ME 680- Spring 014 Representation and Stability Concepts 1
3. Representation and stability concepts 3.1 Continuous time systems: Consider systems of the form x F(x), x n (1) where F : U Vis a mapping U,V R n The function F is also called a vector field. Given an initial condition x 0 U in the domain of the definition of the function F, one gets a solution x(x 0, t), t R + which is called an orbit through x 0 U. The family of solutions generated by the collection of solutions through every point x 0 U is called a flow.
3.1 Continuous time systems.. Thus, the flow is a mapping: n : U with (x ) x(x,t) () t t 0 0 Note that the mapping satisfies the differential equation (1). Thus, d (x,t) F( (x, )) (3) dt t One of the key properties is the semi-group property: 3
3.1 Continuous time systems.. Examples of flow include: Ex 1. t ta (x) e x (4) n n Here : and t k ta (At) e k0 k! Ex. Consider the flow defined by: 1 cos t sin t n n n cn sinnx (u 0,t) (5) n0 1 dn sinnx sin n t cos n t n so that (u,0) u c sinnx n 0 0 n0 dn sinnx 4
3.1 Continuous time systems.. Ex... A careful thought will suggest that the flow in (5) represents solution of a partial differential equation where u(x, t) represents the state of the system. In fact, the flow represents solution of a time-dependent boundary value problem: 4 u u u p 0, t 0, 0 x 1. (6) 4 x x t This represents equation of motion for linear (small) motions for an axially loaded elastic rod: x u(x,t) x=1 P 5
3.1 Continuous time systems.. Ex... The boundary conditions for the pinned-pinned rod are: u u(x,t) 0, x t 0, x 0,1. (7) The initial conditions are: u(x,t) x=1 p u(x,0) c sinnx, n0 n u (x,0) d n sinn x, 0 x 1. (8) t n0 The solution of (6) with the given boundary and initial conditions is d u(x,t) (c sinnx cos t sinnx sin t) (9) n0 with n (n p) n n n n n n x 6
3. Representation and stability concepts 3. Discrete time systems: Consider systems of the form x p1 F(x p ), x n (10) where F( ) : U V is a mapping with, U, V R n The function F( ) is also called a Poincare mapping. Given an initial condition x 1 U, in the domain of definition of the function F( ), one gets a solution x(x 1, p), p R + which is called an orbit through x 1 U. The sequence of points generated as p is increased is also called the forward iterates. 7
3. Discrete time systems Such discrete time systems can arise in two different ways: through the natural definition of discrete-time events for a system where states are only defined at discrete times; and in systems where discrete events are defined for convenience. Ex 3. Consider the system of the ball bouncing on an oscillating table: Y(t) mg ball g As was shown, its dynamic response can be studied by two simultaneous finite-time table equations; they relate relative velocities. ground 8
3. Discrete time systems Thus, the dynamics is governed by: W ( i 1) [sin( i ) sin( i 1)] [W cos( )]( ) ( ) 0 (11) i i i1 i i1 i W e[ {cos( ) cos( )} W ( )] (1) i1 i i1 i i1 i where W() is relative displacement, W is relative velocity, i is time instant of ith impact, W ( i ) is velocity just after ith imact, W ( i+1 ) is velocity just before (i+1)th impact. Y(t) mg ball table g ground 9
3. Discrete time systems Ex 4. Consider the impulsively excited rigid rod: The force P is applied at discrete times t = mτ, m (, ) or m Z(set of integers). The equation of motion of the rod is: d d I b c dt dt [PL (t m )]sin 0 (13) m The equation between impulsive actions of the force is linear and can be solved exactly. At an impulse, the angular velocity undergoes a jump with no change in position. So, develop a mapping relating position and velocity just after an impulse to 10 time instant just after the next impulse is applied.
