Dynamical Systems & Lyapunov Stability

Dynamical Systems & Lyapunov Stability Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University

Outline Ordinary Differential Equations Existence & uniqueness Continuous dependence on parameters Invariant sets, nonwandering sets, limit sets Lyapunov Stability Autonomous systems Basic stability theorems Stable, unstable & center manifolds Control Lyapunov function

Basics of Nonlinear ODE s

Dynamical Systems d x ( t ) ( ) n = f x t, t, x R, t R non-autonomous dt d x ( ) n = f x t, R, R dt autonomous A solution on a time interval t, is a function [ t t ] n xt :[ t, t] R that satisfies the ode.

Vector Fields and Flow n We can visualize an individual solution as a graph xt : t R. For autonomous systems it is convenient to think of ( x) n as a vector field on R - f( x) assigns a vector to each point in R. As t varies, a solution xt traces a path through R n ( x) tangent to the field f. These curves are often called trajectories or orbits. The collection of all trajectories in R of the vector field f( x). n is called the flow f n

Auto at Constant Speed x V s = V V δ d dt ω = β f ( ωβ,, V, δ ) Body Frame y β u Y v θ m, J a b F l Space Frame X F r Notice the three equilibria.

Van der Pol x ( x = ).8 x x x x

Damped Pendulum x x = x x / sin x

Lipschitz Condition The existence and uniqueness of solutions depend on properties of the function f. In many applications f( xt, ) has continuous derivatives in x. We relax this - we require that f is Lipschitz in x. Def : f : R R is locally Lipschitz on an open subset D R if each point x D has a neighborhood U such that for some constant Note: n n n f x f x L x x C always are. L and all x U (continuous) functions need not be Lipschitz, C functions

The Lipschitz Condition A Lipschitz continuous function is limited in how fast it can change, A line joining any two points on the graph of this function will never have a slope steeper than its Lipschitz constant L, The mean value theorem can be used to prove that any differentiable function with bounded derivative is Lipschitz continuous, with the Lipschitz constant being the largest magnitude of the derivative.

Examples: Lipschitz 5 45 4 35 3 =, [ 3, 7 ] L = 4 f x x x 5 4.5 4 3.5 3 L = y 5.5 5 5.5.5 f x = x +, x R -3 - - 3 4 5 6 7 x 5-5 -4-3 - - 3 4 5 4.5 4 3.5 L =.8.6.4 Fails 3..5 =.5, [ 5,5] f x x x.5.8.6.4. =, [, 4] f x x x - -.5 - -.5.5.5 -...4.6.8..4.6.8

Local Existence & Uniqueness Proposition (Local Existence and Uniqueness) Let f( xt, ) be piece-wise continuous in t and satisfy the Lipschitz condition (, ) (, ) f xt f yt L x y { n } for all xy, B= x R x x < r and all t [ t, t]. Then r there exists δ > such that the differential equation with initial condition (, ), x= f xt x t = x B r has a unique solution over [ t, t + δ ].

The Flow of a Vector Field x = f( x), x t = x x x, t this notation indicates 'the solution of the ode that passes through x at t = ' ( xt) More generally, let Ψ, denote the solution that passes through at. The function : n n x t = Ψ R R R satisfies Ψ ( xt, ) t ( (, )), (,) = f Ψ xt Ψ x = x Ψ is called the flow or flow function of the vector field f

Example: Flow of a Linear Vector Field ( xt, ) Ψ = = Ψ(, ) Ψ (, ) = t Example: At x Ax A xt xt e x + x cos t x sin t 3 x R, A=, Ψ ( xt, ) = xcos t xsin t t e x x 3.5 -.5 -. x3 -. - -.5 x.5

Invariant Set A set of points S R n is invariant with respect to the vector field f if trajectories beginning in S remain in S both forward and backward in time. Examples of invariant sets: any entire trajectory (equilibrium points, limit cycles) collections of entire trajectories

Example: Invariant Set x x x d x = x dt x 3 x 3 - -.5.5. x3 each of the three trajectories shown are invariant sets the x -x plane is an invariant set -. - -.5 x.5

