AS if for every trajectory we have as implies is the unique equilibrium point system is locally asymptotically stable LAS near or at if there is an R st 0 k as Basic Lyapunov theory 122 brPage 3br often we change coordinates so that 0 ie we use a ID: 25469 Download Pdf

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AS if for every trajectory we have as implies is the unique equilibrium point system is locally asymptotically stable LAS near or at if there is an R st 0 k as Basic Lyapunov theory 122 brPage 3br often we change coordinates so that 0 ie we use a

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EE363 Winter 2008-09 Lecture 12 Basic Lyapunov theory stability positive deﬁnite functions global Lyapunov stability theorems Lasalle’s theorem converse Lyapunov theorems ﬁnding Lyapunov functions 12–1

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Some stability deﬁnitions we consider nonlinear time-invariant system , where a point is an equilibrium point of the system if ) = 0 is an equilibrium point ) = is a trajectory suppose is an equilibrium point system is globally asymptotically stable (G.A.S.) if for every trajectory , we have as (implies is the unique equilibrium point)

system is locally asymptotically stable (L.A.S.) near or at if there is an R > s.t. (0) k as Basic Lyapunov theory 12–2

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often we change coordinates so that = 0 i.e. , we use a linear system Ax is G.A.S. (with = 0 ⇔< = 1 , . . . , n a linear system Ax is L.A.S. (near = 0 ⇔< = 1 , . . . , n (so for linear systems, L.A.S. G.A.S.) there are many other variants on stability ( e.g. , stability, uniform stability, exponential stability, . . . ) when is nonlinear, establishing any kind of stability is usually ve ry diﬃcult Basic Lyapunov theory 12–3

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Energy and dissipation functions consider nonlinear system , and function we deﬁne as ) = gives dt )) when we can think of as generalized energy function , and as the associated generalized dissipation function Basic Lyapunov theory 12–4

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Positive deﬁnite functions a function is positive deﬁnite (PD) if for all ) = 0 if and only if = 0 all sublevel sets of are bounded last condition equivalent to as example: ) = Pz , with , is PD if and only if P > Basic Lyapunov theory 12–5

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Lyapunov theory Lyapunov theory is used to make conclusions about

trajectories o f a system e.g. , G.A.S.) without ﬁnding the trajectories i.e. , solving the diﬀerential equation) a typical Lyapunov theorem has the form: if there exists a function that satisﬁes some conditions on and then , trajectories of system satisfy some property if such a function exists we call it a Lyapunov function (that proves the property holds for the trajectories) Lyapunov function can be thought of as generalized energy function for system Basic Lyapunov theory 12–6

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A Lyapunov boundedness theorem suppose there is a function that

satisﬁes all sublevel sets of are bounded for all then, all trajectories are bounded, i.e. , for each trajectory there is an such that k for all in this case, is called a Lyapunov function (for the system) that proves the trajectories are bounded Basic Lyapunov theory 12–7

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to prove it, we note that for any trajectory )) = (0)) + )) d (0)) so the whole trajectory lies in (0)) , which is bounded also shows: every sublevel set is invariant Basic Lyapunov theory 12–8

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A Lyapunov global asymptotic stability theorem suppose there is a function such that is

positive deﬁnite for all = 0 (0) = 0 then, every trajectory of converges to zero as i.e. , the system is globally asymptotically stable) intepretation: is positive deﬁnite generalized energy function energy is always dissipated, except at Basic Lyapunov theory 12–9

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Proof suppose trajectory does not converge to zero. )) is decreasing and nonnegative, so it converges to, say, as Since doesn’t converge to , we must have > , so for all )) (0)) (0)) is closed and bounded, hence compact. So (assumed continuous) attains its supremum on i.e. sup a < . Since ))

for all , we have )) = (0)) + )) dt (0)) aT which for T > V (0)) /a implies (0)) , a contradiction. So every trajectory converges to i.e. is G.A.S. Basic Lyapunov theory 12–10

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A Lyapunov exponential stability theorem suppose there is a function and constant α > such that is positive deﬁnite αV for all then, there is an such that every trajectory of satisﬁes k Me αt/ (0) (this is called global exponential stability (G.E.S.)) idea: αV gives guaranteed minimum dissipation rate, proportional to energy Basic Lyapunov theory 12–11

