Phase margin of linear time invariant systems from ... - IEEE Xplore

3 downloads 0 Views 220KB Size Report
Routh array. Empirical formulae for computing the gain cross-over frequency ocg of type 0 systems and of higher-type systems in terms of system gain K and theĀ ...
Phase margin of linear time invariant systems from Routh array V. Krishnamurthi, PhD W.K. Sa'id, PhD N.A. AI-Awad. MSc

Indexing terms: Control theory, Phase margin

Abstract: A simple method to determine the gain margin of linear time invarient systems (LTIS) employing Routh stability criterion was developed in an earlier paper. The term 'stability ratio' was defined connecting the marginal gain K , determined from the Routh array and the actual gain K of the system. This established a direct link between the Routh criterion and the gain margin. It was also shown that the phase cross-over frequency wCpcan be found from the auxiliary equation formed from the elements of the s2-row of the Routh array. Empirical formulae for computing the gain cross-over frequency ocgof type 0 systems and of higher-type systems in terms of system gain K and the marginal gain K , are suggested. The gain cross-over frequency and phase margin calculated using the empirical formulae are in close agreement with exact values, justifying the use of the empirical formulae. It is shown that the Routh stability criterion can be used to measure the relative stability of LTIS. 1

stable system

unstable system

positive gain

'3 -

t

,

I

Introduction

The objective of this investigation is to explore the possible relationship between the Routh's stability criterion and the relative stability of linear systems [l]. The superficial dictum found in elementary courses or text books on control theory, that the Routh-Hurwitz criteria do not provide a quantitative measure of stability, can be checked. In an earlier paper [2], it was shown that: (i) There exists a direct relationship between the Routh stability criterion and gain margin of the system in terms of stability ratio (SR) [2] (ii) Using SR, the system gain for a specified gain margin can be simply determined. (iii) The phase cross-over frequency, U,,,can be found from the Routh array. It is shown that: (a) The gain cross-over frequency wcs can be determined in terms of K , and K ( b )The phase margin can be computed using w C g . The relative stability of LTIS is measured in terms of its gain margin and phase margin. The gain margin is the Paper 80481)(C8), received 21st May 1990 The authors are with the Department of Control and Systems Engineering, University of Technology, Baghdad 35010, Iraq 410

additional gain, in decibels, by which the open loop (OL) gain of a system must be changed so that the system becomes marginally stable. The frequency at which the phase angle of the system is 180" is called the phase cross-over frequency wcp . The graphical method of obtaining the gain margin from the Nyquist and Bode plots is illustrated in Fig. 1 for stable, marginally stable and unstable systems [3].

'

stable system

unstable system

b

Fig. 1

Phase and qain margins of stable and unstable systems

n Polar plots

b Logarithmic plots

If the open loop transfer function (OLTF) of a system is G ( j o ) H (io),then the gain margin is G M = 20 log,, I l / K G ( j w c , ) M j o c , )I

(1)

where wcpis the phase cross-over frequency. A limitation of the gain margin in eqn. 1 is that it is not precisely defined in the case of all-pole second order systems which are stable for all positive values of gain [w,, = CO, K G ( j w ) H ( j w )= 01 and for systems which are globally unstable [w,, = 0, K G ( j w ) H ( j w )= CO]. A second limitation is that the solution of wCpbecomes cumbersome for high order systems. The figure of merit should generally provide the measure of gain margin of all systems without any limitation. To overcome the above limitations, the gain margin was defined as a variable and was termed the stability ratio (SR). The stability ratio of a system was thus defined as the ratio of the marginal gain K , to the actual system gain K SR

=

KJK

(2) I E E PROCEEDINGS-D, Vol. 138, N o . 4 , JULY 1991

Eqn. 2 establishes a direct relationship between the gain margin of a system and the Routh stability criteria. The gain margin, in terms of SR, can be simply evaluated from a knowledge of marginal gain K , obtained from the sl-row test function of the Routh array. For any value of system gain K , the gain margin of SR is immediately known. This avoids the entire process of finding the phase cross-over frequency and the gain margin as required in other methods. In terms of SR, the closed-loop system is stable, marginally stable or unstable according to whether SR is greater than, equal to or less than unity. For a globally unstable system SR is zero. For a system which is always stable, SR is infinity. Thus SR stands precisely defined in all cases, whereas eqn. 1 does not. The gain in terms of SR easily lends itself to determine the system gain K for a specified gain margin. The gain K may be easily obtained from knowledge of SR, since K = K,,,/(SR) The phase-crossover frequency w,, is obtained from the Routh array. From the sl-row test function of the Routh array, the marginal gain K , may be obtained. The roots of the auxiliary equation formed from the ?-row ele. ments with s = jw and K = K , correspond to o c pThis yields the exact value of o C pIt. should be noted that ocp is independent of the system gain K . 2

Gain cross-over frequency and phase margin

The gain margin is one of many ways of representing the relative stability of LTIS. In principle, a system with a large gain margin should be relatively more stable than one with a smaller gain margin. The gain margin alone does not adequately indicate the relative stability of all systems. The two systems represented by the G( jw)H(jo) plots of Fig. 2 appear to have the same gain margin. Plot A corresponds to a more stable system than plot B. The reason for this is that with any change in a system

parameter other than K , it is easier for plot B to pass through or even enclose the ( - 1,j0) point. The phase margin is used as a supplement to the gain margin to strengthen the representation of relative stability of a LTIS. The phase margin is defined as the amount of additional phase angle to be introduced at the gain cross-over frequency oCg at which I G( j o ) H ( j w ) I = 1 such that G(jw)H(jo) locus passes through the critical point ( - 1 , j0) and the system becomes marginally stable. The gain cross-over frequency and the phase margin are shown in Fig. 1 . The procedure to obtain the phase margin of a given system is as follows: Let the transfer function of the system be I "

m

G(s)H(s)= K

2 b,si/ 1ais',

i=O

m