# Lecture 12 Basic Lyapunov theory

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1 EE363 Winter Lecture 12 Basic Lyapunov theory stability positive definite functions global Lyapunov stability theorems Lasalle s theorem converse Lyapunov theorems finding Lyapunov functions 12 1

2 Some stability definitions we consider nonlinear time-invariant system ẋ = f(x), where f : R n R n a point x e R n is an equilibrium point of the system if f(x e ) = 0 x e is an equilibrium point x(t) = x e is a trajectory suppose x e is an equilibrium point system is globally asymptotically stable (G.A.S.) if for every trajectory x(t), we have x(t) x e as t (implies x e is the unique equilibrium point) system is locally asymptotically stable (L.A.S.) near or at x e if there is an R > 0 s.t. x(0) x e R = x(t) x e as t Basic Lyapunov theory 12 2

3 often we change coordinates so that x e = 0 (i.e., we use x = x x e ) a linear system ẋ = Ax is G.A.S. (with x e = 0) Rλ i (A) < 0, i = 1,..., n a linear system ẋ = Ax is L.A.S. (near x e = 0) Rλ i (A) < 0, i = 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 f is nonlinear, establishing any kind of stability is usually very difficult Basic Lyapunov theory 12 3

4 Energy and dissipation functions consider nonlinear system ẋ = f(x), and function V : R n R we define V : R n R as V (z) = V (z) T f(z) V (z) gives d V (x(t)) when z = x(t), ẋ = f(x) dt we can think of V as generalized energy function, and V as the associated generalized dissipation function Basic Lyapunov theory 12 4

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

6 Lyapunov theory Lyapunov theory is used to make conclusions about trajectories of a system ẋ = f(x) (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 V : R n R that satisfies some conditions on V and V then, trajectories of system satisfy some property if such a function V exists we call it a Lyapunov function (that proves the property holds for the trajectories) Lyapunov function V can be thought of as generalized energy function for system Basic Lyapunov theory 12 6

7 A Lyapunov boundedness theorem suppose there is a function V that satisfies all sublevel sets of V are bounded V (z) 0 for all z then, all trajectories are bounded, i.e., for each trajectory x there is an R such that x(t) R for all t 0 in this case, V is called a Lyapunov function (for the system) that proves the trajectories are bounded Basic Lyapunov theory 12 7

8 to prove it, we note that for any trajectory x V (x(t)) = V (x(0)) + t 0 V (x(τ)) dτ V (x(0)) so the whole trajectory lies in {z V (z) V (x(0))}, which is bounded also shows: every sublevel set {z V (z) a} is invariant Basic Lyapunov theory 12 8

9 A Lyapunov global asymptotic stability theorem suppose there is a function V such that V is positive definite V (z) < 0 for all z 0, V (0) = 0 then, every trajectory of ẋ = f(x) converges to zero as t (i.e., the system is globally asymptotically stable) intepretation: V is positive definite generalized energy function energy is always dissipated, except at 0 Basic Lyapunov theory 12 9

10 Proof suppose trajectory x(t) does not converge to zero. V (x(t)) is decreasing and nonnegative, so it converges to, say, ǫ as t. Since x(t) doesn t converge to 0, we must have ǫ > 0, so for all t, ǫ V (x(t)) V (x(0)). C = {z ǫ V (z) V (x(0))} is closed and bounded, hence compact. So V (assumed continuous) attains its supremum on C, i.e., sup z C V = a < 0. Since V (x(t)) a for all t, we have V (x(t)) = V (x(0)) + Z T 0 V (x(t)) dt V (x(0)) at which for T > V (x(0))/a implies V (x(0)) < 0, a contradiction. So every trajectory x(t) converges to 0, i.e., ẋ = f(x) is G.A.S. Basic Lyapunov theory 12 10

11 A Lyapunov exponential stability theorem suppose there is a function V and constant α > 0 such that V is positive definite V (z) αv (z) for all z then, there is an M such that every trajectory of ẋ = f(x) satisfies x(t) Me αt/2 x(0) (this is called global exponential stability (G.E.S.)) idea: V αv gives guaranteed minimum dissipation rate, proportional to energy Basic Lyapunov theory 12 11

12 Example consider system ẋ 1 = x 1 + g(x 2 ), ẋ 2 = x 2 + h(x 1 ) where g(u) u /2, h(u) u /2 two first order systems with nonlinear cross-coupling 1 s + 1 x 1 g( ) h( ) x 2 1 s + 1 Basic Lyapunov theory 12 12

