to a system of higher order [1-4].

      The modeling of complex dynamic system is one of the most important subject in engineering .A High order system are not well suited for analysis and design. It becomes necessary to reduce the order of these high-order systems. Model order reduction technique plays an important role in various engineering applications, especially in large scale

      Where ; 0 and ; 0 are unknown scalar constants.

      The objective of this paper is to realize the kth-order reduced model in the form of (2) from the original system (1).

2. COMPUTATION OF ROM

   The reduction procedure for getting the kth-order reduced models consists of the following two steps:

   Step-1: Determination of the denominator polynomial using modified pole clustering technique [9]:

   The following rules are to be followed for clustering the poles of the original system given in frequency domain:

   • The number of cluster centers to be calculated is equal to the order of the reduced system.

   • The poles are distributed in to the cluster center for the calculation such that none of the repeated poles present in the same cluster center.

   • Minimum number of poles distributed per each cluster center is at least one. There is no limitation for the maximum number poles per cluster center.

   The procedural steps for realizing the denominator polynomial by using the modified pole clustering are:

   Let there be r real poles in ithcluster are 1, 2, where

   |1| < |2| < | |and then modified cluster center peican be obtained by using the algorithm of modified pole clustering suggested in this paper.

   Let mpair of complex conjugate poles in the jthcluster be

   [ 1 ± 1 , 2 ± 2 , . . . . . . . . ± ] where

   |1| < |2| < .

   Now using the same algorithm separately for real and imaginary parts of the complex conjugate poles, the modified cluster center is obtained, which is written as

   = ± ,

   where = + and = .

   Case-3: If some cluster centers are real and some are complex conjugate. For example, (k – 2) cluster centers are real and one pair of cluster center is complex conjugate, then kth- order denominator can be obtained as

   = 1 2 2 1 1

   (5)

   Hence, the denominator polynomial Dk(s) is obtained as Dk(s)=0+1s+……..+ sr (6)

   Step-2: Determination of the numerator of the reduced model by Error minimization technique:

   The obtained reduced order denominator polynomial is used here to obtain the values of reduced order numerator. The given higher order transfer function and the general form of reduced order transfer function are equated.

   0 +1 +2 2…+ 1 1+ = 0+1 +2 2…+ 1 1+

   For finding the modified cluster center, the below algorithm has to be followed:

   0 +1+2 2…+ 1 1+

   (7)

   0 +1 +2 2…+ 1 1+

   Step-1: Let r real poles in a cluster be

   |1| < |2| < | |.

   Step-2: Set j= 1 .

   Step-3: Find pole cluster center

   On cross multiplying the above equation and comparing the

   same powers of s on both sides, we get the following equations:

   00 = 00

   01 + 10 = 01 + 10

   1

   =

   1

   ÷

   02 + 11 + 20 = 02 + 11 + 20

   .

   Step-4:Set j = j+1.

   =1

   | |

   .

   .

   Step-5: Now find a modified cluster center from

   1

   = (8)

   1 1

   |

   = |1 | + | 1 ÷ 2

   Step-6:Is r = j? , if No, then go to step-4 otherwise go to step7.

   Step-7: Take a modified cluster center of the kthcluster as

   =

   After calculating the modified cluster centers, one of the following cases may occur:

   Case-1: If all modified cluster centers are real, then denominator polynomial of the kth-order reduced model can be obtained as

   = 1 2 . . . . . . . ( )

   (3)

   Where 1,2, . . . . . are 1 , 2 , . . . . . modified cluster center respectively.

   Case-2: If all modified cluster centers are complex conugate then the kth-order denominator polynomial is taken as

   On solving the above equations, we can find the unknown numerator coefficients d0,d1,.dq. Hence the reduced order numerator polynomial can be written as:

   = 0 + 1 + 22. . . +1 1 + (9)

3. SLIDING MODE CONTROL

   In control theory, sliding mode control, or SMC, is a control method that alters the dynamics of a system by application of discontinuous control signal that forces the system to "slide" along a cross-section of the system's normal behavior. The state-feedback control law is not a continuous function of time. Instead, it can switch from one continuous structure to another based on the current position in the state space. Hence, sliding mode control is a variable structure control method. The multiple control structures are designed so that trajectories always move toward an adjacent region with a different control structure, and so the ultimate trajectory will not exist entirely within one control structure. Instead, it will slide along the boundaries of the control structures. The motion of the system as it slides along these boundaries is called a sliding mode and the geometrical locus consisting of

   =

   1

   1

   . . .

   /2

   /2

   (4)

   the boundaries is called the sliding (hyper) surface.

