Fractional Order Modeling of Continuous HighOrder MIMO Systems Sunil Kumar Mishra
Dinesh Chandra
Electrical Engineering Department Motilal Nehru National Institute of Technology Allahabad, India
[email protected]
Electrical Engineering Department Motilal Nehru National Institute of Technology Allahabad, India
[email protected]
Abstract: In this work, fractional order modeling as well as integer order modelling of continuous multi-input multi-output (MIMO) system is presented. The parameters of reduced order model (ROM) are obtained by minimizing the objective function which is formed in terms of integral-square-error (ISE) pertaining to step input. In this method the transfer matrix of a MIMO system is considered and PSO has been used for minimizing the objective function. FOMCON toolbox is used for calculating responses of fractional order systems. One numerical example is included to illustrate the proposed approach. Keywords: FOMCON toolbox, Fractional order modeling, ISE, MIMO system, PSO.
I. INTRODUCTION In recent years, evolutionary techniques are used to obtain reduced-order model (ROM) of high-order system. It is also shown in literature that fractional order representation of a system is more realistic than its integer order representation. In principle, if the mathematical model of the high-order system has been established, the output response to the typical inputs are required to be worked out as well as the system characteristics. For higher order systems, implementation, analysis, simulation and various system designs are difficult and complex. So we prefer to use the lower order models, which are easier to implement and to analyze for various system designs. Model order reduction (MOR) is a technique to reduce the order of high order system and produces a reduced-order model such that the characteristic response of reduced-order model is almost same as of high-order system. Different techniques for order reduction of linear continuous MIMO system are available [1-6]. The conventional methods of reduction are widely available in continuous as well as in discrete domain. Most of the methods based on the original continued fraction expansion technique [7-10] fail to retain the stability of the original systems in the reduced order models. In [11], a numerically robust Relative Error Method (REM) for state-space model order reduction is described. In recent years, evolutionary techniques are popular within research community due to their versatility and ability to optimize complex problems. Particle swarm optimization [12] (PSO) is an alternative of the best optimization technique for continuous as well as discrete time system. PSO [13], 978-1-4673-6190-3/13/$31.00 ©2013 IEEE
Genetic algorithm [14-15] are used for MOR. Also in recent studies, fractional calculus has been used to represent control system applications [16-20]. The significance of fractionalorder representation comes from its nature. The fractionalorder differential equations can describe real-world systems more adequately. In [21] a 500 MWe Canadian Deuterium Uranium (CANDU) type Pressurized Heavy Water Reactor (PHWR) is modeled using few variants of Least Square Estimator (LSE) and also fractional order modeling is attempted for reducing the estimated higher order model. In this paper, fractional order modeling as well as integer order modeling of continuous high-order MIMO system using PSO is presented. The transfer matrix of reduced-order MIMO system is obtained by minimizing objective function which is constructed in the form of ISE [9]. In PSO the movement of particle is governed by three behaviors Inertia, Cognitive and Social. Inertia governs the velocity of particle, cognitive helps to local search and social helps to global search. The coefficients of the denominator and numerator polynomials of the ROM are found by two steps. First we choose a transfer function from the transfer matrix and PSO is used to find the denominator and numerator coefficients of ROM using objective function. The denominator coefficients of ROM are retained for further process. In second step, the proposed algorithm is used to find numerator coefficients of remaining transfer functions of transfer matrix. A 6-th order MIMO system is reduced into fractional order model (close to 2nd order model) as well as integer order model (2nd order model) using PSO. The PSO algorithm is used because of its good convergence capability and simple algorithm. All responses for fractional order systems have been calculated using FOMCON toolbox [22] of MATLAB/SIMULINK. Fractional order expression is converted to integer form using oustaloup approximation [17] for each iteration. This work is organized as follows: section-II covers problem statement, section-III presents particle swarm optimization, section-IV contains a numerical example to illustrate the procedure, while the conclusion is provided in section-V.
