A MATLAB Toolbox for Teaching Model Order Reduction Techniques Authors: Ali Eydgahi, Department of Engineering and Aviation Sciences, University of Maryland Eastern Shore, Princess Anne, MD 21853,
[email protected] Jalal Habibi, Department of Electrical and Computer Engineering, University of Tehran, Tehran, I.R. IRAN,
[email protected] Behzad Moshiri, Control & Intelligent Processing, Center of Excellency, ECE Department, Faculty of Engineering, University of Tehran, Tehran, I.R. IRAN,
[email protected]
Abstract This paper presents a MATLAB-based toolbox with a Graphical User Interface (GUI), which can be used to compute reduced models of a large system by using one of the twenty order reduction techniques available in the toolbox. The methods that have been implemented in the toolbox include the Padé, Routh, Cauer, Continued-Fraction Expansion, and algorithms that provide mixtures of these techniques. Upon execution of the toolbox, a GUI will appear with four frames named “Methods”, “High Order System”, “Output Options”, and “Results”. In “Methods” frame, one or more of the reduction techniques can be selected. The high order system is defined by its transfer function numerator and denominator in “High Order System” frame. In this frame, a number of high order benchmark systems, in a popup menu, are provided that can be selected by the user for testing purposes. In “Output Options” frame, the order of the reduced model and the types of the desired output plots are selected. The impulse response and step response are available as an output. In the output plot, the software shows and compares the reduced models responses with nominal system response. The graphs of step and/or impulse responses will be appeared in two different windows. The numerator and denominator of the computed reduced order models will be shown in “Results” frame. Index Terms Control systems, graphical user interface, MATLAB toolbox, model order reduction, simulation. INTRODUCTION The modeling of complex dynamic systems is one of the most important subjects in engineering. A model is often too complicated to be used in real problems, so approximation procedures based on physical considerations or mathematical approaches are used to achieve simpler models than the original one. The subject of model reduction is very important to engineers and scientists working in many fields of engineering, especially, for those who work in the process control area. In control engineering field, model reduction techniques are fundamental for the design of controllers where particular numerically heavy procedures are involved. This would provide the designer with low order controllers that have less hardware requirements. Efforts towards obtaining low-order models from high-degree systems are related to the aims of deriving stable reduced-order models from stable original ones and assuring that the reduced-order model matches some quantities of the original one. In past decades, many papers in the literature have efficiently addressed these two objectives[1]-[27]. Polynomial reduction methods are one of the important groups of the reduction techniques that are applied in the frequency domain. They are used to approximate low-order transfer functions where the model coefficients are chosen according to various criteria. The frequency domain model-reduction techniques that are mainly based on polynomial manipulations are Padé-type approximations, continued fraction expansion methods, model reduction using the Routh stability criterion, model reduction based on the Routh table criterion, model order reduction algorithms using stability equations, mixed methods of approximation, and energy-based methods. Popular software packages such as MATLAB, MAPLE, MATHEMATICA, and PSPICE are essential tools for analysis and design of an engineering system. The benefits of using these packages has been confirmed by the number of new and revised textbooks that have incorporated new exercises and problems based on them. These packages have been found to greatly enhance readers understanding of the materials and can prepare them for design and development of more complex system, much like the ones that are seen in industry. In this paper, a MATLAB-based toolbox for model order reduction with a Graphical User Interface is presented. The toolbox has options for twenty order reduction methods such as Padé approximation, Cauer continued-fraction expansion,
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Routh stability criterion, differntiation method, and a number of different mixed methods of frequency techniques. Some MATLAB codes of order reduction methods have been adopted from [1].
