Robustness of Multiloop Linear Feedback Systems

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 robustness 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 characteristic 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 characteristic 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

Define z to be the complex number at the Thus point of the triangle as in Fig. 3c. Then, by riq) does not encircle the origin for rotating the complex plane with the triangle as a o~cl. In particular, the perturbed contour function of frequency, za (jw) may be obtained r(1) does not encircle the origin, and the per- which is a continuous function IJJ of(Fig. 3d). turbed closed-loop system is stable. 0 This allows the important quantities (13) in and Similar theorenre hold for additive rather (141, that is, theai and 6. to be represented than multiplicative perturbations (with+ G in convenient graphicalfod. As noted in I Fig. 3d, there is an ambiguity to zi depending on substitad for I + G-l) as well as a number of which side the plane is viewed.(To be more other configurations. precise, the z represent a multivalued function of s which coufd be defined on appropriate Riemann This theorem points out the importance of sheets. However, this will be ignored.) The zi singular values. In partic-Y lar, the smallest may be calculated by finding the roots of the singular valueg(I + G(jw) ) gives a reliable quadratic equation frequency-dependent measure of robustness. Stability is guaranteed for all perturbations whose spectral norm is less a.than As will be seen in the examples, eigenvalues do not give a By plotting thez (jw) ii = l,.. .m) for similarly reliable measure. frequencies of interesf a plot analogous to the scalar Inverse Nyquist plot is generated.Wile The singular values also have useful graph-phase does not have the conventional meaning on ical interpretations. Consider the dyadic these plots, the more important notion of expansion n distance from..the critical point preserves its H-l = I + G-l = Oiuivi (12) importance. Gain and bandvidth may alsobe i= 1 interpreted in the conventional manner.These < <'" Lan plots will be referredto as z-plots. a 1-2- where the O.,ui and v. are the singular values, Concepts such as M-circles are obviousalso and left ana right sihgular vectors, respectively in this context. The minimum value Mof is of I + G-'. This is an alternative form of the given by singular value decomposition in equation(2).

14 where k and k are constants. 1 2 Similar results may be obtained for additive Then regions of stability and instability may perturbations by working with I + G rather than be plotted in the (k1, k2) plane as has been done I + C1. In this case a diagram is generated in Figure 6. The open loop point corresponds to which is analogous to the scalar Nyquist diagram. kl = k2 = -1 and noainal dosed loop point point A number of other configurations may be handled corresponds to kl = k2 = 0. Breaking each loop as well. individually examines stability along the kl, k2 axes where robustness is good, but misses the Note that singular values offer no encir- close unstable regions caused by simultaneous clement condition to test for right half-plane changes in kl and k2. Thus, single loop analysis poles. Another test must be made forabsolute is not a reliable way of testing robustness. stability but this presents no obstacle as many simple techniques exist for doing this. Once The second example is a two dimensional feed- stability is deternined the various approaches back system with openbop poles at -1 and -2 and presented in this Section may be used to reliably no transmission zeroes. analyze robustness. The loop transfer matrix is 3. Examples r-47s + 2 56s1 Two simple examples are presented and analyzed using the approaches developed in the previous section. For the purpose of comparison, the methods of loop-breaking, direct eigenvalue analysis of G, and diagonalization by compensation are also used. The advantage of the interpretation of robustness given in this paper is clearly Assume that identity feedback is used, with illustrated. closed-loop poles at -2 and -4. This system may be diagonalized by introducing constant compen- The first example is an oscillator with sation.Let open loop poles at flOj and both closed loop poles c 1 at -1. There are no transmission zeros. The loop transfer function is

and

By closing either loop (the system is synmetric) as in Figure 4, the transfer function (20) for the other loop is

Then letting

which indicates 0 db gain margin in both direc- tions and 90' phase margin in each loop (with the other closed). This is very misleading, however. (21) The z-plot for this example is shown in Figure 5. It may appear somewhat peculiar, since it is not a plot of a rational function. The the system may be rearranged so that important feature is the proximity of the plot to the critical point, indicating a lack of robust- H = G_(I+G)-l ,, ness. = U_GV(I +,UGV)-l = UG(I + G)-ly The apparent discrepcncy between these two = U[E(I + e,- Iv. robustness indications can be easily understood by considering a diagonal perturbation This yields a diagonal system that may be analyzed by scalar methods. In partj.cular under the assumption of identity feedback G represents the new loop transfer matrix. Because U and V represent a similarity transformation, the diagonal elements of G are also the eigenvalues of G so that the decoupling or dominance approach

