Lecture 3 - is it a stable?Adapted from S. N. Sreenach et al., Modeling the dynamics of signaling pathways, Essays in Biochemestry, vol. 45, 2008. 4 Dynamic modeling deterministic,

Lecture 3 - is it a stable? -

1

Review

2

Modeling• A simplified, quantified representation • Dynamical system or process (driven by time) • Answer questions via formal analysis and simulation • Driven by the rise of big data • Good models resemble real-world data

3

Adapted from S. N. Sreenach et al., Modeling the dynamics of signaling pathways, Essays in Biochemestry, vol. 45, 2008.4

Dynamic modeling

deterministic, stochastic

hybrid systems

continuous time (differential equations),

discrete time (difference equation)

discrete event systems

time-driven event-driven

Central concept in modeling• The state of the system! • Dynamic model: state, input, outputs, dynamics • What it is:

• Independent quantities that determine future evolution • What it does:

• Captures effects of the past

5

Phase planes

CDS 110, 10 Oct 07 R. Murray, Caltech

ODEs can also be used to prove stability of a systems

• Try to reason about the long term behavior of all solutions

• Stability ! all solutions return to equilibrium point (more precise defn later)

Example: spring mass system

• Can we show that all solutions return to rest w/out explicitly solving ODE?

• Idea: look at how energy evolves in time

• Start by writing equations in state space form

• Compute energy and its derivative

• Energy is positive " x2 must eventually go to zero

• If x2 goes to zero, can show that x1 must also approach zero (Lasalle, W3)

8

Analyzing Models Using ODEs: Stability

mq + cq + kq = 0

dx

dt=

�x2

� kmx1 � c

mx2

⇥x1 = q

x2 = q

V (x) =12kx2

1 +12mx2

2dV

dt= kx1x1 + mx2x2

= kx1x2 + mx2(�c

mx2 �

k

mx1) = �cx2

2

CDS 110, 10 Oct 07 R. Murray, Caltech

ODEs can also be used to prove stability of a systems

• Try to reason about the long term behavior of all solutions

• Stability ! all solutions return to equilibrium point (more precise defn later)

Example: spring mass system

• Can we show that all solutions return to rest w/out explicitly solving ODE?

• Idea: look at how energy evolves in time

• Start by writing equations in state space form

• Compute energy and its derivative

• Energy is positive " x2 must eventually go to zero

• If x2 goes to zero, can show that x1 must also approach zero (Lasalle, W3)

8

Analyzing Models Using ODEs: Stability

mq + cq + kq = 0

dx

dt=

�x2

� kmx1 � c

mx2

⇥x1 = q

x2 = q

V (x) =12kx2

1 +12mx2

2dV

dt= kx1x1 + mx2x2

= kx1x2 + mx2(�c

mx2 �

k

mx1) = �cx2

2

9 Oct 06 R. M. Murray, Caltech CDS 3

Today: Dynamic Behavior (and Stability)

Goal #1: Stability

! Check if closed loop response is stable

Goal #2: Performance

! Look at how the closed loop system behaves, in a dynamic context

Goal #3: Robustness (later)

system input

control law

SenseVehicle Speed

ComputeControl “Law”

ActuateGas Pedal

Response

depends on

choice of control

(all are stable)


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

-1 0 1-1

-0.5

0

0.5

1

x1

x2

-1 0 1-1

-0.5

0

0.5

1

x1

x2

phaseplot(‘dosc’, ... [-1 1 10], [-1 1 10], ... boxgrid([-1 1 10], [-1 1 10]));9 Oct 06 R. M. Murray, Caltech CDS 3


Goal #1: Stability





system input

control law

SenseVehicle Speed


ActuateGas Pedal

Response

depends on

choice of control

(all are stable)





-1 0 1-1

-0.5

0

0.5

1

x1

x2

-1 0 1-1

-0.5

0

0.5

1

x1

x2

phaseplot(‘dosc’, ... [-1 1 10], [-1 1 10], ... boxgrid([-1 1 10], [-1 1 10]));

matlab >> phaseplot()

6

mathematica >> VectorPlot[]

k

Today• ODEs for (state space) modeling • Linear v. nonlinear systems • Equilibrium points • Types of stability • Local v. global properties • Why care about stability?

7

Linear v. nonlinear systems

8

Differential equations

General form

First order Linear systems

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Exam

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peed

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10

Exam

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#2: S

peed

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trol

Sy

stem+

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reference

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What about higher order linear differential equations?