   1. Linear Control Law

      The system is given by the equation

      = +

      (10)

      The sliding surface can be represented by

      s(,t)= S(t),where S ×

      Also, = +

      (11)

      Once the sliding surface is reached, then S(t)=0

      Therefore, from (7), the linear control laws can be obtained as

      = 1()

      (12)

      which is equivalent to linear control law equation

      = ()1

      (13)

      designed such that the states would remain on sliding surface.

      must be non-singular. Here is a design parameter and

      has a rank of m.

   2. Reachability of the sliding surface

   The second order system is given as

   + , = ()

   (14)

   Sliding mode control for this system may be selected as

   = 1 2 ( )

   (15)

   The linear part of this control input is state feedback law, which is in agreement with the structure of above equation and the discontinuous part is to be arranged so that the system is insensitive to disturbances.

4. PID CONTROLLER

   The PID controller is the most common form of feedback. It was an essential element of early governors and it became the standard tool when process control emerged in the 1940s. In process control today, more than95% of the control loops are of PID type, most loops are actually PI control. PID controllers are today found in all areas where control is used.

   The PID algorithm is described by:

   = + 1 + ()

5. NUMERICAL EXAMPLES

   The proposed method is explained by considering two numerical examples, taken from the literature. The goodness of the proposed method is measured by observing the step responses of the original and reduced model using MATLAB. The step response of the reduced model should be nearly same as that of original higher order system. [13, 14].

   Example-1: Consider an sixth-order system [9]

   35 + 1084 + 12213 + 57922 + 11860 + 8400

   = 6 + 415 + 5714 + 34913 + 100602 + 13100 + 6000

   The poles are: -1, -2, -3, -5, -10,-20.

   Let the 2nd -order reduced model is required to be realized, for this purpose only two real clusters are required.

   Let the first and second cluster consists the poles (-1, -2, -3) and (-5, -10, -20) respectively.

   The modified cluster centers are computed as Pe1=-1.1076

   Pe2=-5.5813

   Using equation (3), the denominator polynomial D2(s) is obtained as

   D2(s)=(s+1.1076)(s+5.5813)

   =s2+6.6889s+6.1818

   Using the equations (8) and (9), the following coefficients are calculated:

   do= 8.6545

   d1= 2.6881

   Therefore, finally 2nd order reduced model is obtained as

   2 2

   = 2.6881 + 8.6545

   + 6.6889 + 6.1818

   (16)

   0

   wherey is the measured process variable, r the reference variable, u is the control signal and e is the control error ( = ). The reference variable is often called the set point. The control signal is thus a sum of three terms: the P term (which is proportional to the error), the I term (which is proportional to the integral of the error), and the D term (which is proportional to the derivative of the error). The controller parameters are proportional gain K, integral time Ti, and derivative time Td.

   Fig. 1. Comparison of the step responses for the given example 1

   TABLE I. COMPARISON WITH OTHER EXISTING METHODS

   Method	Reduced model	ISE
   Proposed method	2.6881 + 8.6545 2 + 6.6889 + 6.1818	0.00019
   Pade-Routh method	0.992 + 1.4017 0.2432 + 1.1434 + 1.0006	0.97

6. EXTENSION FOR DESIGNING OF SLIDING MODE CONTROLLER

   Example 2:Consider a fourth order system described by the transfer function [3]

   72 + 54 + 122 + 3

   = 100 + 180 + 972 + 183 + 4

   First we have to find the reduced denominator polynomial by using modified pole clustering technique:

   Using step-1, two cluster centres from the real poles -1,-2,- 5 and -10 can be formed. The first cluster contains two poles

   i.e -1 and -2. The second cluster contains remaining two poles

   i.e -5 and -10.

   By using the modified cluster center algorithm steps, we get Pe1=-1.1416

   Pe2=-5.7142

   Therefore, by using equation (3) we get D2(s) = (s+1.1416) (s+5.7142)

   =s2+6.8558s+6.5233

   Next for the reduced order numerator coefficients in error minimization technique, using the equations (8) and (9), the following coefficients are calculated:

   do=4.69 d1=0.0045

   Therefore, finally 2nd order reduced model is obtained as

   2 2

   = 4.69 + 0.0045

   6.5233 + 6.8558 +

   For this 2nd order model, a sliding mode controller is designed with a control input which is calculated by the given procedural steps from equations (10)-(15) and is obtained as

   = 171.3951 163.082 (51 + 2)

   This control input is given to the reduced model and a Simulink block is designed for the sliding mode controller.

   Step input is given to the original higher order system, reduced order model, ROM with sliding mode controller as well as PID controller and their responses are shown in the figure 2.

