Analysis of Stochastic Dynamical Systems under Time ... · mechanical systems associated with stochastic force and delay, such as stability analysis when some system parameters are

Analysis of Stochastic Dynamical Systems under Time-delayed Feedback Control

Yanfei Jin

Department of Mechanics

Beijing Institute of Technology

Acknowledgements

Prof. Haiyan Hu, Dr. Wei Li and student Xuan Luo.

Financial support from National Science Foundation of China.

Financial support from China Scholarship Council.

Invitation and help from Dr. Mauro Mobilia.

The organizer Dr. Stephen Griffiths.

Outline

Motivation

Mathematical Models

Dynamics and Control of Stochastic Systems

Linear models

Nonlinear models

Summary

1. Motivation

Passive control: performed by mounting the passive materials (mass, spring, damper) on the structure in order to change its dynamic characteristics (stiffness, damping) .

Active control: performed by using an external controller to cancel the unwanted dynamics.

Linear state feedback control: a linear combination of displacement and velocity of system.

Time delays: exist in all digital controllers, data processing and hydraulic actuators.

1. Motivation

Effects of time delay：1) Deteriorate the control performance or even cause the instability of system ；

2) The properly designed delay may improve the

performance of dynamical systems (e.g. the design of

dynamic absorbers, the control of chaos).

Mathematical Model: Dynamical systems with time delays can be described by a set of delay differential equations (DDEs); both noises and time delays coexist, and the combined effect may be described by the stochastic delay differential equations (SDDEs).

5

1. Motivation

Previous works:

The study of DDE in the frame of functional differential equations (Hale 1977, Qin et al 1989, Diekmann 1995);

The study of stochastic functional differential equations (S.-E.A. Mohammed 1984);

The study of exponential stability and asymptotic properties of neutral SDDE (Mao 2009);

The study of mechanical systems associated with DDE (Stepan 1989, Moiola et al1997, Hu et al 2002);

The Quasi-integrable Hamiltonian systems with time-delayed feedback control and noise (Zhu et al 2012).

1. Motivation

The aim of this talk is to solve some problems in mechanical systems associated with stochastic force and delay, such as stability analysis when some system parameters are to be designed, dynamical behaviors of system, the design of the feedback control.

In the following, we will establish some mathematical models for the controlled mechanical systems with noise to be studied.

2. Mathematical models

Linear dynamic system

8

( ) ( ) ( ) ( ) g(t)mx t cx t kx t f t

Control force: 1 2g( ) ( ) ( )t ux t vx t

f(t): additive and parametric excitation.

Parametric excitation differs from forcing since the action appears as a time varying modification on a system parameter. The classical example of parametric excitation is that of the vertically forced pendulum.

( ) ( sin ) ( )u t t u t 2 2 0 def defg

4

2

l

a

l,


In engineering, the nonlinear dynamical systems with parametric excitation can describe many real problems, such as, nonlinear rolling motions of ships in longitudinal waves, vibrations of flexible beams under axial load, vibrations of a parametrically forced pendulum.

Non-linear dynamic system:

Duffing oscillator

3u( ) ( ) ( ) ( ) 0t u t au t bu t

3u( ) ( ) ( ) ( ) ( ) ( )t u t au t bu t f t g t

A sin

gle

pen

du

lum

in th

e gra

vity

field

A beam under large deflection


van der Pol oscillator

is a self-sustained dynamic system, has a trivial fixed point and a limit cycle attractor. The change rate of the system energy is

e.g. the friction caused by the

relative motion between a mass

and a moving belt may lead to oscillations.

2( ) u( ) [ ( ) 1]u( ) ( ),u t t u t t g t

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

.

=1.0

u

u

( ) ,

,E u u

u

u

10 1

0 1

2 2dissipated-out

feded-in


A quarter car model

The active suspension compensates the motion of vehicle body through the use of hydraulic actuators and controllers.

H.Y. Hu et al, 2002.

where x is the vertical displacement of vehicle body, y is the vertical displacement of the unsprung mass, z is the road disturbance, ks and cs are the stiffness and damping, ft (⋅) is the restoring force of tire.


A half car model

12

Y.F. Jin, L. Luo, Nonlinear Dynamics,

2013: 185-195.


A half car model

13

A six degree-of-freedom (12-dimensions)half-car model with square damping and nonlinear hysteretic stiffness is given

,tMX + CX + KX + g(X,X) = f( )

where is the state vector, is a Gauss random vector with zero mean, is a nonlinear vector representing the constitutive relation of system.

X ( )tf

g(X,X)

3. Dynamics of SDE with Control

For DDE or SDE, classic perturbation methods are used to approximate the solution of system, such as • The method of multiple scales (Nayfey et al, 1979) • The stochastic averaging method (Spanos et al, 1986; Lin et

al, 1986) • The equivalent linearization method (Atalik et al, 1976) In the following, we extended and applied those methods to solve the problems in resonances solution and moment stability analysis, and so on.