9

Linear systems

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10

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ple

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++

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ce

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ch C

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Exam

ple

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peed

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ntro

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trol

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stem+

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rban

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bility

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ce

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te s

en

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g to

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rou

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pu

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n a

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ign

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ility a

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enta

tion fo

r hig

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, un

sta

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raft

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ea

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rco

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n g

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loop

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atu

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eh

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r

X-2

9 ex

perim

ental aircraft

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ct 0

7R

. M. M

urra

y, Calte

ch C

DS

10

Exam

ple

#2: S

peed

Co

ntro

lCo

ntro

lS

ystem

++

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rban

ce

reference

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bility

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rform

an

ce

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dy s

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Difference equations

xk+1 = Adxk + Bduk

yk = Cdxk

Continuous differential equations into discrete difference equations!

11

ODE with derivative input

12

Considerw + 5w + 6w = r + r + 2r

Letu = r

x1 = w � �0ux2 = w � �1u � �0u

Thenx1 = w � �0u= x2 + �1u

x2 = w � �1u � �0u= �5w � 6w + u + u + 2u � �1u � �0u

Letting �0 = 1, �1 = �4

x2 = �5(w � u)� 6w + 2u= �5(x2 � u)� 6(x1 + u) + 2u= �6x1 � 5x2 + 26u= �5(x2 � 4u)� 6(x1 + u) + 2u= �6x1 � 5x2 + 16u

<latexit sha1_base64="o9mNlB+adrWUl7Yl9JNA6Mipcxw=">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</latexit>

• Stability: closed-loop stable?

• Performance: closed-loop behavior?

• Robustness (later)

13

Actuate

Compute

Sense Plant

xu



Goal #1: Stability





system input

control law

SenseVehicle Speed


ActuateGas Pedal

Response

depends on

choice of control

(all are stable)





-1 0 1-1

-0.5

0

0.5

1

x1

x2

-1 0 1-1

-0.5

0

0.5

1

x1

x2


control lawplant input

CONTROL SCHEME

PROCESSSYS

TEM



Goal #1: Stability





system input

control law

SenseVehicle Speed


ActuateGas Pedal

Response

depends on

choice of control

(all are stable)





-1 0 1-1

-0.5

0

0.5

1

x1

x2

-1 0 1-1

-0.5

0

0.5

1

x1

x2


Responses to stable controllers

Calling something a system does not make it stable, controllable, or even analyzable...

14

Equilibrium points

15

16

e.g.

17

18

Stationary conditions for the pendulum

19

-2 0 2 4 6

-2

-1

0

1

2

x1

x 2

find states such that

Givenx = f(x)

f(xe) = 0

No friction case

xe =

±n⇡0

�

d

dt

x1

x2

�=

x2

gl sinx1

�

• Consider with an equilibrium xe ≠ 0

• Note that because

• The equilibrium ze of the new system is ze = 0.

Can always assume xe = 0

20

g(0) = f(0 + xe) = f(xe) = 0

z = x� xe

z = x = f(x)

f(x) = f(z + xe) := g(z)

x = f(x), f(xe) = 0

z = g(z)

e.g.

21

22

Stability of equilibrium points

23

Stable (in the sense of Lyapunov)

24

9 Oct 06 R. M. Murray, Caltech CDS

Equilibrium points represent stationary conditions for the dynamics

The equilibria of the system x = f(x) are the points xe such that f(xe) = 0.

5

Equilibrium Points

!

-2" 0 2"

-2

0

2

x1

x2


Stability of Equilibrium Points

An equilibrium point is:

Asymptotically stable if all nearby initial conditions con-verge to the equilibrium point

! Equilibrium point is an attractor

or sink

Unstable if some initial conditions diverge from the equilibrium point

! Equilibrium point is a source

(or saddle)

Stable if initial conditions that start near the equilibrium point, stay near

! Equilibrium point is a center

-1 0 1

-1

-0.5

0

0.5

1

-1 0 1-1

-0.5

0

0.5

1

0 5 10

-1

0

1

0 5 10

-1

0

1

0 5 10

-1

0

1

-1 0 1

-1

-0.5

0

0.5

1

Stable (in the sense of Lyapunov)




5

Equilibrium Points

!

-2" 0 2"

-2

0

2

x1

x2






or sink



(or saddle)



-1 0 1

-1

-0.5

0

0.5

1

-1 0 1-1

-0.5

0

0.5

1

0 5 10

-1

0

1

0 5 10

-1

0

1

0 5 10

-1

0

1

-1 0 1

-1

-0.5

0

0.5

1

e.p. (equilibrium point) is a center

25

For every ✏ > 0, there exists a � = �(✏) > 0 such that,if kx(0)� xek < �, then kx(t)� xek < ✏, for every t � 0.