Limit Points & Sets A point ( t p) is called an ω-limit point of the trajectory Ψ, if there exists a sequence of time values t + such that q lim Ψ t, p = q t k ( t p) is said to be an α-limit point of Ψ, if there exists a sequence t q R k n of time values lim Ψ t, p = q k k t k such that The set of all ω-limit points of the trajectory through p is the ω-limit set, and the set of all α-limit points is the α-limit set. For an example see invariant set k

Introduction to Lyapunov Stability Analysis

Lyapunov Stability ε,, n x = f x f = f : D R locally Lipschitz The origin is δ a stable equilibrium point if for each ε >, there is a δ ε > such that x < x t < ε t > unstable if it is not stable, and asymptotically stable if δ can be chosen such that x δ x( t) < lim t δ x

Two Simple Results The origin is asymptotically stable only if it is isolated. The origin of a linear system x = Ax At is stable if and only if e N < t > It is asymptotically stable if and only if, in addition At e, t

Example: Non-isolated Equilibria x x = x x x x All points on the x axis are equilibrium points x 4-3 - - 3 x - -4

Positive Definite Functions n A function V : R R is said to be negative (semi-) definite if V ( x) radially unbounded if V ( x) as x T T V ( x) = x Qx Q = Q positive definite if V = and V x >, x positive semi-definite if V = and V x, x is positive (semi-) definite For a quadratic form:, the following are equivalent V x is positive definite the eigenvalues of Q are positive the principal minors of Q are positive

Lyapunov Stability Theorem V x is called a Lyapunov function relative to the flow of x = f x respect to the flow: if it is positive definite and nonincreasing with V ( x) V =, V x > for x V = f x x Theorem : If there exists a Lyapunov function on some neighborhood D is negative definite on of the origin, then the origin is stable. If D then it is asymptotically stable. D V = α V = β < α V = γ < β V

Example: Linear System T T V x = x Qx, Q = Q > where x = Ax T T T T T V = x Qx + x Qx = x QA + A Q x = x Px T QA A Q P + = Lyapunov Equation At P > Ais stable (Hurwitz) : e bounded t > So we can specify Q, compute P and test P. Or, specify P and solve Lyapunov equation for Q and test Q.

Example: Rotating Rigid Body xyz,, body axes; ω, ω, ω angular velocities in body coord's; x y z diag I, I, I I I I > inertia matrix x y z x y z Iz Iy ωx = ωω z y = aωω z y I x Ix I z ωy = ωω x z = bωω x z note abc,, > I y Iy Ix ωz = ωyωx = cωω y x I z z y x

Rigid Body, Cont d ( ωx ωy ωz) Note that a state,, is an equilibrium point if any two of the angular velocity components are zero, i.e., the ω, ω, ω axes are all equilibrium points. x y z Consider a point ω,,. Shift ω ω + ω. ω = aωω x z y b ω = ω + ω ω y x x z ω = c ω + ω ω z x x y x x x x z y x ω x ω =

Rigid Body, Cont d Energy does not work for ω. Obvious? So, how do we find Lyapunov function? We want V V,, =, x ( ωx, ωy, ωz) if ( ωx, ωy, ωz) (,, ) and ( ωx, ωy, ωz) > D V Lets look at all functions that satisfy V =,i.e., that satisfy the pde: V V V aωzωy + ( b( ωx + ωx) ωz) + c( ωx + ωx) ωy = ω ω ω x y z All solutions take the form: bω + bωω + aω cω + cωω + aω f, a a x x x y x x x z 8bcω x V ( ωx, ωy, ωz) = ca + bb + ( ca bb) = ω... x + cωy + bωz + h o t a

Rigid Body, Cont d Clearly, V =, V > on a neighborhood D of V = spin about x-axis is stability This is one approach to finding candidate Lyapunov functions The first order PDE usually has many solutions The method is connected to traditional first integral methods to the study of stability in mechanics Same method can be used to prove stability for spin about z-axis, but spin about y-axis is unstable why?