13 let s use Lyapunov theorem to show it s globally asymptotically stable we use V = (x x 2 2)/2 required properties of V are clear (V 0, etc.) let s bound V : V = x 1 ẋ 1 + x 2 ẋ 2 = x 2 1 x x 1 g(x 2 ) + x 2 h(x 1 ) x 2 1 x x 1 x 2 (1/2)(x x 2 2) = V where we use x 1 x 2 (1/2)(x x 2 2) (derived from ( x 1 x 2 ) 2 0) we conclude system is G.A.S. (in fact, G.E.S.) without knowing the trajectories Basic Lyapunov theory 12 13

14 Lasalle s theorem Lasalle s theorem (1960) allows us to conclude G.A.S. of a system with only V 0, along with an observability type condition we consider ẋ = f(x) suppose there is a function V : R n R such that V is positive definite V (z) 0 the only solution of ẇ = f(w), V (w) = 0 is w(t) = 0 for all t then, the system ẋ = f(x) is G.A.S. Basic Lyapunov theory 12 14

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

16 A Lyapunov instability theorem suppose there is a function V : R n R such that V (z) 0 for all z (or just whenever V (z) 0) there is w such that V (w) < V (0) then, the trajectory of ẋ = f(x) with x(0) = w does not converge to zero (and therefore, the system is not G.A.S.) to show it, we note that V (x(t)) V (x(0)) = V (w) < V (0) for all t 0 but if x(t) 0, then V (x(t)) V (0); so we cannot have x(t) 0 Basic Lyapunov theory 12 16

17 A Lyapunov divergence theorem suppose there is a function V : R n R such that V (z) < 0 whenever V (z) < 0 there is w such that V (w) < 0 then, the trajectory of ẋ = f(x) with x(0) = w is unbounded, i.e., sup t 0 x(t) = (this is not quite the same as lim t x(t) = ) Basic Lyapunov theory 12 17

18 Proof of Lyapunov divergence theorem let ẋ = f(x), x(0) = w. let s first show that V (x(t)) V (w) for all t 0. if not, let T denote the smallest positive time for which V (x(t)) = V (w). then over [0, T], we have V (x(t)) V (w) < 0, so V (x(t)) < 0, and so Z T 0 V (x(t)) dt < 0 the lefthand side is also equal to Z T 0 V (x(t)) dt = V (x(t)) V (x(0)) = 0 so we have a contradiction. it follows that V (x(t)) V (x(0)) for all t, and therefore V (x(t)) < 0 for all t. now suppose that x(t) R, i.e., the trajectory is bounded. {z V (z) V (x(0)), z R} is compact, so there is a β > 0 such that V (z) β whenever V (z) V (x(0)) and z R. Basic Lyapunov theory 12 18

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

20 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 form of the Lyapunov function example: if the linear system ẋ = 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

21 A converse Lyapunov G.E.S. theorem suppose there is β > 0 and M such that each trajectory of ẋ = f(x) satisfies x(t) Me βt x(0) for all t 0 (called global exponential stability, and is stronger than G.A.S.) then, there is a Lyapunov function that proves the system is exponentially stable, i.e., there is a function V : R n R and constant α > 0 s.t. V is positive definite V (z) αv (z) for all z Basic Lyapunov theory 12 21

22 Proof of converse G.E.S. Lyapunov theorem suppose the hypotheses hold, and define V (z) = 0 x(t) 2 dt where x(0) = z, ẋ = f(x) since x(t) Me βt z, we have V (z) = 0 x(t) 2 dt 0 M 2 e 2βt z 2 dt = M2 2β z 2 (which shows integral is finite) Basic Lyapunov theory 12 22

23 let s find V (z) = d dt V (x(t)), where x(t) is trajectory with x(0) = z t=0 V (z) = lim(1/t) (V (x(t)) V (x(0))) t 0 ( = lim(1/t) x(τ) 2 dτ t 0 = lim t 0 ( 1/t) = z 2 t t 0 x(τ) 2 dτ 0 ) x(τ) 2 dτ now let s verify properties of V V (z) 0 and V (z) = 0 z = 0 are clear finally, we have V (z) = z T z αv (z), with α = 2β/M 2 Basic Lyapunov theory 12 23

24 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 verifying 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

25 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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