Asymptotically Stable




5

Equilibrium Points

!

-2" 0 2"

-2

0

2

x1

x2






or sink



(or saddle)



-1 0 1

-1

-0.5

0

0.5

1

-1 0 1-1

-0.5

0

0.5

1

0 5 10

-1

0

1

0 5 10

-1

0

1

0 5 10

-1

0

1

-1 0 1

-1

-0.5

0

0.5

1

26

e.p. is an attractor or sink

Lyapunov stable

There exists � > 0 such that if kx(0)� xek < �,then limt!1 kx(t)� xek = 0.

1.

2.

Exponentially stable

27




5

Equilibrium Points

!

-2" 0 2"

-2

0

2

x1

x2






or sink



(or saddle)



-1 0 1

-1

-0.5

0

0.5

1

-1 0 1-1

-0.5

0

0.5

1

0 5 10

-1

0

1

0 5 10

-1

0

1

0 5 10

-1

0

1

-1 0 1

-1

-0.5

0

0.5

1

Asymptotically stable1.

2. There exist ↵,�, � > 0 such that if kx(0)� xek < �,then kx(t)� xek ↵kx(0)� xeke��t, for t � 0.

Any e.p. that is exponentially stable systems is also asymptotically stable.

↵kx(0)� xeke��t

Unstable




5

Equilibrium Points

!

-2" 0 2"

-2

0

2

x1

x2






or sink



(or saddle)



-1 0 1

-1

-0.5

0

0.5

1

-1 0 1-1

-0.5

0

0.5

1

0 5 10

-1

0

1

0 5 10

-1

0

1

0 5 10

-1

0

1

-1 0 1

-1

-0.5

0

0.5

1

28

e.p. is a source or saddle

Pendulum with friction

29

d

dt

x1

x2

�=

x2

gl sinx1 � k

mx2

�

-2 0 2 4 6

-2

-1

0

1

2

x1

x 2

k = 0

-2 0 2 4 6

-2

-1

0

1

2

x1

x 2

k > 0

Chaotic attractor

30

-2 -1 0 1 2

-2

-1

0

1

2

x1

x 2

d

dt

x1

x2

�=

x21 � x2

2

2x1x2

�

Convergence by itself does not imply Stability

All trajectories converge to xe = 0 but xe is not stable

31


Example #1: Double Inverted Pendulum

Stability of equilibria

! Eq #1 is stable

! Eq #3 is unstable

! Eq #2 and #4 are unstable, but with some stable “modes”

Two series coupled pendula

! States: pendulum angles (2), velocities (2)

! Dynamics: F = ma (balance of forces)

! Dynamics are very nonlinear

Eq #1 Eq #2

Eq #3 Eq #4


Local versus Global Behavior

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

-2! 0 2!

-2

0

2

x1

x2




! Eq #1 is stable

! Eq #3 is unstable






Eq #1 Eq #2

Eq #3 Eq #4








-2! 0 2!

-2

0

2

x1

x2




! Eq #1 is stable

! Eq #3 is unstable






Eq #1 Eq #2

Eq #3 Eq #4








-2! 0 2!

-2

0

2

x1

x2




! Eq #1 is stable

! Eq #3 is unstable






Eq #1 Eq #2

Eq #3 Eq #4








-2! 0 2!

-2

0

2

x1

x2




! Eq #1 is stable

! Eq #3 is unstable






Eq #1 Eq #2

Eq #3 Eq #4








-2! 0 2!

-2

0

2

x1

x2

32

Local v. global behavior• Stability is a local behavior

• define local behavior of the system • feature of each e.p. • system can have stable and unstable e.p.

• Region of attraction • initial conditions that converge to e.p. • local or global feature




! Eq #1 is stable

! Eq #3 is unstable






Eq #1 Eq #2

Eq #3 Eq #4








-2! 0 2!

-2

0

2

x1

x2

33

Key questions• Do equilibrium states exist in the data? • Are equilibrium states unique? • Are equilibrium states optimal? • Are equilibrium states stable?

34

Documents

Lecture 3 - is it a stable?Adapted from S. N. Sreenach et al., Modeling the dynamics of signaling pathways, Essays in Biochemestry, vol. 45, 2008. 4 Dynamic modeling deterministic,