II. PROBLEM STATEMENT Consider an n-th order MIMO system with p inputs and q outputs described in transfer matrix as:
⎡ b1 1 ( s ) ⎢b (s) 1 ⎢ 21 [G ( s )] = Dn (s) ⎢ # ⎢ ⎢⎣ b q 1 ( s )
Where
b1 2 ( s ) b22 ( s ) # bq 2 ( s )
" " # "
b1 p ( s ) ⎤ b1 2 ( s ) ⎥⎥ # ⎥ ⎥ b q p ( s ) ⎥⎦
(1)
Dn ( s ) is the common denominator and bij are the
individual numerator of MIMO system.
ISE = ∫
The general form of gij ( s) of [G ( s) ] in (1) is taken as: g ij =
bij ( s )
where bij ( s) = b1 ( s) n −1 + b2 ( s) n − 2 + " + bn Dn ( s ) = a0 ( s ) n + a1 ( s ) n −1 + " + an
This system is first reduced to fractional order model (close to rth order model) with p inputs and q outputs described by the transfer function as: ⎡ h11 ( s ) ⎢ 1 ⎢ h 21 ( s ) R fractinal ( s ) = Dr (s) ⎢ # ⎢ ⎣⎢ h q 1 ( s )
h12 ( s ) h22 ( s ) # hq 2 ( s )
h1 p ( s ) ⎤ h12 ( s ) ⎥⎥ # ⎥ ⎥ h qp ( s ) ⎦⎥
" " # "
(3)
The general form of rij ( s ) of [ R fractional ( s )] is taken as: rij ( s ) =
hij ( s )
(4)
Dr ( s )
where hij ( s) = h1 ( s) r − 2+α 1 + h2 ( s) r −3+α 2 + " + hr Dr ( s ) = a0 ( s ) r −1+ β 1 + a1 ( s ) r − 2 + β 2 + " + ar
Dr ( s ) is the fractional order denominator polynomial of the transmission matrix for the reduced-order model. The values of α 1, α 2,.... and β 1, β 2...... lies between 0 and 1. Now, this system is also reduced to r -th order model (integer order model) with p inputs and q outputs described by the transfer function as: ⎡ h11 ( s ) ⎢ 1 ⎢ h21 ( s ) R int eger ( s ) = Dr (s) ⎢ # ⎢ ⎣⎢ hq1 ( s )
h12 ( s ) h22 ( s ) # hq 2 ( s )
" " # "
h1 p ( s ) ⎤ h12 ( s ) ⎥⎥ # ⎥ ⎥ hqp ( s ) ⎦⎥
∞
0
(2)
Dn (s )
Step-1: First a transfer function from the transfer matrix is considered and PSO is employed to find the denominator and numerator coefficients of ROM by minimizing objective function. The objective function is the integral square error (ISE) given by
(5)
[ y(t ) − yr (t )] dt 2
(7)
where y(t ) and yr (t ) are, respectively, the step responses of high-order model and reduced-order system. Step 2: The denominator coefficient of ROM is found from the step 1. Now, PSO algorithm is used to found numerator coefficients of remaining transfer functions of transfer matrix. Here also the PSO is used to minimize the ISE by keeping the values of the denominator coefficients, which are found out in step 1. III. PARTICLE SWARM OPTIMIZATION METHOD The PSO method [12] is a wide category of swarm intelligence methods for solving the optimization problems. This technique inspires by bird flocks, fish schools, and animal herds constitute representative examples of natural systems where aggregated behaviors are met, producing impressive, collision-free and synchronized moves. The PSO is population based stochastic optimization and the major advantage of the PSO over other stochastic optimization methods is its simplicity. It is a population based search algorithm. The basic idea of the PSO is the mathematical modeling and simulation of the solution searching activities in multi dimension search space where optimal solution exists. In a PSO system, particles fly around in a multi dimensional search space. During flight each particles adjust its position according its own experience and the experience of the neighboring particles. Update velocity and position of particle are depended upon inertia, cognitive and social factors. Each particle has a memory and hence it is capable of remembering the best previous position in the search space ever visited by it. The position corresponding to the best fitness is known as pbest and the overall best out of all the particles in the population is called gbest . The modified velocity and position of each particle can be calculated using the current velocity and position as follows: Velocity update equation is given by Vi k +1 = W × Vi k + C1 × rand () × ( pbest ,i − X i k ) + C2 × rand () × ( gbest ,i − X i k )
The general form of rij ( s ) of [ Rint eger ( s )] is taken as: rij =
hij ( s )
Dr ( s ) where hij ( s) = h1 ( s) r −1 + h2 ( s) r − 2 + " + hr
(8) (6)
Dr ( s ) = a0 ( s ) r + a1 ( s ) r −1 + " + ar
Dr ( s ) is the r -th order denominator polynomial of the transmission matrix for the reduced-order model. The coefficients of the denominator and numerator polynomials of the ROM are found by the following steps:
Position update equation is given by X
i
k +1
= X
k i
+ V i k +1
(9)
Where, k = number of iterations v = velocity of particle w = inertia weight factor
c1 , c2 = cognitive and social acceleration factors respectively rand () = random numbers uniformly distributed in the range (0, 1)
x = position of particle i = i -th particle.