MODEL ORDER REDUCTION METHODS In this section, brief description of the order reduction techniques that have been implemented in the toolbox along with their advantages and disadvantages are presented. Padé technique[3]: This approach is based on matching a few coefficients of the power series expansion, about s = 0 , of the reduced-order model with the corresponding coefficients of the original model. The power series expansion of a transfer function about s = 0 is an expansion of the form:
H ( s ) = c0 + c1 s + c 2 s 2 + L
(1)
This method is computationally simple and provides perfect agreement between the steady state of the output of the system and the model for polynomial inputs belonging to the class of form ∑ α i t i . Serious drawback of Padé-based approximants is that unstable reduced-order models can be generated from original asymptotically stable high order system. Cauer first, second, and third form[4-6]: Continued-fraction expansion method is one of the most attractive methods for the order reduction of transfer functions. It has many useful properties such as computational simplicity, the fitting of the time moments, and the preservation of the steady-state responses for polynomial inputs of the form ∑ α i t i . There are three basic forms for continued fraction representation of a transfer function. Each form concentrates on a different frequency range for the approximation of the original transfer function. They are referred to as the three Cauer forms. The Cauer first form is represented as:
1
H ( s) = f1 s +
(2)
1 f2 +
1 f3s +
1 f4 +
1 O
Reduced order model can be obtained by truncating the last terms of expansion. It has been shown that the reduced order model obtained from the Cauer first form cannot preserve the steady state value of the original model. The restriction to the original transfer function is that it requires the order of the numerator to be one less than that of the denominator. Otherwise, the Cauer first form does not exist. The reduced order model from the Cauer first form is the matching of the original transfer function by the Markov parameters, which is the power series expansion over 1 s . The Cauer second form is given as:
1
H ( s) = h1 +
(3)
1 h2 + s
1 h3 +
1 h4 1 + s O
It has been shown that this kind of continued-fraction is equivalent to the Maclaurin expansion about s = 0 . This form mathes the steady state reponses, but does not provide a good match for the initial transient response. The Cauer third form also referred to as the mixed Cauer expansion combines the Cauer first and second forms. It is given by:
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1
H ( s) = h1 + f1 s +
(4)
1 h2 + f2 + s
1 h3 + f 3 s +
1 h4 1 + f4 + O s
The Cauer third form can only be applied when the order of the numerator of the original transfer function is one less than that of the denominator. The mixed form is equivalent to the Maclaurin series about s = 0 and s = ∞ . The reduced order model matches the first m time moments and m Markov parameters of the original model, where m is the order of reduced order model. This method has one serious disadvantage. It may produce unstable (non-minimum phase) reduced models even though when the original high-order system is stable (minimum phase). Order reduction based on stability equation[7]: This method is based on forming stability equations of the numerator and denominator polynomials of high-order system, discarding the non-dominant poles and zeros, and obtaining the reducedorder model. This approach preserves the stability of reduced-order model. The disadvantages are related to the fact that this method is founded on heuristic considerations and besides the structural stability that is preserved in the reduced-order model, further correlations between the original and the reduced model are lost. Differentiation method[8]: In this approach, the reciprocals of numerator and denominator polynomials of the highorder transfer function are differentiated suitably many times to yield the coefficients of the reduced order transfer function. This method is computationally simple and is equally applicable to unstable and nonminimum phase systems. Aggregation method[9]: The aggregation method is based on the concept of combining a defined set of state variables of the original system, involving chosen weighting factors. The set of state variables (z ) , often computed from the original
(x) , using an aggregation matrix (L) , by z = Lx . The adequacy of finding the suitable aggregation state vector, and therefore the matrix L , depends upon the objective that the designer wants to reach such as open-loop or system state vector
closed-loop performance, optimisation of certain indexes, etc. Order reduction based on the Routh stability criterion[10]: The system with transfer function M (s ) can be expanded
β parameters as: M ( s ) = β1 f1 ( s ) + β 2 f1 ( s ) f 2 ( s ) + L + β n f1 ( s ) f 2 ( s )L f n ( s ) (5) where n is the degree of high-order system, β i are constants, and the functions f i (s ) are defined as: 1 fi ( s) = (6) 1 αi s + 1 α i +1 + 1 O 1 α n −1s + αns the first term is 1 (1 + α1s ) instead of 1 α1s . Order reduction based on the Routh stability criterion is to compute α i and β i parameters and , then, reconstructing the reduced-order models with certain algorithms after dropping out a few α i and β i parameters. This method cannot be applied to non-asymptotically stable systems. into the so-called α and
SRAM (Simplified Routh Approximation Method)[25-27]: This method obtains the reduced-order models using only
α parameters, where α parameters are defined above. The simulated version of SRAM in this toolbox also retains some time moments of the full system in the reduced model. Order reduction based on real Schur form decomposition[11]: Model order reduction, based on measures of controllability and observability, has been proposed by Moore[12] and extended and generalized by other researchers. In this approach, the system is transformed using balancing transformation such that the input-to-state coupling and the state-tooutput coupling are weighted equally so that state components which are weakly coupled to both input and output are discarded and ones which are strongly coupled to both input and output are retained and used as a low-order approximation for the high order system.