LaSalle Invariance Theorem Theorem : { n x R V x < c} n Suppose : is and let c denote a component of the region Suppose Ω is bounded and within Ω, V x. c V R R C Let E be the set of points within Ω where V x =, Let M be the largest invariant set within E. every sol'n beginning in Ω tends to M. as t. c c c Ω

Example: LaSalle s Theorem d dt x x, x = m kx m cx V V ( x, x ) = mx + kx V x x = cx (, ) M Ω c x E x

Lagrangian Systems (, ) L( xx, ) d L xx T = Q dt x x n x R generalized coordinates x = dx / dt generalized velocities ( ) ( ) T T( xx ) = xm ( x) x ( ) = ( ) + L R R L xx = T xx U x n : is the Lagrangian,,, kinetic energy:, total energy: V xx, T xx, U x

Lagrange-Poincare Systems (, ) L( px, ) d L px x = p, = Q dt p x T (, ) = (, ), (, ) = (, ) d (, ) =, = + L px T px U x T px pm x p pm x pm x p p dt p x T L px T L px T T pm x p M x M x p U x pm ( x) + p p + = Q x x x T T T T T x = p T T M x p p M x U x M ( x) p + p+ = Q T x x x

Example x x x T U =, =, = + x x x x x = cx ( + x ) x x V ( x) = +, V ( x) = cx for c> + x Notice that x the level sets are unbounded for V x = constant V x is not radially unbounded

Example, Cont d 4 4 x x 8 6 4 5 5 5 3 t

First Integrals Definition : A first integral of the differential equation (, ) x = f xt ( xt) ( xt) is a scalar function ϕ, that is constant along trajectories, i.e., ( xt) ϕ, ϕ, ϕ ( xt, ) = f( x, t) + x t Observation : For simplicity, consider the autonomous case x = f x. Suppose ϕ x is a first integral ϕn ϕ ( x) and ϕ x,, x are arbitrary independent functions on a neighborhhod of the point x, i.e., det x ϕ ( x) n x= x Then we can define coordinate transformation x ( x) ( z) z, via ϕ z = ϕ ( x) z = f ( x) z = z constant x x= ϕ The problem has been reduced to solving n differential equations.

Noether s Theorem If the Lagrangian is invariant under a smooth parameter change of coordinates, h : M M, s R, then the Lagrangian system has a first integral Φ ( pq, ) L = dh s ( q) p ds s= s F X, v M M z θω, m x M + m m cosθ v T ( p, q) = [ v ω], cos U q mg θ m cosθ m ω = M + m m cosθ v m ωsinθ v m cosθ m ω + mωsinθ mvsinθ ω F + mg sinθ = X + s hs ( X, θ ) =, Φ ( pq, ) = ( M+ mv ) Momentum in X direction

Chetaev s Method Consider the system of equations x = f xt,, f, t = We wish to study the stability of the equilibrium point x =. ( xt) Obviously, if ϕ, is a first integral and it is also a positive definite function, then V xt, is not positive definite? = ϕ ( xt, ) establishes stability. But suppose ϕ ( xt, ) Suppose the system has kfirst integrals ϕ xt,,, ϕ xt, such that ϕ, t =. Chetaev suggested the construction of Lyapunov functions of the form: k k (, ) = αϕ (, ) + βϕ (, ) V xt xt xt i= i i i= i i k i

Chetaev Instability Theorem Let D be a neighborhood of the origin. Suppose there is a function V x : D R and a set D D such that ) V x is C on D, ) the origin belongs to the boundary of D, D, 3) V x >, V x > on D, 4) on the boundary Then the origin is unstable. of D inside D, i.e., on D DV, x = D r U x( t) D

Example, Rigid Body, Cont d ( ωy ) Consider the rigid body with spin about the y-axis (intermediate inertia), ω =,, Shifted equations: ω = aω ω + ω x z y y ω = bωω y x z ω = c ω + ω ω z y y x Attempts to prove stability fails. So, try to prove instability. V Consider ω, ω, ω = ωω x y z x z { ω ω ω ω ω ω } {( ω ω ω ) ω ω } Let B =,, + + < r and D =,, B >, > r x y z x y z x y z r x z z so that V > on D and V = on D r < ωy ωy + ωy > ( ωx ωy ωz) Br V = aω ω + ω + c ω + ω ω = ω + ω aω + cω z y y y y x y y z x We can take,, in which case V > on D instability B r D T x