Step Response sys_orig sys_red 1
0.8
Amplitude
IV. NUMERICAL EXAMPLE To demonstrate the proposed method, a high-order ( 6-th order) system is taken from literature and the proposed algorithm is employed to obtain 2-nd order model. Consider a system [7-9] having two inputs and two outputs described by the transfer matrix: 1 ⎛ b1 1 G (s) = ⎜ D 6 ( s ) ⎝ b2 1
b1 2 ⎞ ⎟ b2 2 ⎠
(10)
0.6
0.4
0.2
0 0
1
2
3
4
5
6
7
8
9
10
Time (sec)
Fig. 1. Step response of high-order system (g11) and reducedorder model (r11).
Where
Step Response
D6 ( s ) = s6 + 41 s5 + 571 s4 + 3491 s3 + 10060 s2 + 13100 s + 6000
0.45 0.4
(11)
b11 ( s ) = 2 s + 70 s + 762 s + 3610 s + 7700 s + 6000 3
2
b12 ( s ) = s5+ 38 s4+ 459 s3+ 2182 s2+ 4100 s + 2400 b21 ( s ) = s5+ 30 s4+ 331 s3 + 1650 s2+ 3700 s + 3000 b22 ( s ) = s5+ 41 s4+ 601 s3 + 3660 s2+ 9100 s + 6000
sys_red
0.3
(12) (13)
Amplitude
4
0.25
0.2
0.15
(14)
0.1
0.05
(15)
where D6(s) is the common denominator polynomial and b11(s), b12(s), b21(s) and b22(s) are numerator polynomials of the systems. The proposed algorithm is applied to minimize ISE given in (7). A. Fractional Order Model (Close to 2nd Order Model) The coefficients of the denominator and numerator polynomials of the ROM are found by following steps: Step-1: First let a transfer function g11 from the transfer matrix and employ PSO to found the denominator and numerator coefficients of ROM using objective function. The objective function is minimizing ISE. D2(s) = 1.8670s^1.4575+1.8331s^0.7387 +1.2584 (14) h11(s) = 1.4530 s^0.7533 +1.2584 (15)
0
0
1
2
3
4
5
6
7
8
9
10
Time (sec)
Fig. 2. Step response of high-order system (g12) and reducedorder model (r12). Step Response 0.6
0.5 sys_orig 0.4 Amplitude
5
sys_orig
0.35
sys_red
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
Time (sec)
Fig. 3. Step response of high-order system (g21) and reducedorder model (r21). Step Response sys_orig sys_red 1
0.8
Amplitude
Step 2: The denominator coefficient of ROM is found from the step 1. Now, PSO algorithm is used to found numerator coefficients of remains transfer function of transfer matrix. Here also the PSO is used to minimize the ISE by keeping the values of the denominator coefficients, which are found out in step 1. h12(s) = 0.8841 s^0.8573 + 0.5034 (16) h21(s) = 0.5893 s^0.6905 + 0.6292 (17) h22(s) = 1.8444 s^0.8001 + 1.2584 (18) The step responses of high-order system and reduced-order models (fractional order models) are shown in Fig 1-4.