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Calculating the balancing transformation in Moore’s method, tends to be ill-conditioned, especially when the original system is non-minimal or when it has very nearly uncontrollable or unobservable modes. So, an alternative algorithm has been proposed based on finding the eigenspaces associated with large eigenvalues of the cross-gramian matrix using the real Schur-form decomposition. This method avoids calculating the balancing transformation. Stable Padé[13]: This approach combines the Routh-Hurwitz array method and the Padé approximation technique to obtain stable reduced-order models. It is computationally efficient and does not require computation of the poles of the original system for dominant pole retention.This method is not the only method that solves the stability problem of Padé approximation. Condensed continued-fraction method[14]: Much attention has been diverted to overcoming “stability” problem of order reduction techniques based on continued-fraction expansion. Condensed CFE proposed to improve the chances of stability in the reduced model. One of the main causes of instability in the reduced-order models obtained by Cauer continued fractons is thought to be the dependence of the reduced denominator on the full numerator. This is especially true if the numerator has zeros in the right half-plane. Condensed CFE method proposed to use Cauer CFE only on 1 D ( s ) , (where D(s ) is the denominator) instead of G ( s ) = N ( s ) D ( s ) . The effect of
N (s) on the reduced denuminator is thus removed, and the k th-order approximants k −1 of 1 D( s ) can then be multiplied by N (s ) , truncating the numerator after the term in s . This latter operation ensures that the first k time moments are preserved in the reduced model. However it should be remembered that unstable poles
might still occur, and experiences have shown that lightly damped, highly oscillatory systems tend to cause most of the problems. Moment matching technique[15]: The moment matching technique is in fact the matching between the time moments of the impulse response of the original model with those of the reduced order model. This reduction method is not restricted to rational transfer functions. Since other functions, once written in the Maclaurin form, can also be reduced using the moment matching method. This approach, though simple in theory and computations, seems to have serious drawbacks. For example stable reduced-order models may be obtained by high-order unstable system. Truncation approach[16, 17]: From the viewpoint of condensed CFE order reduction approach, if [ p, q ] is taken to be the Padé approximant about s = 0 whose numerator and denominator degrees are
p and q respectively, then the following
k -th order approximants of 1 D( s ) . If 1 1 (7) = D( s ) a 0 + a1 s + L + a n s n then the [m, k ] approximant of 1 D ( s ) , where 0 ≤ m ≤ k − 1 , is equivalent to the two-point Padé/continued-fraction approximant of 1 D ( s ) matching ( k + 1 + m) terms about s = 0 and ( k − 1 − m) terms about s = ∞ . This follows from the fact that the first (n − 1) terms of 1 D ( s ) about s = ∞ are all zero. In the special case of m = 0 , let the approximant be given by 1 (8) e + e1 s + L + ek s k result holds for a
s k . By equating the two series it is easily seen that ei = a i for i = 0,1,2, K , k . Forming the final reduced transfer function by multiplying N (s ) and truncating after
The series expansion of this must be identical to that of 1 D ( s ) for all terms in k −1
the term in s shows that the model is identical to that obtained by simple truncation of terms in the full denominator and numerator, respecively.Yeung in [17] has shown that for the truncation method if the poles of the system are well damped (i.e. |real part| > |imaginary part|), then stability is guaranteed in the reduced model. Factor Divivision (Padé with dominant mode retention)[18]: This approach obtains the numerator of the reducedmodel in a direct way. It avoids to find system time moments and does not solve the Padé equaions, but obtains the reduced models that still retain the initial time moments of the full system. This method retains the dominant modes of the system and has been claimed to be often quicker to use than the similar method of [19] at least when the poles of the high-order system are known. Mixed method 1[20]: Generally speaking, mixed methods are developed to ensure the stability of reduced models. They often use criteria which assure the stability of the reduced-order model if the original system has this property. The numerator is obtained by using Padé method or the continued-fraction expansion. International Conference on Engineering Education
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In first mixed method, the denominator of the reduced-order transfer function is formed by the dominant poles of highorder system, while the numerator is obtained by fitting the first r time moments of reduced-order system and the high-order one. Mixed method 2[21]: In this approach, Routh stability criterion is used to compute the denominator of the reduced model. The numerator is obtained as in the first mixed method. This approach avoids the drawback of the first method in its need to the computation of system poles. Mixed method 3[22]: This method is based on determining the denominator using the Routh array method. The numerator is obtaind by using Padé approximant technique. Mixed method 4[23]: This method takes into account both the adavantages of the stability-equations method for obtaining a reduced-order stable system, and the Padé method for finding the numerator. So this way it fits a number of the Markov parameters and time moments of the original transfer function. Mixed method 5[24]: In this method, the differentiation procedure[8] is used to obtain the reduced-order denominator, thus ensuring stability preservation, while the numerator is obtained by matching the first MacLaurin expansion coefficients of the original system and model transfer functions using Padé method.
