Comments on "Model Reduction Using the Routh Stability Criterion"

A model reduction technique suggested in the above paper' is discnssed. Although the proposed method is very simple and devoid of any tedious calculations it is shown that the applicability of the technique is restricted. For t h e cases for which the procedure works the redaced order model matches the low-frequency response rather poorly. Tke proposed method is thus not suitable for the analysis and design of control systems. Krishnamurthy and Seshadri' have proposed a method of obtaining reduced order models in the frequency domain employing Routh arrays. However, the authors have not given any theoretical justification for their procedure. The proposed technique consists of obtaining the numerator and the denominator polynomials of the reduced order model, respectively, from the numerator and the denominator polynomials of the system. Since the numerator and the denominator polynomials are treated independently, a partial matchng of the Taylor series expansion around s = 0, as proposed by Hutton and Friedland [I], is no longer valid. The Routh table employed by the authors is the conventional Routh table and it is shown in [2] that a direct application of the conventional Routh table yields a reduced order model whose poles approximate the system poles whch are far off from the imaginary axis. The model thus preserves high-frequency behavior, while in a general control application the low-frequency behavior is more relevant. The proposed method is thus not suitable for the analysis and design of control systems. This could have been easily avoided by first performing an inverse transformation [I] and then applying the proposed reduction technique. Since the numerator of the reduced order model is constructed by developing a Routh table from the numerator polynomial of the system, it is evident that the numerator stability array' will break down for all of the following cases [3]. 1) The first column of the Routh table contains a zero because of the presence of a system zero in right-half of the complex plane. 2) The system contains zeros that are symmetrically placed with respect to the imaginary axis, i.e., the numerator polynomial of the Manuscript received November 13. 1978. A. S. Rao is with the Delhi College of Engineering. Delhi, India, on leave at the Department of Electrical Engtneering, Indian Institute of Technology, New Delhi India. S. S. Lamba and S. V. Rao are wth the Department of Electrical Engineering, Indian Institute of Technology, New Delhi, India. 'V. Krishnamurthy and V. Seshadri. IEEE Trans. Auromr. Confr.. vol. AC-23, pp. 729-731. Aug. 1978. Authors' Reply V. KRISHNAMURTHI AND V. SESHADRI The authors thank A. S. Rao et af. for their interest in the reported work. The following points have been raised by them: 1) Taylor series expansion around s = 0 is no longer valid. 2) Reduced order model poles approximate the system poles which are far away from the imaginary axis. 3) The numerator stability array will break down when the system has a zero in the right-hand plane and/or symmetrically placed zeros of the form (s=—a) where a is a constant. 4) No theoretical justification is given for the procedure. a) In the paper', the authors do not claim that the Taylor series expansion is valid for their method. b) This statement is not correct. This has been illustrated in the paper' with a numerical example of order 8. The alpha parameters are differently defined in order to avoid the inverse transformation. Thus, «• •= • i = ! , • • , « . Hence, the reduced order model poles approximate system poles near the imaginary axis only. c) The destabilizing singularities of the numerator are to be separated first before the application of the procedure and thus retained even in the reduced order system (as explained in the paper' for the denominator polynomial). d) Theoretical justification was not included in the paper' due to page restrictions. For theoretical jus~ication, the Ph.D. dissertation of the first author [I] may lundly be referred to. Thus, the procedure is generally applicable and yields desirable results.