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# EE Winter Lecture Basic Lyapunov theory stability positive denite functions global Lyapunov stability theorems Lasalles theorem converse Lyapunov theorems nding Lyapunov functions Some stability

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

## EE Winter Lecture Basic Lyapunov theory stability positive denite functions global Lyapunov stability theorems Lasalles theorem converse Lyapunov theorems nding Lyapunov functions Some stability

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## Presentation on theme: "EE Winter Lecture Basic Lyapunov theory stability positive denite functions global Lyapunov stability theorems Lasalles theorem converse Lyapunov theorems nding Lyapunov functions Some stability"â€” Presentation transcript:

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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