CONTROL SYSTEMS, ROBOTICS AND AUTOMATION - Vol. XII - Input Output Stability - Banks S.P.

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1 CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XII Iut Outut Stability Baks S.P. INPUT OUTPUT STABILITY Baks S.P. Deartmet of Automatic Cotrol ad Systems Egieerig,Uiversity of Sheffield, Sheffield S1 3JD, UK Keywords: stability, iut outut systems, gai, circle theorem, assivity Cotets 1. Itroductio 2. Sigals ad Norms 3. Systems ad Gais 4. The Circle Theorem 5. Passivity 6. Itercoected Systems, Grahs, ad Robustess 7. Coclusios ad Further Develomets Glossary Bibliograhy Biograhical Sketch Summary I this article we defie iut outut stability for a system by cosiderig the relative size of sigals at the iut ad outut. This leads to the otio of gai, which geeralizes the gai of a liear amlifier. The smallgai theorem is the basic result, ad states that the roduct of the gais i a feedback loo should be less tha oe for stability. After cosiderig liear systems, we discuss the case of biliear systems, ad obtai a exressio for the gai of such systems. The circle theorem for feedback systems cotaiig a liear elemet ad a oliear elemet i the feedback loo is covered i detail, ad geeralizatios based o assivity ad multiliers are itroduced. Fially the robustess of stability uder arameter chages ad iut disturbaces is discussed. 1. Itroductio The otio of stability is oe of the most imortat cocets i the whole of systems theory. A overall closedloo system must be stable eve if the lat has bee deliberately desiged to be ustable (as i the case of moder fighter aircraft, for examle). This ituitive idea of stability, that systems should ot roduce sigals of ubouded growth for iut sigals that are bouded i some sese, ca be formalized i a umber of ways; the most oular ad effective are the Lyauov otio of stability based o iteral eergy dissiatio, ad the iut outut defiitio based essetially o the gai of the system. Much of the iut outut theory ca be traced back to the work of Zames, Sadberg, ad Poov. The gai of a liear amlifier is, of course, just the ratio of the outut ad the iut voltages, which is a ositive umber. We shall see that this simle otio of gai geeralizes to may kids of oliear systems, rovided we have a cocet of size o the iut ad outut sigals, ad we Ecycloedia of Life Suort Systems (EOLSS)

2 CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XII Iut Outut Stability Baks S.P. shall do this by itroducig a orm o these sigals. I geeral, we shall see that the gai is the ot ecessarily costat, but ca be a oliear ositive realvalued fuctio. Oce we have defied iut outut stability for a geeral (oeloo) system, it will be clear that it is ot equivalet to Lyauov stability i geeral, although for cotrollable ad observable liear systems the two cocets coicide. Whe we cosider systems i the form of a feedback loo with a subsystem i the feedforward ath ad oe i the feedback ath, such as the oe i Figure 1, the we will show that the most basic result is the celebrated smallgai theorem, which states that if the roduct of the gais of S 1 ad S 2 is less tha oe, the the overall system is iut outut stable. I may cases, oe of the subsystems S 1,S 2 is liear ad we ca obtai geeralizatios of the Nyquist stability criterio i the form of Poov s theorem or the various circle criteria, which state that the frequecy resose of the liear art must lie i a certai circle whose radius ad ceter are secified by the oliear art. Fially, most systems are ot kow comletely: some arameters may be ukow, the model may be oly aroximate, ad there may be ukow disturbaces i the system. It is therefore ecessary to kow that the stability of some omial system imlies that of systems erturbed aroud the omial oe ad also what size of erturbatios ca be allowed. This is the theory of robustess, ad we shall metio the most basic results. 2. Sigals ad Norms Figure 1. A simle feedback system A sigal i a system is take as some vectorvalued fuctio t f(): t R R of time ito Euclidea sace (although it is sometimes useful to cosider comlex vectorvalued fuctios). Such a sigal may be a iut or a outut of a system. (Note, however, that for distributed arameter systems, described by artial differetial or fuctioal differetial equatios, the image sace must be some ifiitedimesioal vector sace.) I order to defie the stability of a system, we must have some otio of size of a sigal i order to measure the relative size of the outut ad iut sigals to the system. This is usually doe by meas of a `orm. First, o the image sace R of the sigals we lace some dimesioal orm, such as x 2 1/2 2 xi or i= 1 = 1/ x i, 1 x = > i= 1 Ecycloedia of Life Suort Systems (EOLSS)

3 CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XII Iut Outut Stability Baks S.P. or x = max x. i i The first is the most commo (the stadard Euclidea orm) ad we usually omit the subscrit 2. Now, give a sigal f(t) we may defie its orm i may ways. Agai, here are some of the most commo (ote that the sigal may be very geeral: all that is really required is the formal mathematical coditio of measurability so that the itegrals are well defied): 2 f = f() t dt 2 0 1/2 or ( ) 1/ f = f() t dt,1 < or f = t [0, ) 0 su f( t) (where the orm f(t) of the fuctio value is a suitable orm such as oe of those above). Agai, the first ad last are the most commoly used; the first is essetially a measure of the total eergy i the sigal, while the last measures absolute size at each time. The first has the desirable roerty that (by Parseval s theorem) it has the same value (modulo 2π) as that of the Fourier trasform of the sigal: f () t dt = F( jω ) dω 2π (i.e. ower ca be measured i the state sace or the frequecy domai). We must also cosider the saces of all sigals for which these orms are fiite, so we itroduce the saces L (0, ) = { f : f < },1. However, sice we wat to allow sigals for which these orms are ot ecessarily bouded, we also itroduce more geeral saces of fuctios f, whose trucatio f T, defied by f T f () t 0 t T () t = 0 t > T belogs to the aroriate sace, i.e. Ecycloedia of Life Suort Systems (EOLSS)