0.6
0.4
0.2
0
0
1
2
3
4
5
6
7
8
9
10
Time (sec)
Fig. 4. Step response of high-order system (g22) and reducedorder model (r22). B. Integer Order Model (2nd Order Model)
The coefficients of the denominator and numerator polynomials of the ROM are found by following steps:
Step Response
0.5 sys_orig sys_red 0.4
Amplitude
Step-1: First let a transfer function g11 from the transfer matrix and employ PSO to found the denominator and numerator coefficients of ROM using objective function. The objective function is minimizing ISE. D2(s) =0.1994 s2+ 0.9172 s + 0.6849 (19) h11(s) = 0.2926 s + 0.6894 (20)
0.3
0.2
0.1
0
Step 2: The denominator coefficient of ROM is found from the step 1. Now, PSO algorithm is used to found numerator coefficients of remains transfer function of transfer matrix. Here also the PSO is used to minimize the ISE by keeping the values of the denominator coefficients, which are found out in step 1.
0
1
2
3
4
5
6
7
8
9
10
Time (sec)
Fig. 7. Step response of high-order system (g21) and reducedorder model (r21). Step Response sys_orig sys_red
(21) (22) (23)
The step responses of high-order system and reduced-order models (integer order models) are shown in Fig 5-8. A comparison of proposed method with the other well-known order-reduction techniques available in literature is given in the Table I. The comparison is made by calculating the ISE. The ISE is calculated for each element of the transfer function matrix of the reduced-order model with respect to the corresponding high-order system. Step Response sys_orig sys_red
0.8
Amplitude
h12(s) = 0.2282 s + 0.2740 h21(s) = 0.1279 s + 0.3424 h22(s) = 0.4000 s + 0.6849
1
0.6
0.4
0.2
0
0
1
2
3
4
5
6
7
8
9
10
Time (sec)
Fig. 8. Step response of high-order system (g22) and reducedorder model (r22). Table 1 Comparison of ISEs Reduction method r11 (s ) r12 ( s ) r21 ( s ) r22 ( s)
1
Proposed method
7.3055 x
3.5472 x
9.3971 x
(fractional order)
10-4
10-5
10-5
Proposed method
1.7309 x
2.1225 x
3.7400 x
(integer order)
10-4
10-4
10-5
C. B. Vishwakarma
0.001515
7.845
0.000299
0.004681
0.007328
1.066123
Amplitude
0.8
0.0083
0.6
0.4
0.0101
0.2
0
0
1
2
3
4
5
6
7
8
9
×10−5
and Prasad[15]
10
Time (sec)
Fig. 5. Step response of high-order system (g11) and reducedorder model (r11).
Safonov and
0.590617
0.037129
chiang[11]
Step Response 0.45 0.4 sys_orig 0.35
V. CONCLUSION
sys_red
Amplitude
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
Time (sec)
Fig. 6. Step response of high-order system (g12) and reducedorder model (r12).
An approach based on the ISE minimization employing PSO has been presented to derive reduced-order models for linear time invariant MIMO dynamic systems. The parameters of reduced-order models are derived by minimizing the objective function, which is ISE, formed in terms of errors between step responses of high-order systems and reduced-order models. The PSO algorithm is used because of its good convergence capability and simple algorithm. In this work two methods have been proposed. The first method is related to fractional order modeling (close to 2nd order model) and the second method is related to integer order modeling (2nd order model) of a continuous high order system
(6th order). The systematic procedure of proposed method is illustrated by a numerical example. The step response of reduced-order model is almost same as of high-order system shown in fig. 1-4 for reduced fractional order system and in fig. 5-8 for reduced integer order system. In table 1 comparison of ISE values of proposed model with other methods which are given in literature. It is also shown in table-1 that proposed methods perform better than other methods. In this work only PSO is used for calculating the optimized parameters of the reduced order models but comparison of this technique with other techniques might be the subject of future work.
[11].
[12].
[13].
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