15 and eigenvalue or characteristic loci approach into the properties ofmultiloop feedback systems. would generate the same Nyquist or Inverse Nyquist One possible difficulty with the approach is plot shinFigure 7. Only a singlelocus is that it can lead to overly pessimistic views of shown sincethe contours of l/(s+l) and2/(9+2) robustnessbecause it considersperturbations areidentical. The temptingconclusion that might which may notbe physically possible. This bereached from these plots is thatthe feedback problem exists as well with gain and phase margin system is ermninently robustwith apparent margins evaluations. Of course, some of thisdifficulty of f mdb ingain and 90' in phase. The closed- can be alleviated byexamining the specific loop pole locations wouldseem to support this. perturbationsleading to instability. These may beeasily computedfrom equation(12). On the Thisconclusion, however, would be wrong. otherhand, it mightbe argued that a little The z-plot for I + G-1 is shown in Figure 8 and healthypessimism would be refreshing in the field there is clearly a serious lack ofrobustness. of multivariable linear control research. The (kl, k2) - plane stability plot for this example is shown inFigure 9. Neitherthe The results in this paper concerning domi- diagonaldominance nor eigenvalue approaches indi- nancemethods and use of characteristic loci of cate the close proximity ofan unstable region. the loop transfer matrix are not meant to imply This failure can be attributed to two causes. thatdesign procedures employing these methods are useless. However, simplydesigning "in the First,the eigenvalues of a matrixdo not, in frequency domain" is no guarantee that resulting general,give a reliablemeasure of its distance controllers will have no undesirable properties. (in a parametricsense) from singularity, and so computing theeigenvalues of G(s)(or I+G(s)) does Although for simplicity's sake only rational notgive an indication ofrobustness. Using transfer functions were considered the results eigenvalues rather than singular values will always in this paper shouldextend to nonrational detectunstable regions that lie along the kl = k2 transferfunctions. In practical application it diagonal,but may miss regionssuch as the one in shouldbe possible to use frequency response data Figure 9. directly. The results may alsobe extended to include nonlinear perturbations by exploiting Second, when cmpensationand/or feedback is the general stability theory developed by Safonov used toachieve dominance, the "new plant"includes [13]*. Inthis setting, nonlinearities may be thiscmpensation andfeedback. Because of this, loosely viewed as lineartime-invariant elements no reliable conclusions may be drawnfrom this withtime-varying parameters. A mathematically "new plant"concerning the robustness of the final more rigoroustreatment of these issues may be design with respect to variations in the actual found in Zames ([14], [lS]),as well as in [13]. plant. It is importantto evaluate robustness where there is uncertainty. A limitation to the robustness theorem as stated is the requir.ement that L(s) bestable. Another important property of multiloop In practice, it is not uncommon forphysical feedback is that,unlike scalar feedback, pole systemsto have parameter variations which cause locations alone are not reliable indicators of polesto migrate across the imaginary axis. It robustness. This was demonstratedin the last appears likely that if such pole movements are example and may beexplained as follows. Consider restricted to some regionof the complex plane, a statefeedback prob1.a where the plant is and another restriction is made on thesystem controllable from eachof two inputs. One input zeroes, a modifiedrobustness theorem may be may be used to place the poles far into the right obtained.This would depend,of course, on the half plane and the other used to bring them back nature of theother sources ofsystem uncertainty tothe desired location. Such a high-gain as well as the nature of thefeedback employed. control design of "opposing"loops will be extremely sensitive to parameter variations The significance of theapproach presented regardlessof the nominal pole locations. herein seems to be in the natural way it gives multivariable interpretations to many important It is interesting to examine the5-plot of classicalcontrol concepts. Preliminary H = G(I + G)-1 for the second example shown in multivariablefeedback designs using methods based Figure10. Recall that the singular values of on theseresults are most encouraging[18], and H are equal to the inverses of the singular this too appears to be a promisingarea for valuesof I + G-I. There is a ratherlarge peak research. inthe frequency response at approximately 3 radians.This could not occur in scalar-unity feedbacksystems without there being a pole relativelynear the imaginary axis. It canhappen inmultiloop systems because of the high gains possiblewithoutcorresponding large pole movement.