   Fig. 2. Comparison of the step responses for the given example 2 TABLE II. COMPARISON WITH OTHER EXISTING METHODS

   Method	Reduced model	ISE
   Proposed method	4.69 + 0.0045 6.5233 + 6.8558 + 2	0.000025
   Pade-Pole clustering method	6.3920 + 0.9523 8.8778 + 7.9999 + 2	0.002025

7. CONCLUSION

   In this proposed order reduction method for the linear single-input-single-output high order systems, the determination of denominator polynomial of the reduced model is done by using the modified cluster method while the numerator coefficients are computed by error minimization technique. The proposed algorithm has been explained with a numerical example. The step responses of the original and reduced system of second order are shown in the Figure 1 and

   2. This method is simple, proficient and takes little computational time. This method secures the stability and avoids steady-state error between the step responses of the original and reduced modls. The results of SMC for second order system show the stabilized performances and robustness of SMC.

8. REFERENCES

1. V. Singh, D. Chandra and H. Kar, Improved Routh-Pade approximants: A Computer aided approach, IEEE Trans. Autom. Control, 49(2),pp.292-296, 2004.

2. S.Mukherjee and R.N. Mishra, Reduced order modeling of linear multivariable systems using an error minimization technique, Journal of Franklin Inst., 325 (2), pp.235-245, 1988.

3. Sastry G.V.K.R Raja Rao G. And MallikarjunaRao P., Large scale interval system modelling using Routh approximants, Electronic Letters, 36(8), pp.768-769, 2000.

4. R. Prasad, Pade type model order reduction for multivariable systems using Routh approximation, Computers and Electrical Engineering, 26,pp.445-459, 2000.

5. G. Parmar, S. Mukherjee and R. Prasad, System reduction using factor division algorithm and eigen spectrum analysis, Applied Mathematical Modeling, Elsevier, 31, pp.2542-2552, 2007.

6. R. Prasad and J. Pal, Stable reduction of linear systems by continued fractions, Journal of Institution of Engineers IE(I) Journal-EL, 72,pp.113-116, 1991.

7. R. Prasad, S.P. Sharma and A.K. Mittal, Improved Pade approximation for multivariable systems using stability equation method, Journal of Institution of Engineers IE (I) Journal-EL, 84, pp.161-165, 2003.

8. A.K. Sinha, J. Pal, Simulation based reduced order modeling using a clustering technique, Computer and Electrical Engineering., 16(3), 1990, 159-169.

9. J. Pal, A.K. Sinha and N.K. Sinha, Reduced-order modelling using pole clustering and time-moments matching, Journal of The Institution of Engineers (India), Pt El, 76,1995, 1-6.

10. Sastry,G.V.K.R and Krishnamurthy.V,State Space Model Reduction using SRAM Electronic Letters,Jrl.of IEE(U.K),vol 22,pp 1300-1320,1987.

11. Sastry .G.V.K.R, Raja Rao.G,andGabriel.T,Large scale MIMO systems reduction using Pade Polynomial Derivative Techniques,ASME International Conference on Applied Modelling and Simulation,Banff,CANADA,1997.

12. Nagendra.N, SudheerKumar.M and Madhubabu.T. Design of Discrete Controller via Novel Model Order Reduction Method, International Journal of Engineering Research and Applications, Vol. 2, No.4, pp.2331-2334, 2012.

13. Mr.M.Sudheer Kumar ,Asst Prof, Mr.N.Nagendra and Mr.T.Madhu Babu,2/2 M.E(C.S) have published a paper on A Novel Mixed Method for Model Order Reduction of Linear Systems using PSO and polynomial method in the proceedings of International Conference on Electrical and Electronics Engg,pp 175-178 held on 9th June 2012 at Tirupathi.

14. Mr.M.SudheerKumar,AsstProf,Dept of EEE has presented a paper on A New Mixed Method for the Order Reduction of Linear Continuous Systems Using Mihailov Criterion and Factor Division Method for MIMO systems in the National Conference NCACEE-11 at RIT,Pondicherry,Yanam on 11th November 2011.

M.Prajeeth received the B.Tech degree in Electrical and Electronics Engineering from P.V.P Siddhartha Institute of Technology, Vijayawada, A.P, India in 2012. He is pursing

M.E (control Systems) in the department of Electrical and Electronics Engineering, ANITS Engineering College, Visakhapatnam, AP, and India. His research interests include large scale system modelling, advanced control systems, Non-linear control, Modelling and simulation.

M.Sudheer Kumar received the B.Tech degree in Electrical and Electronics Engineering from Chaitanya Engineering College,Kommadi,Visakhapatnam, A.P, India in 2008. He has completed M.E (Power Electronic Drives and Control) in the department of Electrical and Electronics Engineering, A.U, Visakhapatnam, AP, and India in 2012. His research interests include Large scale system modelling & design and Advanced control systems.