Viability for a Hybrid Control System Yan Gao(? ?)

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Viability for a Hybrid Control System Yan
Gao(? ?)

• University of Shanghai for Science and Technology
• Email gaoyan_at_usst.edu.cn

The 4th China-Europe Summer School on Complexity
Science  12th August,2010
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Outlines

• Hybrid System
• Viability of a Control System
• Determining Viability
• Multi-Lyapunov ( Nonsmoooth Lyapunov) Function
• Population Control for a City

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1 Hybrid System

• Dynamical system with continuous variable
  (state) and discrete 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.

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• 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.

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A hybrid system
x
t
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• Modelling of hybrid control
• As a complex system, hybrid systems has
  various forms.
• A representative model for hybrid control
  system is as the following.

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continuous part
is discrete part
is state is control
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• 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.

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A switched system
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• Trajectory of switched control system is
  nonsmooth.
• Nonsmoothness appears at switching point.

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• 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.

(1)
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• The earliest publication for hybrid system is
  
• Witsenhausen,IEEE Transactions on Automatic
  Control,1966.
• The notion of hybrid system was proposed
  formally in 80s .

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• Some periodicals have some special issue on
  hybrid system
• IEEE Transactions on Automatic Control
• Automatica
• Lecture Notes in Computer Science

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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.

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x(t)
x0
Fig1 Viability for a
continuous system
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• .
• Fig2 Viability
  for a hybrid system

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• There are three topics in research on viability
• 1.Determining viability
• 2.Finding a variable solution
• 3.Design of viable region.

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Applications of viability

• Safety region design
• Stability and stabilization for a control system
• Differential game ( dynamical game)
• Actually,viable region is a safety region.

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• 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.

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is viable region in
Fig3 The bigest varible
region
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3.Viabity Condition

• Consider a differential inclusion
• 
  (2)
• Where is a mapping from to its
  subset.
• 
  

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• 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
  

,
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• Differential inclusion (2) reduces to a
  nonlinear control system if
  
  
• that is

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

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Tangent cone of K at
K
K
K
Tangent cone of K at
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Viability conditionThe closed set is
viable for the differential inclusion (2) if and
only if



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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
functionThere is function, which is a Lyapunov
function for each model.

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• In most applications, there are no common
  Lyapunov function.
• Multi- Lyapunov function, one is a nonsmooth,
• Lyapunov function, for instance, a piecewise
  smooth function.

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• 5. Population Control for a City
  
• Continuous model of population is

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

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We get imigration to Shanghai
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Thank You Very Much!