1
2
x1, x2
ode45(‘dosc', [0 10], [1 0]); function time interval initial condition
0
-1
0
2
4
6
8
10
time, sec
Phase Portraits (2D systems only)
Phase plane plots show 2D dynamics as vector fields & stream functions • Plot f(x) as a vector on the plane; stream lines follow the flow of the arrows phaseplot(‘dosc’, ... d x1 x2 [-1 1 10], [-1 1 10], ... x = − x − x dt 2 1 2 boxgrid([-1 1 10], [-1 1 10]));
•
Equilibrium points (2) 200 • (0,0): unstable ⇒ species don’t die out 180 • ~(50,35): unstable ⇒ no steady state 160 population 140 Invariant curves (3) 120 • Start on curve, stay on curve x2 100 • “Limit cycle” ⇒ population of each 80 species oscillates over time 60 • This is a global feature of the dynamics 40 (not local to an equilibrium point)
20 0 0
stable unstable
50
100 x1
150
200
Input/Output Performance
• Return to system with inputs • How does system response to changes in input values? • Transient response: • What happens right after a new input is applied • Steady state response:
0 1
Example #1: Double Inverted Pendulum
• Two series coupled pendula • States: pendulum angles (2), velocities (2) • Dynamics: F = ma (balance of forces) • Dynamics are very nonlinear Eq #1 Eq #2
•
Example #2: Predator Prey (ODE version)
• Continuous time (ODE) version of predator prey dynamics:
ɺ x1 = br x1 − ax1 x2 ɺ x2 = ax1 x2 − d f x2
•
Continuous time (ODE) model MATLAB: predprey.m (from web page)
Principle: Choice of model depends on the questions you want to answer
u(t) q2 q1 m1 k1 k2 m2
k3
b
function dydt = f(t,y, k1, k2, k3, m1, m2, b, omega) u = 0.00315*cos(omega*t); dydt = [ y(3); y(4); -(k1+k2)/m1*y(1) + k2/m1*y(2); k2/m2*y(1) - (k2+k3)/m2*y(2) - b/m2*y(4) + k3/m2*u ];
Eq #3
Eq #4
•
Stability of equilibria • Eq #1 is stable • Eq #3 is unstable • Eq #2 and #4 are unstable, but with some stable “modes”
Local versus Global Behavior
Stability of Equilibrium Points
• • An equilibrium point is: 0.5 Asymptotically stable if all nearby initial conditions con- 0 verge to the equilibrium -0.5 point -1 -1 • Equilibrium point is an 1 attractor or sink Unstable if some initial 0.5 conditions diverge from the 0 equilibrium point -0.5 • Equilibrium point is a -1 source (or saddle) -1 Stable if initial conditions 1 that start near the 0.5 equilibrium point, stay near 0 • Equilibrium point is a center -0.5
Step Response
• Output characteristics in response to a “step” input • Rise time: time required to move from 10% to 90% of final value • Overshoot: ratio between amplitude of first peak and steady state value • Settling time: time required to remain w/in p% (usually 2%) of final value • Steady state value: final value at t = 1
-1 -1 1
1 0 -1 0
t →∞
5
10
lim x (t ) = xe ∀ x (0) − xe < ε
0 1
1 0 -1 0
t →∞
•
5
10
lim x (t ) = ∞ for some x (0)
0 1
1 0 -1 0 5 10
•

x (t ) − xe <ε ∀t , x (0) − xe < δ ε
-0.015 0
Mass spring system (L2.1)
200 300 400 500
100
• Nearly unstable systems (slow convergence) often exhibit “ringing” (highly oscillatory response to [non-periodic] inputs)
• Stability is a local concept • Equilibrium points define the local behavior of the dynamical system • Single dynamical system can have stable and unstable equilibrium points Region of attraction • Set of initial conditions that converge to a given equilibrium point
1 1
0.5 x x
0.5
2
0
2
0
-0.5
-0.5
-1 -1
0 x 1
1
-1 -1
0 x 1
1
Equilibrium Points
Equilibrium points represent stationary conditions for the dynamics
2
x2 0
-2 -2π 0 2π
x1
0.005 0.015
Transient
Steady State
0.01
• What happens a long time after the input is applied • Stability vs input/output performance
0
-0.005
-0.01
• Systems that are close to instability typically exhibit poor input/output performance
CDS 101: Lecture 3.1 Stability and Performance
Richard M. Murray 13 October 2003 Goals: Describe different types of stability for an equilibrium point Explain the difference between local/global stability, and related concepts Describe performance measures for (controlled) systems, including transients and steady state response Reading: Åström and Murray, Analysis and Design of Feedback Systems, Ch 3