Further Comments andConclusions 4. *Recently, it hasbeen shown thatthe results in this paper may bederived from those in The approach to the analysis of robustness [13], and therobustness results in 1131can presented here appears to yield useful insight beexpressed in terms of singular values [17]. Acknowledgements G. Zames, "On the Input-Output Stability of Time-Varying Nonlinear FeedbackSystems - I would like tothank all those whose Part II", IEEE Trans. On Automatic Control, criticisms and comments helped to mold this paper. Vol. AC-11, NO. 3, pp. 465-476, July, 1966. I would particularly like to thank Drs. G. Stein, M.G. Safonov, and C.A. Harvey for their continued B.S. Garbo, et al, Matrix Eigensystems technical input. Routine - EISPACK GuiQExtension, Lecture Notes in Computer Science, Volume51, I would also like to thank the Math Lab Group, Springer-Verlag, Berlin, 1977. Laboratory for Computer Sciences, MIT for use of their invaluable tod, MACSYMA, a large symbolic M.G. Safonov, "Singular Valuesand Multi- manipulation language. The Math Lab Group is loop Nonlinear System Robustness", Honeywell supported by NASA under grant NSG 1323 and DOE Memo MR 12513. under contract #E(ll-1)-3070. G. Stein and J. Doyle, "SingfilarValues References and Feedback: Design Examples",16 th Annual Allerton Conferenceon Communication, B.C. Kuo, Automatic Control Systems, Prentice- Control, and Computing, Monticello, Ill. Hall, 1967. October 4-6, 1978. J.W. Brewer, Control Systems, Prentice-Hall, 1974. I.M. Horowitz, Synthesis of Feedback Systems, Academic Press, 1963. H.H. Rosenbrock, Computer-Aided Control System Design, Academic Press, 1974. Figure 1. Multiloop FeedbackSystem A.G.J. MacFarlane andI.. Postlethwaite, "The Generalized Nyquist Stability Criterion and Multivariable Root Loci", Int. J. Control, Vol. 23, No. 1, January, 1977,pp. 81-128.

A.G.J. MacFarlane and B. Kouvaritakis, "A Design Technique for Linear Multivariable Feedback Systems", Int.J. Control, Vol. 23, No. 6, June, 1977,pp. 837-874. G. Strang, Linear Algebra and It's Figure 2. PerturbedSystem Applications, Academic Press, 1976. J.H. Wilkinson and C. Reinsch, Linear Algebra, Springer-Verlag, 1971. J.H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, 1965.

G.E. Forsythe and C.B. Moler, Computer 0.u. Solutions of Linear Algebraic Systems, 11

Prentice-Hall, 1967. 2. A.G.J. MacFarlane and M. Karcanias, "Poles and Zerosof Linear Multivariable Systems: tJ A Survey of Algebraic, Geometric, and Complex Variable Theory", Int. J. Control, July, 1976, Vol. 24, NO. 1, pp. 33-74.

K. Knopp, Theory of Functions, Dover, 1947. M.G. Safonov, "Robustness and Stability Aspects of Stochastic Multivariable Feedback System Design", Ph.D. dissertation, Mass. Inst. Tech., Sept. 1977. Figure 3 Construction of z-plot G. Zames, "On the Input-Output Stabilityof Time-Varying Nonlinear Feedback Systems- Part I",~IEEE Tran. on Automatic Control, Vol. AC-11, No. 2, pp.228-238, Apr. 1966.

17 r------

Figure 4. Analysis of Loop-Breaking

Figure 6. Example 1 Stability Danain Figure 5. Example 1 r-plot

Figure 7. Example Nvquist and InverseNyquist Diagram

Figure 8. Example 2 z-plot

Frequency (radians)

Figure 9. Example 2 StabilityDaMin Figure 10. Closed-LoopFrequency Response -l/o(I+G-1 )