Viability for a Hybrid System Control1 Viability for a Hybrid Control System Yan Gao( 高岩 ) University of Shanghai for Science and Technology Email: [email protected]

Viability for a Hybrid System Control 1

Viability for a Hybrid Control System

Yan Gao( 高岩 )

University of Shanghai for Science and Technology

Email: [email protected]

The 4th China-Europe Summer School on Complexity Science

12th August,2010



Outlines

Hybrid System

Viability of a Control System

Determining Viability

Multi-Lyapunov ( Nonsmoooth Lyapunov) Function

Population Control for a City



1 Hybrid System Dynamical system with continuous variable (state) and di

screte variable (state).

Usually, there is an interaction between continuous state and discrete state .

In some cases, discrete state is on logic.

Discrete state generate a jump.



Examples: Mechanical system: continuous motion interrupted by

collisions.

Electrical circuit: charging of capacitor, interrupted by switched opening or closing.

Chemical system: continuous evolution of chemical interaction is control by valves and pumps.

A car with 5 gears, gear shift control generate a jump for velocity.



A hybrid system

t

x



Modelling of hybrid control

As a complex system, hybrid systems has various forms.

A representative model for hybrid control system is as the following.



RRf n : continuous part

is discrete part

is state

is control


),( uxrx xu

Jxuxrx

Jxuxfx

),,(

),,(.


Swiched control system

A hybrid system without jump is said to be switched.

Even no jumps in a switched system, but dynamical models changed, it has multi-models.



A switched system University of Shanghai for

Science and Technology

1x

2x

0x


Trajectory of switched control system is nonsmooth.

Nonsmoothness appears at switching point.



An important switched is linear switched system has of the form:

There are m models for the systerm(1).We can choose the switching rule.

mixAtx i ,,1,)( (1)



The earliest publication for hybrid system is :

Witsenhausen ， IEEE Transactions on Automatic Control,1966.

The notion of hybrid system was proposed formally in 80’

s ．



Some periodicals have some special issue on hybrid system

IEEE Transactions on Automatic Control

Automatica

Lecture Notes in Computer Science



2. Viability

Definition of viability:

Consider a control system and a set, if for any initial point in this set, there exists a solution of the control system such that it stays in the set for ever, then the set is said to be viable under this control system.



x0

x(t)

Fig1 Viability for a continuous system



．

Fig2 Viability for a hybrid system


0x0

1x1

2x2

3x3

K

2x

1x


There are three topics in research on viability:

1.Determining viability;

2.Finding a variable solution;

3.Design of viable region.



Applications of viability

Safety region design

Stability and stabilization for a control system

Differential game ( dynamical game)

Actually,viable region is a safety region.







Usually a given a set (region) is not viable.

We try to find a subset, particular a bigest one which is viable.

Viable region: the bigger, the better.



1D D

D

1D

Fig3 The bigest varible region


is viable region in


Consider a differential inclusion

(2)

Where is a mapping from to its subset.

, nx t F x x R

F x nR

3.Viabity ConditionUniversity of Shanghai for

Science and Technology


Both linear and nonlinear control systems are special cases of the differential inclusion (2).

For instance, (2) reduces to a linear control system if

That is

For instance, (2) reduces to a nonlinear control system if

, }{)( UuBuAxxF

BuAxtx '



Differential inclusion (2) reduces to a nonlinear control system if

that is

}),({)( UuuxfxF

),(' uxftx



Definition Let be nonempty. The tangent cone of at is defined by

where

0)(inflim)(

0hvxdRvxT K

h

nK

nRK K Kx

syydKs

K

inf



1x

2x

K1x

K

1x

2x K

Tangent cone of K at 2x

Tangent cone of K at


Viability condition:The closed set is viable for the differential inclusion (2) if and only if

nRW

.,)( WxxTxF W



4.Nonsmooth Lyapunov Function

Stability of for a control system is based on Lyapunov function.

For switched system, since it is multy-model,switched system has two methods:

Common Lyapunov function:There is function, which is a Lyapunov function for each model.



In most applications, there are no common Lyapunov function.

Multi- Lyapunov function, one is a nonsmooth,

Lyapunov function, for instance, a piecewise smooth function.



5. Population Control for a City

Continuous model of population is 　

)()(

txdt

tdx 0)0( xx

where is population, is a parameter, which depend on birth rate and health level.

)(tx



Migration for the city is taken as discrete.

We suppose jump happen once a year

Example: Population of Shanghai in 2007 is 18580,000, ． We hope to control the population in 2012 within 21000000 ．

96.0



We get imigration to Shanghai:

1400002008 T 1800002009 T

2000002010 T 2400002011 T

2800002012 T



Thank You Very Much!

Documents

Viability for a Hybrid System Control1 Viability for a Hybrid Control System Yan Gao( 高 岩 ) University of Shanghai for Science and Technology Email: [email protected]

Viability for a Hybrid System Control1 Viability for a Hybrid Control System Yan Gao( 高岩 ) University of Shanghai for Science and Technology Email: [email protected]