WA1 - 10:45 ROBUSTNESS OF MJLTILOOP LINEAR FEEDBACK SYSTEMS* J. C. Doyle Consultant Honeywell Systems and Research Center 2600 Ridgway Parkway Minneapolis, Minnesota 55413 ABSTRACT critical role in evaluating system robustness. The Bode plot or scalar Nyquist or Inverse Nyquist This paper presents new a approach to the diagram (polar plots of the loop transfer function) frequency-domain analysis of multiloop linear feed-provides a meansof assessing these quantities at back systems. The properties of the return a glance. For multiloop systems, individual loops difference equation are examined using the conceptsmay be analyzed using scalar techniques, but these of singular values, singular vectors and the methods fail to capture the essentially multi- spectral norm of a matrix.A number of new tools variable nature of manysystem. For example, for multiloop systems are developed which are scalar methods may ignore variations which analogous to those for scalar Nyquist and Bode simultaneously affect multiple loops. analysis. These provide a generalization of the scalar frequency-domain notions such as gain, There are a number of other possible ways bandwidth, stability margins and M-circles, and to extend the classical frequency-domain tech- provide considerable insight into system robust- niques. One involves using compensation or feed- ness. back to decouple(or approximately decouple) a multiloop system into a set of scalar systems which may be treated w#th scalar techniques (i.e., 1. Introduction "Diagonal Dominance", bsenbrock[4]). Another A critical property of feedback systems is method uses the eigenvaluesof the loop transfer their robustness; that is, their ability to main-matrix (G(s) in Figure 1) as a function of fre- tain performance in the face of uncertainties. quency (i.e., "Characteristic Loci", MacFarlane, In particular, it is important that a closed-loop et. al. [SI, [SI). While these methods may system remain stable despite differences betweenprovide legitimate tools for dealing with some the model used for design and the actual plant.multivariable systems, they can lead to highly These differences result from variations optimistic conclusions about the robustness of in modelled parameters as well as plant elements multiloop feedback designs. Examples in Section which are either approximated, aggregated, I11 will demonstrate this. or ignored in the design model. The robust- ness requirements of a linear feedback design This paper develops an alternative view of are often specified in terms of desired gain multiloop feedback systems which exploits the and phase margins and bandwidth limitations concepts of singular values, singular vectors, and associated with loops broken at the input to the spectral norm of a matrix. ([7] - [IO]). This the plant actuators ([I],[2]). These approach leads to a reliable method for analyzing specifications reflect in part the classical the robustness of multivariable systems. notion of designing controllers which are adequate for a set of plants constituting Section 2 presents a basic theorem on a frequency-domain envelope of transfer robustness and sensitivity properties of linear functions [3]. The bandwidth limitation multiloop feedback systems. Multivariable provides insurance against the uncertainty which generalizations of the scalar Nyquist, Inverse grows with frequency due to unmodeled or Nyquist and Bockanalysis methods are then developed aggregated high frequency dynamics. from this same result. Two simple examples are analyzed in Section 3 Thus, the classical notions of gain, band- Using the tools of Section2. Pspromised,the inade- --width, and stability margins have played a quacies of the existing approaches outlined earlier * This work was performed for Honeywell Systemswill be made clear. and Research Center, Minneapolis,MN, with partial support from the Office of Naval Section 4 contains a discussion of some of the Research Contract N-00014-75-C-0144 implications of this work. Mailing Address: MS "7-2367, Honewell Systems and Research Center The goal of this paper is to focus on the 2600 Ridgway Parkway, Minneapolis,MN 55413 analysis of robustness and sensitivity aspects of linear multiloop feedback systems. Some new approaches emerge vhich yield important insights 2. Basic Results into their behavior. The mathematical aspects of these topics are fairly mundane at sobest, rigor Consider the feedback system in Fig.2 where and generality are most always sacrificed for G(s) is the rational loop transfermatrh and L(s) simplicity. is a perturbation matrix, nominally zero, which represents the deviation ofG(s) from the true Preliminaries and Definitions plant. While this deviation is unknown, there