3.1 Moment stability analysis- linear SDDE

Consider a linear SDOF system with delayed state feedback and Gaussian white noise:

( ) 2 ( ) ( ) ( ) ( ) ( ( )) ( )x t x t x t ux t vx t h x t t

The zero solution of a DDE is delay-independent stable if the system is asymptotically stable for any given time delay.

I. Additive noise (h(x)=constant):

2/10 12/1 1Fig.1


The critical value of time delay is given by

02)1()],)2(

arccos()1(2[1

02)1()],)2(

arccos(2[1

2

222

2

2

222

2

uvvu

uvuk

uvvu

uvuk

c

The delay-independent stability criteria are known to be rather conservative. If the (u,v) do not fall into this stable region, the real part of at least one characteristic root changes its sign as time delay varies.

)4(2

1 2

2,1 qpp 22 24 vp 21 uq

When the time delay takes the above critical value, a Hopf bifurcation will take place.


The critical value of time delay is given by

II. Multiplicative noise (h(x)=x):

Fig.2 Regions of the second-order moment stability for =0.5.

Itô in

terpretatio

n

0)4()1(2)],)24(2

2

1arccos()1(2[

1

0)4()1(2)],)24(2

2

1arccos(2[

1

22

222

22

22

222

22

uvvu

vuvuk

uvvu

vuvuk

c

5.02 0.12

2 1 2( ) 2 ( ) ( ) ( ) ( ) ( ) ( )x t x t x t ux t vx t x t t


First, introducing the following harmonic transformation:

1 2

1 2

( )sin ( ) sin ( ) ( )( ) ,

( )cos ( ) cos ( ) ( )( ) .

( ) ( )

g f t h t tA t

g f t h t tt

A t A t

( )t t ( ) ( )cosx t A t ( ) ( )sinx t A t

Using the stochastic averaging method, the corresponding Ito SDE can be obtained

1 2

1 11 1

1 2

2 22 2

( , ) ( ( , )) ( ),

( , ) ( ( , )) ( ).

dA a A dt b A dB t

d a A dt b A dB t


The nth-order moment is asymptotic stable if

2

1 sin ( 2)cos

2 8

u n Dv

Fig 3. Regions of stability for the second-order moment.

1 21

2 2

3[ ] [ ] ( ).

2 16 8

B D DdA Adt AdB t

3.2 Dynamics and control- nonlinear SDE

Resonances in the delayed Duffing oscillator with narrow-

band random excitation

2 3

1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) cos( ( ))

x t x t x t x t

ux t vx t f x h t W t

，

where W(t) is the standard Wiener process.

Herein, the external dynamic force ( )t is modeled by a cosine-function with a deterministic amplitude h and an angle ( ) ( )t t W t whose constant rotation speed is superimposed by white noise ( )W t of intensity .

( ) cos( ( ))t h t W t

Wedig, W.V, Struct. Saf. 8, 13–25, 1990.

The power spectrum of (t) is

422422

42222

)4(

)4(

2

1)(

hS


The second-order filtered white noise model (the stationary output of a stable second-order linear system due to white noise input) usually used to describe narrow-band random excitation.

The power spectrum is

2 2 2 2

2 2( ) 2

( ) ( )

b bS a

b a b a

a and b are the real part and imaginary part of the eigenvalue of the second-order linear filter.


Additive noise: primary resonance

2 2 2

0 0 1

2 2 2

0 0 1

, .

, .

E E E

E E E

2 2

2 2

1 2

1 1 2 3

2

2

1 1 2

1 2 3

2 2

2 1 2 3

1 2

1 1 2 3

,4 ( )

,4( )

(2 ).

4 ( )

BE

B B B B

BE

B B B

B B BE

B B B B

2 2 4

20 0

1

3 9

4 64e e

2 2 4

20 0

2

3 9

4 64e e

2 2 2 1 2

1 1 0 0

32[ ( ) ]

8e ef

2 2 2 1 2

2 2 0 0

32[ ( ) ]

8e ef 4 3c e ef When , jumps

and hysteresis phenomena can be eliminated.


Stochastic bifurcation

Stochastic bifurcation: dynamical bifurcation (D-bifurcation) and phenomenological bifurcation (P-bifurcation). • The Largest Lyapunov exponent (invariant measures) gives the almost-

sure stability condition and defines the D-bifurcation point. • The qualitative changes of stationary probability density of the system

response can describe the P-bifurcation.

In the following, we illustrate the stochastic bifurcation in a business cycle model with random parametric excitation .

3( ) ( ) ( ) (1 ) ( ) (t) ( )x t vx t ux t x t x w t ，

W. Li, W. Xu, J.F. Zhao, Y.F. Jin, Chaos, Solitons and Fractals, 31, 2007: 702–711.


• We used the stochastic averaging method of Hamiltonian system (e.g. multiple scale, general stochastic average) to study the corresponding Ito SDE.

• Largest Lyapunov exponent: or Before the D-bifurcation, the trivial solution is asymptotically stable with probability one. The D-bifurcation occurs when the largest Lyapunov exponent vanishes.

