Automatica, Vol. 13, pp. 189-192. Pergamon Press, 1977. Printed in Great Britain Brief P aper Graphical Stability Criteria for Nonlinear Multiloop Systems* J. D. BLIGHTt and N. H. MCCLAMROCH~: Key Word Index--Feedback; nonlinear systems: stability criteria; Popov criterion; multivariable systems. Summary--In this paper a description of a certain class of multiloop systems, called the Standard Multiloop Form, is introduced. This description is expressed explicitly in terms of certain scalar subsystems and can be shown to include many of the common descriptions of multiloop systems. The stability criteria presented in this paper involve the individual Nyquist plots of the linear scalar subsystems and a certain positivity condition on the nonlinear subsystems. The method allows for relatively convenient computations. The deriva- tion depends on the hyperstability concept introduced by V. M. Popov. I. Introduction NONLINEAR stability theory has received the attention of researchers for many years. In 1961 V. M. Popov introduced a method of stability analysis based on the use of the frequency domain[l], which greatly simplified the analysis for systems having a particular structure, namely, systems having a linear time-invariant 'plant', one nonlinear element, and a single feedback loop. Since the introduction of Popov's method and the associated Circle Condition, many resear- chers have generalized the Popov method to include systems having multiple nonlinearities and multiple feedback loops, e.g. [ 1-6]. It is the purpose of this paper to introduce a method of analysis which leads to a convenient graphical inter- pretation in the frequency domain. There have been a few recent results[7-10] where an attempt has been made to preserve a graphical interpretation; in these works various system representations were considered but the difficulty of handling large-scale multiloop systems remains. 2. System description Stability criteria will be derived for systems having a particular structure called the Standard Multiloop Form. 2.1 Definition. A system of equations of the form ~+(t) = A,x,(t) + b~(t) y,(t) = c',x,(t) .f,(t)= ~ ~,j[u~(t) - ~, ~b,,(y,, t), t] for i = 1..... n (2.1) is said to be in the Standard Multiloop Form. The function x,(t) is an n~-vector, and A(t) and y,(t) are scalar functions. The function udt) is a scalar input function to the ith subsystem. The time varying and nonlinear functions ff,j and ~btjare assumed to be continuous functions of their arguments *Received 7 August, 1975; revised 5 March 1976; revised 26 July 1976. The original version of this paper was presented at the 6th IFAC Congress which was held in Boston, Cambridge, MA, U.S.A. during August 1975. The published Proceedings of this IFAC Meeting may be ordered from: Instrument Society of America, 400 Stanwix Street, Pitts- burgh, PA 15222, or John Wiley, Baffins Lane, Chichester, Sussex, U.K. This paper was recommended for publication in revised form by associate editor B. D. O. Anderson. t United Pacific Controls, Inc., Portland, OR 97214, U.S.A. ~Computer, Information and Control Engineering, The University of Michigan, Ann Arbor, MI 48109, U.S.A. with ~b,j(0) = 0 and $,j(0) = 0. It shall be assumed throughout that n > 1. The transfer function for the ith linear subsystem is given by by Y,(s) _ c'j(sI - A,)-'b,, (2.2) ~,(s) = F--~ - and a block diagram of the system is shown in Fig. 1. It may be shown that a great many multiioop feedback systems may be placed in this form through suitable selection of the nonlinearities +,j and ~b0[ll, 12]. In this work the stability of the multiloop system is considered only for zero inputs. 2.2 Definition. The system (2.1) is globally stable with degree "r if, for u+(t)-=0, i= l ..... n, and for any x,(0), i = I ..... n, there exist numbers K+ > 0 such that IIx,(t)ll <- K, e-", t -> 0, i= 1.... ,n. The definition implies global asymptotic stability in the sense of Liapunov if 3, > O. 3. Stability criteria The stability criteria are based upon a variation of a lemma developed by V. M. Popov for single-loop systems [1, 11]. 3.1 Basic lemma. If the linear systems J~,(t) = A+x+(t) + b~(t) y+(t) = c'~x+(t) + d.~,(t), i = I ..... n with transfer functions G,(s) c'~(sl- A+)-'b+ + d+ are each irreducible, i.e. controllable and observable, and the func- tions G,(s) have all poles in Re s < 0, and the Nyquist plots of G,(s) lie in Re s->0, then there exist ~+ >0 and v, >0, i = 1..... n such that ~',..,~,,llx+(t)lP <- ~,_, v+llx,(O)ll + + *)y,O) d~" (3.1) i=l i=li= i= holds for all t > 0. 189 Yi I ~-*ij (Yj) i= I ...... FIG. 1. Block diagram of the ith subsystem of the Standard Multiloop Form.