DESCRIPTION OF THE TOOLBOX Upon execution of the toolbox, a GUI will appear with four frames named 'Methods', 'High Order System', 'Output Options', and 'Results'. Figure 1 shows the workspace of GUI. In 'Methods' frame, one or more of the reduction techniques can be selected. The high order system is defined by its transfer function numerator and denominator in 'High Order System' frame. In this frame, a number of high order benchmark systems, in a popup menu, are provided that can be selected by the user for testing purposes. In 'Output Options' frame, the order of the reduced model and the types of the desired output plots are selected. The impulse response and step response are available as an output. In the output plot, the software shows and compares the responses of the reduced models with nominal system response. By clicking 'Run' push button, the software will start the computation of the reduced order model. Step and/or impulse responses graphs will be appeared in two different windows. The numerator and denominator of the computed reduced order models will be shown in 'Results' frame. Computing another model or changing some coefficients of the original model can easily be done in GUI. A 'Reset' push button is also provided to clear all frames and start computations for a new system. An example is provided to demonstrate the performance of the order reduction algorithms of the toolbox.
EXAMPLE Here we present an illustrative example to show the operation of the order reduction toolbox. Some methods inherently put some restrictions on high order system. For example Cauer first and third forms restrict the relative degree of high order system to be equal to one. So, as notified previously, some standard high order systems have been considered in the toolbox as benchmark. Assume the high-order system from benchmark of the form:
H ( s) =
s 4 + 35s 3 + 291s 2 + 1093s + 1700 s 9 + 9 s 8 + 66s 7 + 294s 6 + 1029s 5 + 2541s 4 + 4684s 3 + 5856s 2 + 4629s + 1700
(9)
and let’s use Cauer second form, Routh stability criterion, Schur decomposition, and condensed CFE methods to find the reduced-order models of order 5. Figure 2 shows the impulse responses for the nominal system and for reduced-order models obtained using selected methods. It can be seen that the Schur decomposition method produces most accurate model.
REFERENCES [1] Fortuna, L., Nunnari, G. and Gallo, A., "Model order reduction techniques with application in electrical engineering", Springer-Verlag, 1992. [2] Eydgahi, A.M., “A mixed method for the simplification of high order systems”, Proceedings of American Control Conference, Atlanta, GA, June 15-17, 1988, pp. 1064-1067.