1 21lim lnH ( )t

tt

2 2

2

1 1lim ln ( ) ( )

2tX t X t

t


After the D-bifurcation and before P-bifurcation, the trivial solution is unstable and the stationary joint probability density is unimodel with the peak at the origin.

Joint and marginal probability density at u=0 .


• The averaged FPK: In this case, its exact stationary solution can be obtained. But in most cases, for example We can use numerical method (e.g. the method of path integration) to get the joint stationary probability density of the amplitude and phase. A. Naess, V. Moe, Structural Safety and Reliability, 1998: 795–801.

22

2

1[m(H) ] [ (H) ]

H 2 H

pp p

t

2 2

1 2 2[m ( , ) ] [m ( , ) ]

2

p pa p a p

t a


After the P-bifurcation, the trivial solution is still unstable, while the stationary joint probability density becomes crater-like.

Joint and marginal probability density at u=0.32 .

The Design of Feedback Control

The control performance for two distinct time delays is better than that of only one time delay in displacement feedback, if velocity feedback gain is positive. The control performance for only one time delay in displacement feedback is superior to that of two distinct time delays when the velocity feedback gain is negative.


The approximate analytical results obtained by multiple scale method are well coincide with the numerical results.


Numerical results

( ) 4x t

( ) 4x t

[ ,0]t

( ) 3.5x t

( ) 4x t

[ ,0]t

It is seen that the steady-state solution may converge to the smaller steady-state solution or the larger one, depending on its initial condition. Figures also illustrate the jumps.


2 2

1

1 2

2 2

( ) ( ) [1 ( )] ( )

[ ( ) ( )] ( ),

x t x t x t x t

ux t vx t t


Moment stability of a delayed van der Pol oscillator with external Gaussian white noise

The stability conditions of the first and second-order moments require

2 2sin cos 0u v

stable


-1.8 -1.6 -1.4 -1.2

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

stochastic averaging method

numerical simulation

EA

v

-1.8 -1.6 -1.4 -1.2

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

stochastic averaging method

numerical result

EA

2

v


Stochastic optimal control of a half-car model under random road excitation

The equivalent linear system can be written as

,t* *MX + C X + K X = f( )

The road excitation can be generated by passing a white noise input through a linear first-order filter given as

( ) ( ) 2 ( ),t v t v t f If Iw

State space representation:

1 2 3 4 5 6 12

Tx x x x x x x

aX

Equivalent linear equations ( ) ( ) ( ) ( )t t t ta a

X = FX + GU Dw

The overall system performance index

1

01 1 1 1 1 2 2 3 3 4 4E ( ) ( ) ( )

t

tJ t t J J J J dt

T

a aX S X

2

4 1

2

2

[ ]

[ ]

J E u

E u

2

3 1

2

3

[( ) ]

[( ) ]

f

r

J E y h

E y h

2

1 2

2

4

[ ]

[ ]

J E y

E y

2

2 1 2

2

3 4

[( ) ]

[( ) ]

J E y h

E y h


1

01 1 1

( ) ( ) ( )E ( ) ( ) ( ( ) ( ) )

( ) ( ) ( )

tT T

Tt

t t tJ t t t t dt

t t t

aT

a a a

A N XX S X X U

N B U

The control law is described by minimizing the performance index as following

( ) ( ) ( )t t t a

U C X1( ) ( )T Tt C B N G S

36

0 2 4 6 8 100

1

2

3

4

5

6

Time (s)

J 1 (

m/s

2)2

active

passive

0 2 4 6 8 100

1

2

x 10-4

Time (s)

J 2 (

m2)

active

passive

J1 and J2 in active suspension systems changes slower, ride comfort is improved.


0 2 4 6 8 100

1

2

3

4

5

6x 10

-5

Time (s)

J 3 (

m2)

active

passive

0 2 4 6 8 100

1

2

3

4

5

6

7

8

Time(s)

Over

all

Per

form

ance

J

active

passive

J3 in active suspension systems changes quicker because we consider the ride comfort more. The overall performance of suspension system improves with active control.

37

0 2 4 6 8 100

0.5

1

1.5

2

2.5

Time (s)

J 1 (

m/s

2)2

Simulation

Linearization

0 2 4 6 8 100

1

x 10-4

Time (s)

J 2 (

m2)

Simulation

Linearization


0 2 4 6 8 100

1

2

3

4

5x 10

-5

Time (s)

J 3 (

m2)

Simulation

Linearization

0 2 4 6 8 100

1

2

3

4

5x 10

5

Time (s)

J 4 (

N2)

Simulation

Linearization

Summary

The perturbation methods can be successfully applied to solve the problems in the resonances and the moment stability of the SDDE.

The stability regions can be plotted. It is important for the design of parameters in the controlled mechanical systems.

Our researches study the dynamics and control in SDEs with time delays or hysteresis and may serve as a guide in many complex situations.

Documents

Analysis of Stochastic Dynamical Systems under Time ... · mechanical systems associated with stochastic force and delay, such as stability analysis when some system parameters are