CDS 101 Precourse Phase Plane Analysis and Stability Melvin Leok Control and Dynamical Systems California Institute of Technology Pasadena, CA, 26 September, 2002. [email protected] http://www.cds.caltech.edu/~mleok/ Control and Dynamical Systems 2 Introduction ¥ Overview ² Equilibrium points ² Stability of equilibria ² Tools for analyzing stability ² Phase portraits and visualization of dynamical systems ² Computational tools 3 Equilibrium and Stability ¥ Equilibrium Points ² Consider a pendulum, under the influence of gravity. ² An Equilibrium Point is a state that does not change under the dynamics. ² The fully down and fully up positions to a pendulum are ex- amples of equilibria. 4 Equilibrium and Stability ¥ Equilibria of Dynamical Systems ² To understand what is an equilibrium point of a dynamical systems, we consider the equation of motion for a pendulum, g θÄ + sin θ = 0; L which is a second-order linear di®erential equation without damp- ing. ² We can rewrite this as a system of ¯rst-order di®erential equations by introducing the velocity variable, v. θ_ = v; g v_ = ¡ sin θ: L 5 Equilibrium and Stability ¥ Equilibria of Dynamical Systems ² The dynamics of the pendulum can then be visualized by plotting the vector ¯eld, (θ_; v_). ² The equilibrium points correspond to the positions at which the vector ¯eld vanishes. 6 Equilibrium and Stability ¥ Stability of Equilibrium Points ² A point is at equilibrium if when we start the system at exactly that point, it will stay at that point forever. ² Stability of an equilibrium point asks the question what happens if we start close to the equilibrium point, does it stay close? ² If we start near the fully down position, we will stay near it, so the fully down position is a stable equilibrium. 7 Equilibrium and Stability ¥ Stability of Equilibrium Points ² If we start near the fully up position, the pendulum will wander far away from the equilibrium, and as such, it is an unstable equilibrium. 8 Types of Stability ¥ Lyapunov Stability ² An equilibrium point is Lyapunov Stable if whenever we start su±ciently close to the equilibrium, we will stay close to the equi- librium. Examples of Lyapunov stable and unstable behavior 9 Types of Stability ¥ Asymptotic Stability ² An equilibrium point is Asymptotically Stable if it is Lya- punov stable, and for any solution that starts su±ciently close to the equilibrium point will converge to the equilibrium point. 10 Tools for Analyzing Stability ¥ Potential Energy near the Equilibrium ² When the system only experiences forces that can be expressed in terms of a potential energy, looking at the potential energy near the equilibrium can give one information about the stability of that point. Energy minimum Energy maximum Stable Unstable ² More generally, such stability analysis methods are known as Lya- punov Function methods. 11 Tools for Analyzing Stability ¥ Eigenvalue Analysis ² An analytic method of analyzing stability is related to Eigenvalue Analysis in linear algebra. ² As an example, consider the following scalar linear di®erential equa- tion, x_ = ax; Which we readily verify to have the solution, at x(t) = x0e : ² Notice that the behavior of the equilibrium at the origin, x = 0, depends on the value of the parameter a. 12 Tools for Analyzing Stability ¥ Eigenvalue Analysis ² If a > 0, we see that the solution diverges from 0, and the origin is unstable. 25 20 15 10 5 0 0 0.5 1 1.5 2 2.5 3 13 Tools for Analyzing Stability ¥ Eigenvalue Analysis ² If a < 0, we see that the solution converges to 0, and the origin is stable. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 14 Tools for Analyzing Stability ¥ Eigenvalue Analysis ² In general, if we are given a system of coupled ¯rst-order linear di®erential equations of the form, x_ = Ax; where x 2 Rn is a n-vector, and x 2 Rn£n is a n £ n matrix, the stability of an equilibrium can be analyzed by determining the eigenvalues of the matrix A. 15 Tools for Analyzing Stability ¥ What about nonlinear systems? ² We can do this analysis for linear systems of di®erential equations, but what happens in the case of nonlinear systems of di®erential equations, which we may not be able to solve exactly? ² Notice that the notion of stability is only concerned with what hap- pens in a small neighborhood of the equilibrium point, and as we zoom in closer and closer, the vector ¯eld starting looking like that of a linear system, so we do the obvious thing: Linearization: We approximate the nonlinear system by a linear system. Eigenvalue Analysis: We evaluate the eigenvalues of the lin- earization to obtain information about the stability of the nonlinear system. 16 Visualizing Dynamical Systems ¥ Hamiltonian (Energy) Methods ² The pendulum example we considered is special in that it is con- servative, and hence, by looking at level sets of the energy, we can also get a sense of how the system behaves. 0.5 y y−cos(x) 10 8 6 4 2 0 −2 4 2 0 −2 4 6 0 2 −4 −2 y −6 −4 x 17 Visualizing Dynamical Systems ¥ Phase Portraits ² Instead of plotting position or velocity against time, in a time- series plot, we can often gain insight by a Phase portrait, where we plot velocity against position as a parametric plot. ² Returning to the pendulum example, we have the following phase portrait, 18 Visualizing Dynamical Systems ¥ Phase Portraits ² Periodic solutions show up as closed orbits. ² We can see from the nearby trajectories whether a equilibrium point is stable or unstable. ² Phase portraits allow us to get a sense of the di®erent types of behavior which may occur in a dynamical system. ² In the pendulum example, we clearly see the distinction between oscillating modes, and whirling modes. 19 Visualizing Dynamical Systems ¥ Phase Portraits ² It might seem to you that the whirling motion of a pendulum is a periodic or- bit, but how do we see that from the phase portrait? ² If we recall that we need to make the identi¯cation θ = ¼ = ¡¼, we can wrap the phase plane into a cylinder, and the whirling modes become closed curves as expected of periodic orbits. 20 Visualizing Dynamical Systems ¥ More Phase Portraits ² Consider the more complicated example of a damped pendulum. The phase portrait is more complicated, and is shown below, θ ' = ω D = 0.1 ω ’ = − sin(θ) − D ω 4 3 2 1 ω 0 −1 −2 −3 −4 −10 −8 −6 −4 −2 0 2 4 6 8 10 θ 21 Visualizing Dynamical Systems ¥ Extended Phase Portraits ² The time evolution of a θ ' = ω D = 0.1 damped pendulum is more ω ’ = − sin(θ) − D ω interesting. 90 ² We can combine time-series 80 70 plots and phase portraits, by 60 50 t looking at the Extended 40 30 Phase Portrait, which is 20 10 a parametric plot of posi- 0 tion, velocity and time. 2 1 3 0 2 1 −1 0 ² −1 The time-series and phase −2 −2 ω portrait are projections of θ the extended phase portrait. 22 Computational Tools ¥ MATLAB and PPLANE6 ² A good program for phase plane analysis is PPLANE6, which is written for MATLAB. The homepage is, http : ==math:rice:edu=~dfield= 23 Computational Tools ¥ MATLAB and PPLANE6 24 Computational Tools ¥ Numerical Integration ² How does a computer compute the solution of a nonlinear di®er- ential equation? ² Given the equation, x_ = f(x); we could think of computing the solution at a discrete set of time intervals, spaced at ¢t = 0:1. ² We could then make the approximation, ¢x x_ = ; ¢t from which we have, xn+1 ¡ xn = ¢x = ¢tf(xn): ² This method is known as the Forward Euler method. 25 Computational Tools ¥ Numerical Integration ² A more accurate and stable numerical integration algorithm is the Runge-Kutta method, which is very popular. It is given by, k1 = f(xn)¢t k2 = f(xn + k1=2)¢t k3 = f(xn + k2=2)¢t k4 = f(xn + k3)¢t 1 x = x + (k + 2k + 2k + k ) n+1 n 6 1 2 3 4 ² Numerical integration algorithms in software like MATLAB are more sophisticated, but are based on algorithms like the Runge- Kutta method above. 26 Resources ¥ Related Courses at Caltech ² CDS 140 Introduction to Dynamics ² CDS 201 Applied Operator Theory ² ACM 110 Introduction to Numerical Analysis ¥ Webpages ² Control and Dynamical Systems Homepage http : ==www:cds:caltech:edu= ² MATLAB Homepage http : ==www:mathworks:com ² PPLANE6 Homepage http : ==math:rice:edu=~dfield= Control and Dynamical Systems.