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When i]? exceeds Hh range, the fp tends tr become VP nearly contlat 3 show io fy « PY Sidems Bheg adtun aay ; 4 Byrds -ersib) the phensaann + N' Bahnation dur to Limitations sf physical e at these comp need = Se B® common shenyrmency 19 elechonte amplife eget Are | ahiralin 4 He 07? sersove fir ss Peli, velocity, tem pe ate. « ey,[ea Reel dove HB a reGer in Lich ondpt B ane fe oa ge hops. Some systems do nxt expend FO smell inp Sanels 7 faa potradew wonye 4 oe opt & Ker calbal dead zone. ele ms ie. Sv iP FHA — Kretien enish, Then tere ve motion beha cen Contndieg suclnces . Th fhedionel fon called jiscous Kickin. Vises Fie B prepehen) to se veloc Halt denfacep. 2? (rece) ws ate Fein B Larear in nokuve « oO Vig @ Mein) Ayan be ag Bpeax Fikes: @ Cronly sy eh censtork xeloeeling Pros bt enisly mm born B idkated. eH 4 R xFd t, maiatein Consto. hm . © fasanr- wea fee FT he yh. mebon. Bus men fore wet mee an heck fom vak HS Ae maietain It in motion Bo Stetien fra B “e ? 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Such conditions ar Span con to x, uniformly int equation (3.3) if Fa. 3) \itions for the existence and uni -¢ and uniqueness that /lx, 0 is Lipschitz continuous with ontinno of ‘ontinuous in 4. A point x* € Ry, is an Torey Intuitively and somewhat crudel s o icly speaking, we say an equi i solutions which start near x* (meaning that the intial Set aig eat ote c tial conditions are i ‘ remain near x* for all time. nditions are ina hood of x’ for instance, The equilibrium point x* i id to be Locally Asymptotically stabl furthermore, all solutions starting near tend towards x8 AS locally stable and, We say somewhat crude because the ti kinds of additional subtleties. Nonethel equilibrium point when the pendulum point when it is pointing straight up. If the pendul is locally asymptotically stable. By shifting the, equilibrium point of ition (3.3) introduces all J pendulum has a locally stable fown and an unstable equilibrium ped, the stable equilibrium point the system, we may assume that the terest occurs at x* multiple equilibrium points exist, we will need to study the stability of each by apprapriaggly shifting the origin. 3.1 Stability in the sense of Lyapuny ~ ‘The equilibrium point x* = 0 o} Wostable (in the sense of Lyapunov) at t= to if for any € > 0 there exists a d(to, € ) > 0 Sygh that wae (34) ICV? S ixwoiice, veet- wy stability mild requirement on equilibrium points. In particular, it does ot sagas sino ei ohn han sree stability is ds Se ea stant fo. Uniform stability is a concept which arte f t the equilibrityn point is not losing stability. We inst that for a uniformly stable equilibrium MD Definition 3.1 not be a function of fo, so that eavation. (GA) may hold for al in the following definition: fe stability is made preci ymptotic stability ium point x* = 0 of (3:3) is asymprotteaTy stable at t= toif += 0 is stable, and 0 is locally attractive: fen there exists (fo) such that 5) Ibxcta I <= fim x(0) = 0 a, asymptotic stability is defined at fo- As in the previous definit 88Beart Uniform asymptotic stability requires: Lx *=Oisuniformly stable,and sar k 2.x *=Ois uniformly locally attractive; ie., there exists 4 independent ee of 0 for which equation (3.5) holds. Further, it is required that the convergence in equ: (3.5) is uniform. %, Finally, we say that an equilibrium point is unstable if it is not stable. This a of a tautology than it sounds and the reader should be sure he or she ean negate the definition of stability in the sense of Lyapunov to get a definition of instability. In robot Cy are almost always interested in uniformly asymptotically stable equilibria. If we wis ve the robot to a point, we would like to actually converge (o that point, not merely, & waist nearby. Figure ion below illustrates the difference between stability in the sense of Lyay and asymptotic stability, Cera stem ns le if it is stable for all initial many applications it can be theorems and indicate where it is uniformity are only important for bility implies uniform stability Definitions 3.1 and 3.2 are /ocal definitions: they describe t equilibrium point. We say an equilibrium point x* is globe conditions x € Rj. Global stability is very d difficult to achieve. We will concentrate on local silty f Possible to extend the results to the global case. Noti time-varying systems. Thus, for time-invariantesstems, and asymptotic stability implies co stability. “4 Peery (b) Asymptotically stable (c) Unstable (saddle) Figure:3.20 Phase porvaits for stable ghd unstable equilibrium points. 