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[3] Bultheel, A. and Van Barel, M., " Padé techniques for model reduction in linear system theory: a survey", Journal of Computational and Applied Mathematics, Vol. 14, 1986, pp. 401-438. [4] Chen, C. F. and Shieh, L. S., "A novel approach to linear model simplification", Int. J. Control, Vol. 8, 1968, pp. 561-570. [5] Shieh, L. S. and Goldman, M.J., "Continued Fraction Expansion and Inversion of the Cauer Third Form", IEEE Trans. Circuits and Systems, May 1974, pp.341-345. [6] Rathore, T. S., Singhi, B. M. and Kibe, A. V., "Continued fraction inversion and expansion", IEEE Trans. Automat. Contr., Vol. 24, No. 2, April 1979, pp. 349-350. [7] Chen, T. C., Chang, C. Y. and Han, K. W., "Reduction of transfer functions by the stability-equation method", J. Franklin Inst., 1979, pp. 389-404. [8] Gutman, P., Mannerfelt, C. F., Molander, P., "Contributions to the model reduction problem", IEEE Trans. Automat. Contr., Vol. 27, April 1982, pp. 454-455. [9] Aoki, M., "Some approximation methods for estimation and control of large scale systems", IEEE Trans. Automat. Contr., Vol. 23, February 1978, pp. 173-182. [10] Hutton, M. F. and Friedland, B., "Routh approximations for reducing order of linear time invariant systems", IEEE Trans. Automat. Contr., Vol. 20, 1975, pp. 329-337. [11] Aldhaheri, R.W., "Model order reduction via real Schur-form decomposition", Int. J. Control, Vol. 53, No. 3, 1991, pp. 709-716. [12] Moore, B. C., "Principal component analysis in linear systems: controllability, observability and model reduction", IEEE Trans. Automat. Contr., Vol. 26, January 1981, pp. 17-31. [13] Pal, J., “Stable reduced-order Padé approximants using the Routh-Hurwitz array”, Electronic Letters, Vol. 15, No. 8, April 1979, pp. 225-226. [14] Lucas, T. N., "Model reduction by condensed continued-fraction method", Electronic Letters, Vol. 21, No.16, August 1985, pp. 680-681. [15] Lal, M. and Mitra, R., "Simplification of large system dynamics using a moment evaluation algorithm ", [16] Shamash, Y., "Truncation method of reduction: a viable alternative", Electronic Letters, Vol. 17, 1981, pp. 97-99. [17] Yeung, K. S., "Stability of reduced model obtained by truncation", ibid. , Vol. 17, 1981, pp. 374-375. [18] Lucas, T. N., "Factor Division: a useful algorithm in model reduction", IEE Proceedings, Vol. 130, Pt. D, No. 6, November 1983, pp.362-364. [19] Shamash, Y., "Linear system reduction by Padé approximation to allow retention of dominant modes", Int. J. Control, Vol. 21, 1975, pp. 257-272. [20] Davison, E. J., Chidambara, M. R., "On a method for simplifying linear dynamic systems", IEEE Trans. Automat. Contr., 1967, pp. 119-121. [21] Shamash, Y., "Model reduction using the Routh stability criterion and the Padé approximation technique", Int. J. Contr., Vol. 21, No. 3, 1975, pp. 475484. [22] Pal, J. and Ray, L.N., "Stable Padé approximant in multivariable systems using a mixed method", Proc. IEEE, Vol. 68, No. 1, 1980, pp.176-178. [23] Chen, T. C., Chang, C. Y. and Han, K. W., "Model reduction using the stability-equation and the Padé approximation method", J. Franklin Inst., 1980. [24] Lepschy, A., Viaro, U., "A note on model reduction problem", IEEE Tans. Automat. Contr., Vol. 28, April 1983, pp. 525-527. [25] Sastry, G. V. K. R. and Krishnamurthy, V., "Relative stability using simplified Routh approximation method (SRAM)", J. Inst. Electron. & Telecommun. Eng., Vol. 33, 1987. [26] Sastry, G. V. K. R. and Krishnamurthy, V., "Biased model reduction by simplified Routh approximation method", Electronic Letters, Vol. 23, No. 20, September 1987, pp. 1045-1047. [27] Sastry, G. V. K. R. and Krishnamurthy, V., "State-space models using simplified Routh approximation method", Electronic Letters, Vol. 23, No. 24, November 1987, pp. 1300-1301.
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FIGURES AND TABLES FIGURE. 1 WORKSPACE OF THE GRAPHICAL USER INTERFACE OF THE TOOLBOX.
FIGURE. 2 IMPULSE RESPONSES FOR HIGH-ORDER SYSTEM OF EXAMPLE AND REDUCED-ORDER MODELS USING FOUR METHODS OF TOOLBOX.
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