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Example consider system where |≤| |≤| two ﬁrst order systems with nonlinear cross-coupling + 1 + 1 Basic Lyapunov theory 12–12

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let’s use Lyapunov theorem to show it’s globally asymptotically sta ble we use = ( required properties of are clear ( , etc.) let’s bound ) + (1 2)( where we use | (1 2)( (derived from |−| we conclude system is G.A.S. (in fact, G.E.S.) without knowing the trajectories Basic Lyapunov theory 12–13

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Lasalle’s theorem Lasalle’s theorem (1960) allows us to conclude G.A.S. of a syste m with only , along with an

observability type condition we consider suppose there is a function such that is positive deﬁnite the only solution of ) = 0 is ) = 0 for all then, the system is G.A.S. Basic Lyapunov theory 12–14

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last condition means no nonzero trajectory can hide in the “zero dissipation” set unlike most other Lyapunov theorems, which extend to time-varyin systems, Lasalle’s theorem requires time-invariance Basic Lyapunov theory 12–15

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A Lyapunov instability theorem suppose there is a function such that for all (or just whenever there is such that < V (0) then, the

trajectory of with (0) = does not converge to zero (and therefore, the system is not G.A.S.) to show it, we note that )) (0)) = < V (0) for all but if , then )) (0) ; so we cannot have Basic Lyapunov theory 12–16

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A Lyapunov divergence theorem suppose there is a function such that whenever there is such that then, the trajectory of with (0) = is unbounded, i.e. sup (this is not quite the same as lim Basic Lyapunov theory 12–17

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Proof of Lyapunov divergence theorem let (0) = . let’s ﬁrst show that )) for all if not, let denote the smallest positive time

for which )) = . then over [0 , T , we have )) , so )) , and so )) dt < the lefthand side is also equal to )) dt )) (0)) = 0 so we have a contradiction. it follows that )) (0)) for all , and therefore )) for all now suppose that k i.e. , the trajectory is bounded. (0)) k is compact, so there is a β > such that whenever (0)) and k Basic Lyapunov theory 12–18

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we conclude )) (0)) βt for all , so )) , a contradiction. Basic Lyapunov theory 12–19

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Converse Lyapunov theorems a typical converse Lyapunov theorem has the form if the trajectories of system

satisfy some property then there exists a Lyapunov function that proves it a sharper converse Lyapunov theorem is more speciﬁc about the f orm of the Lyapunov function example: if the linear system Ax is G.A.S., then there is a quadratic Lyapunov function that proves it (we’ll prove this later) Basic Lyapunov theory 12–20

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A converse Lyapunov G.E.S. theorem suppose there is β > and such that each trajectory of satisﬁes k Me βt (0) for all (called global exponential stability , and is stronger than G.A.S.) then, there is a Lyapunov function that proves

the system is exponen tially stable, i.e. , there is a function and constant α > s.t. is positive deﬁnite αV for all Basic Lyapunov theory 12–21

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Proof of converse G.E.S. Lyapunov theorem suppose the hypotheses hold, and deﬁne ) = dt where (0) = since k Me βt , we have ) = dt βt dt (which shows integral is ﬁnite) Basic Lyapunov theory 12–22

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let’s ﬁnd ) = dt =0 )) , where is trajectory with (0) = ) = lim (1 /t ) ( )) (0))) = lim (1 /t d d = lim /t d −k now let’s verify properties of and ) = 0 = 0 are clear

ﬁnally, we have ) = αV , with = 2 β/M Basic Lyapunov theory 12–23

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Finding Lyapunov functions there are many diﬀerent types of Lyapunov theorems the key in all cases is to ﬁnd a Lyapunov function and verify that it has the required properties there are several approaches to ﬁnding Lyapunov functions and v erifying the properties one common approach: decide form of Lyapunov function ( e.g. , quadratic), parametrized by some parameters (called a Lyapunov function candidate try to ﬁnd values of parameters so that the required

hypotheses hold Basic Lyapunov theory 12–24

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Other sources of Lyapunov functions value function of a related optimal control problem linear-quadratic Lyapunov theory (next lecture) computational methods converse Lyapunov theorems graphical methods (really!) (as you might guess, these are all somewhat related) Basic Lyapunov theory 12–25

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