89pasie theorem of Lyapunov Lev (8.1) be anon-negative function with derivative V along the trajectories ofthe system, LYVadis locally positive definite and V(x, 1) <0 locally in.x and for all , then the 1 in ofthe system i locals stable (nthe sense of Lyapuen) LIFV (sis locally positive definite and decrescent. and Vix, <0 locally in.x and for all {then the origin of the system is uniformly locally stable (in the sense of Lyapunov). 4. (6.1) is locally positive definite and decrescent, and ~ V(x, 1) locally positive » Ucfinite then the origin of the system is uniformly locally: asymptotically stable > 4. If V (&. 0) is positive definite and decrescent, and ~ V(x. 0) is positive defi kag the origin of the system is globally uniformly asymptotically stable. e Theorm-1 Ow Consider the system =0N Suppose there exists a scalar funetion v(x) which fay (eM number €> 0 satisfies the following properties for all x in the region trots (a) V(x)>0; x # 0 that is v(x) is positive ange salar function. WV) =0 (c)V(x) has continuous partial derj ith respect fo all component of X dv (a) S <0 (ie dvidtis negaiive’ fefinite scalar function) Then the system is ee ‘Theorem-2 If the property Ieoreme! is replaced with (d) = <0,x #0 (ie dvidt is negative Nonpathen the system is asymptotically stable. Gus since continuous ¥ function™0 except at {cept that x will eventually approach the origin We satisfies the condition shall avoid the rigorous of ‘all the conditions of theorem-2 hold and in addition. Vx) + as lIxll + Then the system is asymptotically stable in-the-large at the origin.Instability Itmay be noted stabi established by direct recourse to inear system can he established hy y be noted that instability in a nonlinear s) ied Oy area Tee i h instabilin theorem af the cet method! The basic instability theorem is presented below: the instability D re eet Sb Consider a system = £0); £0) = 0 oP Suppose there a exist a scalar function W(x) which, for real numbeNN@>0 , satisfies the following properties for all x in the region [IX] <¢ : omnivo:x#o © may =o (ING) has continuous partial derivatives with respect Lah ponen of Then the system is unstable at the origin. : Direct Method of Lispunoy & the Linea : S of liapunov provides @ simple approach to Stability analysis. 1 must be emphagi%ed that compared to the results presented, no new. Hout © obtained by the use of ireptimethod for ie stability analysis of linear systems, However, the study of lincarsaysto using the direct method is quite useful because it extends our thinking tor ies on s Consider a linear suena 'm described by the state equation KEAX In case of linear systems, the direc ++ (3.6) The linear syst gmplotically stable in-the-large at the origin if and only it symmetric, TeARite matrix Q, there exists @ symmetric povine which is the uni solution » A'P+ Pa=—a P 3 bove theorem, let US assume Higa ctinte matrix P exists which isthe unique seleren of e4n.3.8). C given any ite matrix P that a symmetric ‘onsider thescalar function Vix) =x Ps Note that Vix) > 0 forx * 9 ViO=0 And The time derivate of V(x) isV(X) = RT PX + XTPK Using eqns. (3.6) and (3.7) we get V(x) = sTATPs +s"PAs =xT(ATP + PA)x eet | “xX Qx since Q is positive definite, V(x) is negative definite, Norm of x may be defined . yy be defined as IXI = (xT Px)! Then Sw yo Te P ‘The system is therefore asymptotically stable in-the large at the oF In order to show that the result is also necessa i how that \ Fy, suppose that i stable and P is negative definite, consider the scalar function ri ieee Vox) = xTPX Mo vo >[errcaxtr] There is contradiction since V(x) ‘Therefore S ‘Eqn. (3.8) satisfies instability theorem. fe Thus the conditions for th definiteness of P are necessary and sufficient for asymptotic stability of the sfsttin of eqn. (3.6). Methods of construc paramos functions for Non linear Systems As has been Sa, “the liapunov theorems give only sufficient conditions on system stability and fubkggtmore there is no unique way of constructing a liapunov function except in ear systems where a liapunov function can always be constructed and both sufficient conditions Established .Because of this draw back a host of methods, fomie available in literature and many refinements have been suggested to enlarge the ‘which the system is found to be stable. Since this treatise is meant as a first re of the student to the liapunoy direct method, only two of the relatively simpler niques of constructing a liapunov's function would be advanced here, Krasoyskii's method Consider a system Define a liapunoy function as 92ve=f'Pr .. Where P=a symmetric positive definite matrix. Now V=ftpr+ tpt ;_ of aX OX Fi Of Oh hy ax, Ox, OXy Of, df Dy Ox; Ox, Ox, af, fy x, Ox, ‘Substituting in eqn (3.10), we have : = + ETRE 2 TOTP + PYE Let Qe AS a=resry Since V is positive d or the system to be asymptotically stable, Q should be negative definite. If in ad ‘ iz) ~@ as IIXIl — 2, the system is asymptotically stave in-the- large.