is usually some knowledge asto its size. A A brief discussion of singular values and reasonable measure of robustness for a feedback vectors follows. Although the concepts apply system is the magnitude of the otherwise more generally, only square matrices will be arbitrary perturbation vhich may be tolerated considered in this paper. A more thorough without instability. The folloving theorem discussion of these topics may be found[7] -in characterizes robustness in this way. The [lo1 * "magnitude" of L(s) is taken to be the spectral nom. Only stable perturbations are considered The singular valuescyi of a complexn x n since no feedback design be may made robust wi&h matrix A are the*non-negatfve square roots of the respect to arbitrary unmodeled unstable poles. eigenvalues of A A where*A is the conjugate transpose of A. Since A A is Hermitian, its Robustness theorem: Consider the perturbed eigenvalues are reaA. The (right) eigenvectors v system in Figure2 with the following assumptions of A*A and of AA are the right and left sing-i ular vectors, respectively,of A. These may be i) G(s) and L(s) are nxn rational square chosen such that matrices diul = Avi , i = 1,... n ii) det (G(s)) 2 0 (1) Ul _< u2 _< u iii) L(s) is stable ... -n and the 6,) and (vi) form orthonormal sets of iv) the nominal closed loop system vectors. H = G(I+G)-' It is well known that is stable. A=UCV* (2) Under these assumptions the perturbed system is stable if Where U and V consist of the left and right singular vectors, respectively, andC = diag. (Ol,. ,(J). The decomposition in(2) is called the singdar value decomposition. for alls in the classical Nyquist &contour (defined below) Denote Proof: -u (A) = min 11x1 I=1 It is wellknown [4] that sinceG is invertible and 'l.. - lklI = I IAl l2 = an (4) where 'is) is the nominal closed-loop character- istic polynomial and@ (s) is the transmission zero polynomial cfG [ill. The singular valuesare important in that they characterize the effect that A has as a map- For the perturbed system ping on the magnitude of the vectorsx. They may be thought of as generalizing to matrices the notion of gain. The singular values also give a measure Of how "closeA is to being singular .(in a parametric sense). In fact, the quantity where Q2(s) is the perturbed closed-loop charac- teristic polynomial andQ3(s) is the character- -d /is istic polynomial ofL(s). is known as the condition number with respect to inversion [g]. The eigenvalues of A do not in Let D be a large contour in the s-plane general give such information. Ifh is an consisting of the imaginary axis from -jR to +jR, eigenvalue of A, then together with a semicircle of radious R in the right half-plane. The radious R is chosen large - enough so that all finite rootsof I/J~(s)have 05 1x1 '0 magnitude less thanR. and it is possible for the smallest eigenvalue to be much larger thana. *It is possible that this requirement may be relaxed somewhat. See Section 4. 13 Let the contourr be the image of D under The quantities ai, u and v may be viewed as the map JI (e) det (I +OG(s)-l). Since H is functions of -p&a varihle, or in parti- stable, ii followsfrw the principle of the cular as functions of frequency. argument thatro will not encircle the origin. Since Define the map 1* H = (I + G-l)-l = C - v u ai i i y(q,s)=9,(s) det (I+G(s)-~+~L(s)), q real (8) the values l/a(jw) and l/a (jw) give the maximum and mih~~mpossiblg magnitude and let y(q,s) map D into the- Contourr(q) for responses toan input sinusoid at frequencyw. fixed q, o:q:l. The mapy(q,s) may be written as n Eigenvalues give no such information.In this y(q,s)= *1(~)+3(~) + qel(s)+. ..+s enb) sense, a plot of these singular values vs. frequency may be thoughtof as a multivariable +3(s) generalization of the Bode gain plot. Plots of this type will be referredas to0-plots. Another useful graphical interpretation analogous to the scalar Inverse Nyquist diagrammay be Clearly, since r(o) = r , it does not encir- constructed by noting that cle the origin. Since the r8ots of9, are a1 ebraic functions ofq, they are continuous in * G-l=Ccrii~ -I q f121. Thus the only way that the perturbed iii contour r(l) can encircle the origin is for * - Bigivi det (I+G(s)-' + qL(s)) = 0 (10) where Big, = 0.u - vi with 6, real andI lgil I = 1 for sme s in D and someq on the interval o<q<l. for all i. ('deigivs do not necessarily form (Recall that $ (5) has no right ha1 plane roots). an orthonormal set.) When (10) is sitisfied thena (I*-€-+ qL) must also be zero. However, as a consequence (5) of The quantities in(14) at some frequencyw are related as in diagram in Fig. 3a. Since vio is of unit length a complex plane may be con- structed as in, Fig.