3. Representation and stability concepts 3.3 Equations of motion and dynamics: We have already considered equations of motion for some systems. In finitedimensional continuous time systems, the equations of motion are of the form n x F(x), x U, t (14) where as for finite-dimensional discrete time systems, the dynamics is defined by iterates of mappings n x p1 F(x p ), x U, p (15) For systems defined by infinite-dimensional state variables, the formulation has to consider some function space. 11
3.3 Equations of motion and.. As an example of a system defined in a function space, consider the dynamics of a axially loaded beam: s (s) s=l P x u(s,t) Assuming an inextensible rod, the strain energy of the system can be shown to be L L 1 V EI ( ) ds P[L cos ds] (16) s 0 0 where the horizontal component of displacement of a material point, u(s, t) is: u(s, t) = s x Then, L 0 u(s, t) = s cosψds (17) 1
3.3 Equations of motion and.. Accounting for changes in kinetic energy of the rod, and using Hamilton s principle, one can show that the equation of motion of the rod (neglecting inertia terms) has the form (s) s x u(s,t) s=l EI P sin 0 (17) s along with the appropriate boundary conditions (for pinned ends): P s 0, s 0,L (18) 13
3.3 Equations of motion and.. Clearly, any solution of the system must belong to the set of functions that satisfy the boundary condition. So, we define a function space: H {uc u(0) u(l) 0} (18) This is he space of all twice differentiable functions that satisfy the boundary condition. It is a linear vector space, an example of a Banach space. The operation in (17) takes such a function and gives a function that is only a continuous function of s. So, the resulting function is not a member of H, even if it may form a linear vector space. It belongs to the space K = {u C 0 14
3.3 Equations of motion and.. Equation (17) then can be written as G(u, ) 0, u H (19) Here, G is a mapping from H into K, i.e., G:H K (0) Such mappings arise naturally in infinite-dimensional systems physical systems which are dependent on space variables, in addition to time. 15
3. Representation and stability concepts 3.4 Definitions of stability: First, consider the concept of a metric distance in the space of variables one is using to define the system under consideration. As an example, consider the beam system (equation (6)): 4 u u u p 0, t 0, 0 x 1. (6) 4 x x t Let us define velocity of a point on the beam: v(x,t) u t and the state vector: w(x, t) = u, v T Then, a couple of examples of norms are: L u u 1/ s 0 s (w,0) [ {v ( ) ( ) u }ds] (1) 16
or 3.4 Definitions of stability.. sup x [0,L] L u 1/ 0 s (w,0) [u] [ {v ( ) }ds] () There are quite a few definitions of stability. The most significant and important one is called Lyapunov stability. Lyapunov stability: Consider the geometric picture in the Fig. The flow is defined by φ t : U K. For a specific trajectory, we consider the solution started at an initial condition u 0, t 0 ); φ t (u 0, t 0. 17
3.4 Definitions of stability.. Consider also the solution started at a neighboring initial condition v 0, t 0 ); φ t (v 0, t 0 At a time t, the two solutions are at φ t u 0, t 0 andφ t (v 0, t 0 ). The stability of the trajectoryφ t (u 0, t 0 ) is related to the deviations of all other solutions from φ t (u 0, t 0 ). Thus, the definition: The solution φ t u 0, t 0 is stable (in the sense of Lyapunov) if and only if, for any initial time t 0 and any prescribed positive number ε > 0, there exists a number > 0 such that for all t > t 0 ( (u,t ), (v,t )) (u,v ) t t t 0 0 0 0 0 0 0 0 0 0 t implies that ( (u,t ), (v,t )) t 0 0 0 0 t 18
3.4 Definitions of stability.. Asymptotic stability: If the solution φ t (u 0, t 0 ) is stable (in the sense of Lyapunov) and in addition, ρ(φ t (u 0, t 0 ), φ t v 0, t 0 for t, then the solution φ t (u 0, t 0 )is called asymptotically stable. In some cases, this definition is too restrictive. For example, consider motion of a nonlinear pendulum. 19
3.4 Definitions of stability.. For the nonlinear pendulum, periodic oscillations around the bottom equilibrium are common, and the period of oscillations depends on the amplitude, or initial conditions. Shown here are the phase plane trajectories φ t (ψ 0 ) and φ t (ψ 0 + δ) corresponding to two different initial conditions ψ 0 and ψ 0 + δ started on the displacement axis. The orbit φ t (ψ 0 ) has period T, so that φ 0 (ψ 0 ) = φ T (ψ 0 ) = φ T (ψ 0 ).... 0
3.4 Definitions of stability.. The orbit φ t (ψ 0 + δ), even though periodic, will have period > T since energy is larger. Thus, at time T, the solution on this orbit is at φ T (ψ 0 + δ) with φ T (ψ 0 + δ) φ 0 (ψ 0 + δ) As time marches, the solutionφ t (ψ 0 + δ) deviates more from the solutionφ t (ψ 0 )at each instant of time though the two orbits remain close to each other for all time. Thus, need a different way to characterize stability of periodic solutions Poincare (orbital) stability. 1
3.4 Definitions of stability.. An attractor is a set of states (points in the phase space), invariant under the dynamics, towards which neighboring states in a given basin or domain of attraction asymptotically approach in the course of dynamic evolution. An attractor is defined as: the smallest unit which cannot be itself decomposed into two or more attractors with distinct basins of attraction. This restriction is necessary since a dynamic system may have multiple attractors. So, if φ t (.) is the flow, by invariant we mean a set A such that φ t (A) A) Furthermore, A must be asymptotically stable, i.e., there is a neighborhood V.
3.4 Definitions of stability.. With A V such that for every x V, lim ( (x),a) 0 t The region V domain of attraction of A. y (1 y ) y y 0 t 3