Stability of Linear Systems - Summary Consider the linear system x = Ax Choose V x T T T V x = x A P + PA x : = x Qx a) if their exists a positive definite pair of matrices PQ, that satisfy = T x Px T A P + PA = Q (Lyapunov equation) the origin is asymptotically stable. b) if P has at least one negative eigenvalue and Q>, the origin is unstable. c) if the origin is asymptotically stable then for any Q >, there is a unique solution, P >, of the Lyapunov equation. sufficient condition necessary condition

Second Order Systems Consider the system T T T Mx + Cx + Kx =, M = M >, C = C >, K = K > (, ) T T E x x = x Mx + x Kx d E x x x T Mx x T Kx x T Cx Kx x T Kx dt (, ) [ ] d E ( x, x ) = dt = + = + + x T Cx Some interesting generalizations: T T ) C, ) C C, 3) K K The anti-symmetric terms correspond to circulatory forces (transfer conductances in power systems) they are nonconservative. The anti-symmetric terms correspond to gyroscope forces they are conservative.

Example V, δ -ja Assume uniform damping Assume e= Designate Gen as swing bus -jb -jd 4 -jc V 4, δ 4 Eliminate internal bus 4 e-jf V 3, δ 3 3 θ + γθ = P b sin θ b θ θ sin sin 3 θ + γθ = P + b sin θ θ b θ 3 θ = δ δ, θ = δ δ, P = P P, P = P P 3 3

Example Cont d This is a Lagrangian system with ( θ, θ ) = θ θ cos( θ ) cos( θ θ ) cos( θ ) U P P b b b T 3 3 ( ω, ω ) = ( ω + ω ), Q = [ γω γω ] To study stability choose total energy as Lyapunov function ( ω, ω ) ( θ, θ ) V = T + U V = γω γω T( ω ω) ( ω ω) U ( θ θ ) Note: T, = and, >, Equilibria corresponding to, a local minimum are stable. G. V. Arononvich and N. A. Kartvelishvili, "Application of Stability Theory to Static and Dynamic Stability Problems of Power Systems," presented at Second All-union Conference on Theoretical and Applied mechanics, Moscow, 965.

Example, Cont d Since π θ < π and π θ < π we should consider ( θ θ ) U, as a function on a torus U : T R 3 - - -3 - - P =, P =, b =, b =, b = 3 3 - - -3-3 - - 3

Example, Cont d 3 - - - - - -3-3 - - 3 P =.5, P =, b =, b =, b = 3 3

Example Cont d 3 - - - - - -3-3 - - 3 P = π /5, P =, b =, b =, b = 3 3

Example Cont d 3 5.5 -.5 - - - - -3-3 - - 3 P = π /5, P =, b =, b =.5, b = 3 3

Control Lyapunov Function Consider the controlled system n x = f xu,, x D R containing x=, u R Find u x such that all trajectories beginning in D converge to x=. Definition : A control Lyapunov function (CLF) is a function V : D R with V =, V x >, x, such that V x D, u( x) V ( x) = f ( x, u( x) ) < x Theorem : (Artsteins Theorem) A differentiable CLF exists iff there exists a 'regular' feedback control u x. m

Example Consider the nonlinear mass-spring-damper system: v = q, m + q v + bv + k q + k q = u 3 v d q ( 3 ) dt v = k q kq bv u + m + q suppose the target state is v=, q=, Define r = v + αq, α >. A CLF candidate is V = r. This is positive definite wrt the target state 3 u bv kq kq V = rr = ( v + αq) v = v + αq κv, κ = > m( + q ) 3 u bv kq kq + α = κ = + + + α + κ m( + q ) d q v closed loop dt v = κv αq 3 q v u bv kq kq m q q v