4 CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XII Iut Outut Stability Baks S.P. e T L (0, ) = { f : f <, for all T > 0}. These are the exteded L saces. Note that L e (0, ) L (0, ) for all 1. Bibliograhy TO ACCESS ALL THE 17 PAGES OF THIS CHAPTER, Click here Baks S.P. (1981). The circle theorem for oliear arabolic systems. Iteratioal Joural of Cotrol 34, [This aer geeralizes the circle theorem to artial differetial equatios defied by diffusio systems.] Baks S.P. (1988). Mathematical Theories of Noliear Systems, 320. Lodo: PreticeHall. [This is a geeral moograh o the geometric theory of oliear systems, ad also cotais oliear distributed arameter systems.] Baks S.P. ad AlJurai S.K. (1996). Pseudoliear systems, Lie algebras ad stability. IMA Joural of Mathematical Iformatio ad Cotrol 13, [This aer itroduces a ew Lie algebra aroach to stability theory.] Baks S.P. ad Chaae B. (1988). A geeralized frequecy resose for oliear systems. IMA Joural of Mathematical Iformatio ad Cotrol 5, [This resets a very geeral frequecydomai theory for oliear systems.] Baks S.P. ad McCaffrey D. (1998). Lie algebras, structure of oliear systems ad chaotic motio. Iteratioal Joural of Bifurcatio & Chaos 8(7), [This alies the Lie algebra aroach i the third referece to Lyauov stability ad chaos.] Baks S.P., Riddalls C., ad McCaffrey D. (1997). The Schwartz kerel theorem ad the frequecydomai theory of oliear systems, Archives of Cotrol Sciece, 6(42), [This gives a direct derivatio of the Volterra series for geeral oliear systems.] Cook P. (1986). Noliear Dyamical Systems, 216. Lodo: PreticeHall. [A itroductory book o oliear systems, icludig the circle theorem.] Curtai R.F. ad Zwart H.J. (1991). A Itroductio to IfiiteDimesioal Liear Systems Theory, 698. New York: SrigerVerlag. [This moograh resets a detailed itroductio to systems theory, icludig iut outut stability for distributed arameter systems.] Desoer C.A. ad Vidyasagar M. (1975). Feedback Systems: Iut Outut Proerties, 260. New York: Academic Press. [A very detailed book o most asects of iut outut systems, ad multilier theory, icludig ocausal multiliers.] Georgiou T.T. ad Smith M.C. (1990). Otimal robustess i the ga metric. IEEE Trasactios of Automatic Cotrol 35, [A discussio of robustess of iut outut system stability.] Khalil H.K. (1992). Noliear Systems, 340. New York: Macmilla. [A recet book o geeral oliear systems ad stability.] Ecycloedia of Life Suort Systems (EOLSS)

5 CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XII Iut Outut Stability Baks S.P. Poov V.M. (1961). Absolute stability of oliear systems of automatic cotrol. Automatio ad Remote Cotrol 22, ,. [A semial aer o iut outut stability.] Sadberg I.W. (1964). O the L 2 boudedess of solutios of oliear fuctioal equatios. Bell Systems Techical Joural 43, [Oe of the first aers i the West o geeralized gai of systems.] Sastry S. (1999). Noliear Systems: Aalysis, Stability ad Cotrol, 630. New York: Sriger Verlag. [A thorough coverage of oliear stability theory.] Teel A.R., Georgiou T.T., Praly L., ad Sotag E. (1995). Iut outut stability. Cotrol Hadbook, Ed. W.S. Levie. Florida: CRT Press, [A comrehesive review of iut outut stability.] Vidyasagar M. (1993). Noliear Systems Aalysis, 428. Lodo: PreticeHall. [Geeral oliear systems theory.] Yosida K. (1970). Fuctioal Aalysis, 435. New York: SrigerVerlag. [Classic text o fuctioal aroach to aalysis.] Zames G. (1966a) O the iut outut stability of timevaryig oliear feedback systems. Part I: Coditios usig cocets of loo gai, coicity ad ositivity. IEEE Trasactios of Automatic Cotrol, 11, [Semial aer o iut outut stability.] Zames G. (1966b) O the iut outut stability of timevaryig oliear feedback systems. Part II: Coditios ivolvig circles i the frequecy lae ad sector oliearities, IEEE Trasactios of Automatic Cotrol, 11, [Part 2 of the last referece.] Biograhical Sketch S.P. Baks gaied his B.Sc. i 1960, M.Sc. i 1971, ad Ph.D. i 1974, all i mathematics. After comletig a seior research fellowshi i the Cotrol Theory Cetre, Uiversity of Warwick, ad a short stay i British Aerosace, he joied the Deartmet of Cotrol Egieerig (ow the Deartmet of Automatic Cotrol ad Systems Egieerig), Uiversity of Sheffield as a lecturer i 1980 ad became a full rofessor i He has ublished over 130 aers ad six books ad is curretly editor of IMA Joural of Mathematical Cotrol ad Iformatio ad a associate editor of Comutatioal ad Alied Mathematics. Ecycloedia of Life Suort Systems (EOLSS)